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Gravity
Saxon Algebra 1 math terms
Terms in this set (2000)
GCF of list of common variables raised to powers
the variable raised to the smallest exponent in the list.
factor by grouping
1) arrange terms so the first two terms have a common factor & the last two have a common factor 2) for each pair of terms, factor out the the pair's GCF 3) if there is now a common binomial factor, factor it out 4) if no common binomial factor, begin again, rearranging terms differently. if no rearrangement works it can't be factored.
factoring trinomials
1) use the form x(sq) + bx + c 2) factor out the GCF and then factor a trinomial of the form x(sq) + bx + c To factor ax(sq) + bx + c, try various combinations of factors of ax(sq) and c until a middle term of bx is obtained when checking
perfect square trinomial
a trinomial that is the square of some binomial:
x(sq)+4x+4= (x+2)sq
formulae: 1) a(sq)+2ab+b(sq)=(a+b)(sq); a(sq)-2(ab)+b(sq)=(a-b)(sq)
factor trinomials by grouping
ax(sq)+bx+c 1) find two numbers whose product is a*c and whose sum is b 2) rewrite bx, using the factors in step 1 3) factor by grouping ex: 3x(sq)+14x-5 step 1: 15&-1 step 2: 3x(sq)+14x-5 = 3x(sq)+15x-1x-5 step 3: 3x(x+5) -1(x+5) = (x+5) (3x-1)
difference of squares
a(sq)-b(sq) = (a+b) (a-b)
ex: x(sq)-9=x(sq)-3(sq) = (x+3)(x-3)
sum or difference of cubes
a(cu)+b(cu) = (a+b)(a(sq) -ab+b(sq))
ex: y(cu)+8= y(cu)+2(cu) =(y+2) (y(sq)-2y+4) OR
a(cu)-b(cu)= (a-b)(a(sq)+ab+b(sq)
ex: 125z(cu)-1= (5z(cu)-1(cu)= (5z-1)(25z(sq)+5z+1)
standard form of quadratic equations
formula: ax(sq)+bx+c=0, with a not being 0 ex: x(sq)=16 in standard form is x(sq)-16=0 y=2y(sq)+5 in standard form is 2y(sq)+y-5=0
zero factor theorem
if a & b are real numbers and if ab=0, then a=0 or b=0
ex: if (x+3)(x-1)=0, then x+3=0 or x-1=0
to solve quad expressions by factoring
1) write the equation in standard form: ax(sq)+bx+c=0
2) factor the quadratic
3)set each factor containing a variable = to 0
4) solve the equations
5)check in the original equation
linear inequality in 1variable
an inequality that can be written in the form ax+b<c where a,b,c are real numbers and a is not 0
interval notation
instead of open circle, parenthesis are used.
instead of closed circle, brackets are used.
-infinity is all the numbers less than x to infinity.
graphing
picturing the solutions of inequalities on a number line. the picture is called the graph
to graph x<or=3
shade the numbers to the left of 3 and place a bracket at 3 on the number line. the bracket indicates that 3 is a solution:3 is less than or =to 3. in interval notation we write
(-infinity,3] *may be easier to graph the inequality first then write it in interval notation. to help, think of the number line as approaching -infinity to the left or +infinity to the right. then simply write the interval notation by following your shading from left to right
addition property of inequality
if a, b, and c are real numbers, then a<b & a+c<b+c are equivalent inequalities
multiplication property of inequality
1) if a,b,c are real numbers, and c is positive, then a<b & ac<bc are equivalent inequalities 2) if a,b,c are real numbers, and c is negative, then a<b & ac>bc are equivalent numbers. *the direction of the inequality symbol must be reversed for the inequalities to remain equivalent
solving linear inequalities in one variable
1) clear the inequality of fractions by mult. both sides of ineq. by LCD of all fractions in the inequality. 2) remove grouping symbols such as ( ) by using distributive property 3) simplify each side of inequality by combining like terms
4) write the inequality w variable terms on one side and numbers on the other by using addition prop of inequalities 5) get the variable alone by using multiplication prop of inequalities
solving linear equalities in one variable
1) clear the equality of fractions by mult. both sides of ineq. by LCD of all fractions in the equality.
2) remove grouping symbols such as ( ) by using distributive property
3) simplify each side of equality by combining like terms
4) write the equality w variable terms on one side and numbers on the other by using addition prop of equalities
5) get the variable alone by using multiplication prop of equalities
finding slope given 2 points on a line
slope= change in y (vertical change)/change in x (horizontal change) to do this, choose two points of a line. label the two x-coordinates of two points x(1), x(2) [read "x sub one" and "x sub 2], and label the corresponding y-coordinates y(1) and y(2) the vertical change(rise) btwn these points is the diff in the y-coordinates: y(2)-y(1). the horizontal change(run) btwn the points is the diff of the x-coordinates: x(2)-x(1). the slope of the line is the ratio of y(2)-y(1) to x(2)-x(1), and we use the letter m to denote slope in this formula: m=y(2)-y(1)/x(2)-x(1)
slope of a line
the slope(m) of a line containing points (x(1), y(1)) & (y(1), y(2)) is given by: m = rise/run = y(2)-y(1)/x(2)-x(1) as long as x(2) does not equal x(1)
slope-intercept form
y=mx+b
substitution method
substituting 2nd equation into 1st equation
linear equation in 3 variables
1) write ea. equation in standard form: Ax+By+Cz=D 2)choose a pr of equations& use the equations to eliminate a variable
3) choose any other pr of equations & eliminate the same variable as in step 2
4) two equations & two variables should be obtained by step 2 & step 3. solve this system for both variables
5) to solve for the 3rd variable, substitute the values of the variables found in step 4 into any of the original equations containing the third variable
6) check the ordered triple solution in all three orig. equations
problem-solving linear equations
1) understand the problem. read and re-read it. choose 2 variables to represent the two unknowns. construct a drawing if possible. propose a solution and check.
2) translate the problem into two equations
3)solve the system of equations
4) interpret the results: CHECK the proposed solution in the stated problem and state your conclusion
product rule for exponents
if m and n are positive numbers, and a is a real number, then a(m) * a(n) = a(m+n) {add exponents but keep common base}
power rule for exponents
if m and n are positive integers and a is a real number, then multiply exponents and keep the base a(m)(n)= a(mn)
power of a product rule
if n is a positive integer and a & b are real numbers, then (ab)(n) = a(n)b(n)
power of quotient rule
if n is a positive integer and a & c are real numbers, then (a/c)(n) = a(n)/c(n), and c does not equal 0
quotient rule for exponents
if m & n are positive integers & a is a real number, then a(m)/a(n)=a(m-n) as long as a is not 0
zero exponent
a(0) = 1, as long as a is not 0
degree of a term
the sum of exponents on the variables contained in the term
degree of a polynomial
the greatest degree of any term of the polynomial
adding polynomials
combine all like terms
subtracting polynomials
change the signs of the terms being subtracted, then add ex: a-b = a+(-b)
FOIL method
F: product of FIRST terms O: product of OUTER terms I: product of INNER terms L: product of LAST terms; then combine like terms
squaring a binomial
= to the square of the first term plus or minus twice the product of both terms plus the square of the second term (a+b)(sq) = (a)(sq) + 2ab +2b(sq) OR (a-b)(sq) = a(sq)-2ab+b(sq)
multiplying the sum & diff of two terms
the product of the sum & difference of two terms is the square of the first term minus the square of the second term (a+b) (a-b) = a(sq)-b(sq)
A linear equation is...
variables x_1, x_2,...,x_n, any equation that can be written in the form: a_1x_1+a_2x_2+...+a_nx_n=b; a_1, a_2, ..., a_n are real numbers
A system of linear equations (a linear system) is...
x_1,x_2,...,x_n a collection of linear equations that is the value of the same set of variables
A solution to the linear system x_1, ..., x_n is...
any ordered n-tuple (S_1,...,S_n) of real numbers if you substitute S_1 for x_1, S_2 for x_2, ..., S_n for x_n, then every equation becomes a true relation
Solving a linear system means...
to find the solution set, i.e. the set of all possible solutions
Two linear systems are equivalent if...
they have the same solution set
Solving a two variable system amounts to...
finding the intersection of two lines
Any linear system can have either...
1) no solutions 2) exactly 1 solution 3) infinite many solutions
A matrix is...
a rectangular array of numbers aligned in rows and columns
Solving a linear system:
Replace the system by an equivalent system that is easier to solve
Gauss elimination method
1) keep the x_1 term in one equation, eliminate x_1 from all others 2) keep the x_2 term in the second equation, eliminate x_2 from all next ones 3) and so on
Replacement:
add a multiple of one equation to another
Interchange:
swap 2 equations
Scaling:
multiply all terms in one equation by a nonzero constant
Elementary row operation:
row replacement, row interchange, row scaling
Two matrices A and B are row equivalent if...
there is a sequence of row operations that transform one into another
Fundamental question about a linear system:
Is the system consistent, i.e. does a solution exist, if it does, is this a unique solution?
Any row/column is...
a row/column with 1 nonzero entry in it
The leading entry of a nonzero is...
the leftmost nonzero entry
A is in Echelon from if it has 2 properties:
1) All nonzero rows have to lie above any row of all zeros 2) The leading entry of any nonzero row is to the right of the leading entry of any row above it
A is in Reduced Echelon form if...
1) All nonzero rows have to lie above any row of all zeros 2) The leading entry of any nonzero row is to the right of the leading entry of any row above it 3) The leading entry of any nonzero row is 1 4) The entry is the only nonzero entry in the column
Any given matrix A is row equivalent to...
exactly one matrix B in Reduced Echelon form; B = rref(A)
Row reduction algorithm
1) Locate the first pivot column/first pivot position 2) Locate a pivot, choose a nonzero entry in the pivot column to be the pivot, if necessary do row interchange to move the pivot to the pivot position, if necessary do row scaling to make the pivot equal to 1 3) Create zeros below the pivot use row replacement to make all entries below pivot equal to 0 4) Cover the row containing the pivot and any row above it, apply steps 1-3 to the remaining submatrix 5) Start from the right most pivot and then move upward and to the left, if a pivot does not equal 1, make it equal 1 with row scaling, use row replacement to create zeros above each pivot
The basic variables are...
the variables corresponding to the pivot column in the coefficient matrix
The free variable is...
the non-pivot column of the coefficient matrix
A linear system is consistent if and only if
the last column of the augmented matrix is not a pivot column, a consistent linear system has 1 solution precisely when it has no free variables
How to solve a linear system
1) Write down the augmented matrix B of the system 2) Row reduce B to Echelon form; if the last column of B is pivot, the system is inconsistent; if the last column of B is not pivot, it is consistent 3) Complete the row reduction to C = rref(B) 4) Write down the new system corresponding to C; locate basic/free variables 5) Use the new system to express all variables in terms of free variables
A matrix with one column is called a...
column vector
R is...
the set of all real numbers
R^n is...
the set of all vectors in n entries with real coefficients
The set of all possible linear combinations of v_1, ..., v_p is called...
the subset of R^n spanned by v_1, ..., v_p, denoted by Span(v_1,...,v_p)
The vector equation x_1a_1+x_2a_2+...+x_na_n = b and the system with the augmented matrix [a_1,a_2,...a_n|b] are equivalent if and only if
the corresponding system is consistent
The product Ax = x_1a_1+x_2a_2+...+x_na_n and is only defined when...
the number of columns in A equals the number of entries in x
To compute the i-entry of Ax...
multiply every entry in the i-row of A by corresponding entries in x, take the sum of these products
The identity matrix is...
I_n, nxn matrix
Best way to solve any system of linear equations is to...
row reduce the augmented matrix
If the system is consistent, these statements are true:
1) The equation Ax=b is always consistent for all bER^m 2) The column a_1, a_2,...,a_n fo A span R^m 3) the coefficient matrix has a pivot in every row
The operation of reading any vector vER^n to p + v is called...
the translation by p
The system Ax=b is called homogeneous if...
b=0
The system Ax-b is not homogeneous if...
b!=0
The homogeneous system is always...
consistent, x=0 is always a solution
x=0 is the...
trivial, zero solution
The homogeneous system Ax=0 has a nontrivial solution if and only if...
the system has infinite many solutions, the system has at least one free variable
To find the parametric vector form for the solution set...
1) Row reduce the augmented matrix 2)Get a parametric description of the solution set 3) Write your solution x as a column vector; single out the coefficient for each variable in a column
v_1...v_p are linear dependent if...
there is a nontrivial linear relation between them; i.e. fi we can find some scalars c_1, c_2, ..., c_p, not all zeros, where c_1v_1 + c_2v_2 +...+c_pv_p = 0
v_1...v_p are linear independent if...
there is no non-trivial linear relation between them, i.e. has only the trivial solution
v_1,v_2,...,v_p are independent if and only if...
the system has no free variables
v_1, v_2, ..., v_p are dependent if and only if...
the system has >= 1 free variable
If you have more vectors than the number of entries in each of them, i.e. p>n then they are...
dependent
If v_1,v_2,..,v_pER^n then the set {v_1,v_2,v_p} is dependent if and only if
one of the vectors is a linear combination of the preceding ones
A transformation T:R^n-R^m is linear if...
T respects the vector addition and scalar multiplication; i.e. T(u + v) = T(u) + T(v), T(cu)=cT(u)
Augmented matrix transformation is...
linear
1-1
same images
onto
range (-infinity, infinity)
T is onto if and only if...
the equation Ax=b is always consistent, the matrix A has a pivot in every row, the column of A spans R^m
T is 1-1 if and only if...
the homogeneous equation Ax=0 has only the trivial solution, has no free variables, has a pivot in every column, the n column of A are linear independent
Relation:
A relationship between sets of infomation
Verticle line test:
Given the graph of a relation, if you can draw a verticle line that crosses the graph in more than one place, the relation is not a function
Unordered lists:
Are sets of information; {2, 4, 8}
f (x) :
Means plug a value for " X " in a formula " f "
The arguement of a function:
Is the " X " in f(x)
Symmetric about the Y-axis:
Whatever the graph is doing on one side of the Y-axis is mirrored on the other side
f(x) = X²:
Parabola pointing up; graphs at zero
f(x) = X² + 2 :
Shifts parabola up two units
f(x) = X² - 3 :
Shifts parabola down three units
f(x) = ( X + 2 ) ² :
Shifts parabola left 2 units
f(x) = ( X-3 )² :
Shifts parabola right 3 units
f(x) = | X | :
V pointing up
f(x) = | X + 2 | :
Shifts V two units left
f(x) = | X - 3 | :
Shifts V three units right
f(x) = | X | - 3 :
Shifts V down three units
f(X) = | X | + 1 :
Shifts V up one unit
f(X) = - | X + 7 | :
Reflection (V pointing down) shifted seven left
f(x) = X³ :
A vertical line that bends to the right, goes horizontal, then bends to the left, turning vertical again; this crosses at the (0,0)
f(x) = ( X + 3 ) ³ :
Shifts graph left 3 units
f(x) = ( X - 6 ) ³ :
Shifts graph right 6 units
f(x) = X³ + 3 :
Shifts graph up 3 units
f(x) = X³ - 5 :
Shifts graph down 5 units
f(x) = - X³ :
Reflection of X³
f(x) = ¼ X :
Stretch of a parabola
f(x) = 4 X² :
Shrink of a parabola
f(x) = √ X :
Graphs as ½ of a horizontal parabola pointing towards ∞
f(x) = ±√ X :
Not a function because it would have two answers; + 9 or - 9
f(x) = - √ -x :
Graphs -x, -y to infinity
f(x) = √ -x :
Graphs -x, +y to infinity
f(x) = - √ x :
Graphs +x, -y to infinity
Reflection about the "X" axis:
The - sign outside the arguement indicates what?
Reflection about the "Y" axis:
The sign inside the arguement indicates what?
- f(x):
Example of a reflection about the X axis
f(-x) :
Example of a reflection about the y axis
2f(x):
Example of a vertical stretch
f(½x):
Example of a horizontal stretch
f(2x):
Example of a horizontal shrink
½ f(x) :
Example of a vertical shrink
√5 √5 = :
√ 5 × 5
Union symbol:
U
Intersection symbol:
∩
(f/g)(x)= :
f(x)/g(x)
f ° g = :
f (g (x))
g ° f = :
g(f(x))
One to One function:
No two ordered pairs in a function have the same second componet (y-value)
Horizontal line test:
A function is one-to-one ONLYif every horizontal line intersects the graph at ONLY one point
ⁿ√60 where n=6:
60 1/6
³√ a = :
a 1/3
a/b divided by c/d:
a/b × d/c = ?
11x/2-x divided by 11/2-x:
11x/2-x × 2-x/11 = ?
Are the inverse of each other:
√ and ( )²
Exponet rule:
√16+9 ╪ 4 + 3
Linear Function formula:
f(x)=mx + b ; where m ╫ 0
Constant function:
f(x)= mx + b , where m = 0
Linear graph in 2 variables:
A + By = C , where A or B ╪ 0
Slope of a Line:
M= Y2 - Y1 / X2 - X1 , where X1 ╪ X2
Slope-intercept form:
Y=mx + b
Point slope form:
y - y1 = M ( X - X1 )
Horizontal line:
Y = b ; slope 0
Vertical line:
X = s ; slope undefined
Parallel lines:
Two lines with equal slopes are...
Perpendicular lines:
Two lines whose slopes are negative reciprocal of each other are...
What will the solution of this absolute value equation look like |exp.| > a ?
exp. < -a or exp. > a
What will the solution of this absolute value equation look like |exp.| = a ?
exp = a or exp = -a
What will the solution of this absolute value equation look like |exp.| < a ?
-a < exp < a
What is the first step in this equation : 2|2x+1| -4 = 16 ?
Get rid of the variables outside of the absolute value.
What do you do when you have a negative denominator? ex b-2/-5
Move it up top. answer: 2-b/5 (Make sure to change signs)
How would you describe this inequality in a solution set: x < 5 ?
{x|x<5}
How would you describe this inequality in interval notation: x ≥ 2 ?
[ 2 , ∞ )
How would you express this inequality in a solution set: -4 < t ≤ 5/3 ?
{t| -4 < t ≤ 5/3}
Describe this inequality in interval notation: x < 5/3 ?
( -∞ , 5/3)
What do you do when you divide a negative number in an inequality?
Switch the signs
For what inequality symbols do you use brackets?
≤ and ≥
For what inequality symbols do you use parenthesis?
< and >
What is the solution for |x - 5| < 0?
∅ The absolute value of a number can never be < 0. No solution.
What is the solution for |x - 5| ≤ 0?
{5} Absolute value of a number can never be < 0 but it can be 0 when x =5. |5 - 5| = 0
What is the solution for |x - 5| ≥ 0?
R. Any real number, because the absolute value of a number is always ≥ 0.
How would you word the function (front) part of this equation? P(x) = 7.25x
P is a function of x
Is this a linear equation or a linear function: y = 2x - 1 ?
A linear equation.
Is this a linear equation or a linear function: f(x) = 2x - 1 ?
A linear function
What is the standard form for a linear equation?
ax + by = c Both x and y are on the right side of the equation.
How do you find the y-intercept in a linear equation? ax + by = c
Set x = 0 to isolate the y.
How do you find the x-intercept in a linear equation? ax + by = c
Set y = 0 to isolate the x.
The graph of y = b is?
A horizontal line through b on the y axis. (Slope = 0)
The graph of x = b is?
A vertical line through b on the x axis. (Slope undefined)
How is slope defined?
The slope of a line measures the tilt of the line.
How does slope tell us the tilt? y/x
Because y/x = rise/run
How do we view lines on the graph to get the correct slope?
Left to right
How is a positive line viewed?
It runs upwards. m>0
How is a negative line viewed?
It runs downwards. m<0
If m=0 what type of line would be viewed on the graph?
A horizontal line.
How do we calculate slope from two given points? (x¹,y¹) (x²,y²)
y² - y¹/x² - x¹ to get the slope. (rise/run)
What is the formula for slope-intercept form?
y = mx + b
What is the point slope equation? Used when you have a slope and a random point.
y - y¹ = m(x - x¹)
How do you find the equation of a line if you are given two points, but no y-intercept or slope? Such as: (4 , 3) (6 , -2)
You would have to find the slope first, x² - x¹/y² - y¹, and then use the point slope equation, y - y¹ = m(x - x¹)
What do parallel lines have in common?
Equal slopes. m¹ = m²
What is the difference in slopes for perpendicular lines?
They have negative reciprocals. ex. Line¹: m= 2/4, would mean Line²: m= -4/2
What is the equation of the line through (0 , 2) that is perpendicular to the graph of the line: y = 1/2x - 4 ? (Think of the perpendicular rule)
y = -2/1x + 2 because (0 , 2) gives us the y intercept and -2/1 is the reciprocal of the line in the question.
When graphing linear inequalities, for what symbols would you use a solid line in the graph?
≤ and ≥
When graphing linear inequalities, for what symbols would you use a dotted line in the graph?
< and >
What is the first step in graphing a linear inequality? y ≥ 1/2x - 3
Decide what type of line it will be. In this case it's a solid line because of the ≥ sign.
How do you graph the line of a linear inequality? (Also the 2nd step) y ≥ 1/2x - 3
Set the inequality as an equation y ≥ 1/2x - 3 turns into y = 1/2x - 3, and plot the line.
How should you decide what side will be shaded in while graphing a linear inequality? y ≥ 1/2x - 3
Test it by putting in a point, typically (0 , 0) and see if it's true. 0 ≥ 1/2(0) -3 is 0 ≥ 3, which means true. Shade on the side of the point. If false shade on opposite side.
What three methods could be used to solve systems of linear equations? Equation¹: 2x + y = 6 Equation²: 3x - 2y = 16
By graphing, the substitution method, and the addition method.
These two equations would be easiest to solve by what method Equation¹: y = -2x + 6 Equation²: 3x - 2y = 16?
The substitution method, substitue first equation in place of y in the second: 3x - 2(-2x + 6) = 16 and solve.
These two equations would be easiest to solve by what method Equation¹: 4x + 2y = 6 Equation²: 3x - 2y = 16?
The addition method. If set up as if adding them together the y's would cancel. Then you could solve for x and then solve for y.
A linear system where two lines cross (one answer) would be what type of system?
A consistent system.
A linear system where two lines are parallel (no answer) would be what type of system?
An inconsistent system.
A linear system where two lines are directly on top of each other (infinite answers) would be what type of system?
A dependent system.
When solving a linear system of equations, if the substitution or addition method resulted in 2 = 2, what would you have?
A dependent system, two lines on top of each other. ∞ Solutions
When solving a linear system of equations, if the substitution or addition method resulted in 2 ≠ 4, what system would you have?
An inconsistent system, parallel lines. No solution.
What will the solution of this absolute value equation look like |exp.| > a ?
exp. < -a or exp. > a
What will the solution of this absolute value equation look like |exp.| = a ?
exp = a or exp = -a
What will the solution of this absolute value equation look like |exp.| < a ?
-a < exp < a
What is the first step in this equation : 2|2x+1| -4 = 16 ?
Get rid of the variables outside of the absolute value.
What do you do when you have a negative denominator? ex b-2/-5
Move it up top. answer: 2-b/5 (Make sure to change signs)
How would you describe this inequality in a solution set: x < 5 ?
{x|x<5}
How would you describe this inequality in interval notation: x ≥ 2 ?
[ 2 , ∞ )
How would you express this inequality in a solution set: -4 < t ≤ 5/3 ?
{t| -4 < t ≤ 5/3}
Describe this inequality in interval notation: x < 5/3 ?
( -∞ , 5/3)
What do you do when you divide a negative number in an inequality?
Switch the signs
For what inequality symbols do you use brackets?
≤ and ≥
For what inequality symbols do you use parenthesis?
< and >
What is the solution for |x - 5| < 0?
∅ The absolute value of a number can never be < 0. No solution.
What is the solution for |x - 5| ≤ 0?
{5} Absolute value of a number can never be < 0 but it can be 0 when x =5. |5 - 5| = 0
What is the solution for |x - 5| ≥ 0?
R. Any real number, because the absolute value of a number is always ≥ 0.
How would you word the function (front) part of this equation? P(x) = 7.25x
P is a function of x
Is this a linear equation or a linear function: y = 2x - 1 ?
A linear equation.
Is this a linear equation or a linear function: f(x) = 2x - 1 ?
A linear function
What is the standard form for a linear equation?
ax + by = c Both x and y are on the right side of the equation.
How do you find the y-intercept in a linear equation? ax + by = c
Set x = 0 to isolate the y.
How do you find the x-intercept in a linear equation? ax + by = c
Set y = 0 to isolate the x.
The graph of y = b is?
A horizontal line through b on the y axis. (Slope = 0)
The graph of x = b is?
A vertical line through b on the x axis. (Slope undefined)
How is slope defined?
The slope of a line measures the tilt of the line.
How does slope tell us the tilt? y/x
Because y/x = rise/run
How do we view lines on the graph to get the correct slope?
Left to right
How is a positive line viewed?
It runs upwards. m>0
How is a negative line viewed?
It runs downwards. m<0
If m=0 what type of line would be viewed on the graph?
A horizontal line.
How do we calculate slope from two given points? (x¹,y¹) (x²,y²)
y² - y¹/x² - x¹ to get the slope. (rise/run)
What is the formula for slope-intercept form?
y = mx + b
What is the point slope equation? Used when you have a slope and a random point.
y - y¹ = m(x - x¹)
How do you find the equation of a line if you are given two points, but no y-intercept or slope? Such as: (4 , 3) (6 , -2)
You would have to find the slope first, x² - x¹/y² - y¹, and then use the point slope equation, y - y¹ = m(x - x¹)
What do parallel lines have in common?
Equal slopes. m¹ = m²
What is the difference in slopes for perpendicular lines?
They have negative reciprocals. ex. Line¹: m= 2/4, would mean Line²: m= -4/2
What is the equation of the line through (0 , 2) that is perpendicular to the graph of the line: y = 1/2x - 4 ? (Think of the perpendicular rule)
y = -2/1x + 2 because (0 , 2) gives us the y intercept and -2/1 is the reciprocal of the line in the question.
When graphing linear inequalities, for what symbols would you use a solid line in the graph?
≤ and ≥
When graphing linear inequalities, for what symbols would you use a dotted line in the graph?
< and >
What is the first step in graphing a linear inequality? y ≥ 1/2x - 3
Decide what type of line it will be. In this case it's a solid line because of the ≥ sign.
How do you graph the line of a linear inequality? (Also the 2nd step) y ≥ 1/2x - 3
Set the inequality as an equation y ≥ 1/2x - 3 turns into y = 1/2x - 3, and plot the line.
How should you decide what side will be shaded in while graphing a linear inequality? y ≥ 1/2x - 3
Test it by putting in a point, typically (0 , 0) and see if it's true. 0 ≥ 1/2(0) -3 is 0 ≥ 3, which means true. Shade on the side of the point. If false shade on opposite side.
What three methods could be used to solve systems of linear equations? Equation¹: 2x + y = 6 Equation²: 3x - 2y = 16
By graphing, the substitution method, and the addition method.
These two equations would be easiest to solve by what method Equation¹: y = -2x + 6 Equation²: 3x - 2y = 16?
The substitution method, substitue first equation in place of y in the second: 3x - 2(-2x + 6) = 16 and solve.
These two equations would be easiest to solve by what method Equation¹: 4x + 2y = 6 Equation²: 3x - 2y = 16?
The addition method. If set up as if adding them together the y's would cancel. Then you could solve for x and then solve for y.
A linear system where two lines cross (one answer) would be what type of system?
A consistent system.
A linear system where two lines are parallel (no answer) would be what type of system?
An inconsistent system.
A linear system where two lines are directly on top of each other (infinite answers) would be what type of system?
A dependent system.
When solving a linear system of equations, if the substitution or addition method resulted in 2 = 2, what would you have?
A dependent system, two lines on top of each other. ∞ Solutions
When solving a linear system of equations, if the substitution or addition method resulted in 2 ≠ 4, what system would you have?
An inconsistent system, parallel lines. No solution.
Natural Numbers (Counting Numbers)
1, 2, 3, 4, 5, 6, 7, 8, 9, etc.
Whole numbers that are not negative.
Integers
-3, -2, -1, 0, 1, 2, 3, etc.
These are the natural numbers, their additive inverses (negatives), and 0.
Rational Numbers
2/1, 1/3, (-1/4), 22/7, 0, 1.2, etc.
Every integer is a rational number. Rational numbers can be expressed as decimals that either terminate (end) or repeat a sequence of digits.
Irrational Numbers
Irrational Numbers are numbers which are no rational numbers. They cannot be expressed as the ratio of two integers and has a decimal representation that does not terminate or repeat.
Real Numbers
2, -10, -131.3337, 1/3, etc.
Real Numbers can be represented by decimal numbers. Real numbers include both the rational and irrational numbers.
Classify: 5
Natural Number, Integer, Rational Number.
Classify: -1.2
Rational Number.
Classify: 13/7
Rational Number
Order of Operations is...
PEMDAS
Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction.
Example: a/0
Not defined.
Example: 0/a
0.
What is Mean?
The quotient of the sum of several quantities and their number; an average.
What is Median?
The median (middle) value of a range of values.
What is a Cartesian (rectangular) coordinate plane?
What does Relation mean?
A relation is a set of ordered pairs.
What is an Ordered Pair?
(x, y)
What is the Domain?
x.
What is the Range?
y.
What is the Distance Formula?
What is the Midpoint Formula?
What is the Standard Equation of a Circle?
What is a Circle?
A Circle is a collection/set of points that are on a fixed distance (radius) from a fixed point (center).
What must you ALWAYS be careful with?
Your signs!
Absolute Value is...
Absolute Value is the distance from zero; always positive.
What is Function Notation?
f(x)=y
What is a Function?
A Function is a relation in which every element in a first set (domain) is paired to a unique element on a second set (range).
What is the Domain of a Function?
The Domain of a Function is the set of values that makes the function well defined. If the function is not well defined at a point, then that point is not in the domain.
What is the Range of a Function?
The Range of a Function f(x) is the set {y=f(x)}.
What is the Vertical Line Test?
If every vertical line intersects a graph at no more than one point, then the graph represents a function.
Increasing & Decreasing Functions
f increases on l if, whenever x^1<x^2, f(x^1)<f(x^2).
f decreases on l if, whenever x^1<x^2, f(x^1)>f(x^2).
What are the "Library of Functions?"
How do we read? And how should I read a graph?
From left to right. Read a graph the same way.
What is Interval Notation and how do you use it?
What is the U (Union) Symbol?
Joins two pairs. You write this in between the inequality of the interval notation.
What is the Average Rate of Change?
The Average Rate of Change of f from x^1 to x^2 is...
y^2-y^1/x^2-x^1
What is the Difference Quotient?
What is the Point-Slope Form of an Equation of a Line?
What is the Slope Formula?
What is the Slope-Intercept Form of an Equation of a Line?
Also known as a Linear Function.
What is the Standard Form?
With Verticle Lines...
The Slope is undefined!
The General Form of the Equation of a Line is...
What is an Equation?
An Equation is a statement that two mathematical expressions are equal.
What are the three types of Equations?
1. Contradiction - No solution.
2. Identity - There is a solution.
3. Conditional - Satisfied by some, but not by all values of the variable.
What is a Linear Equation?
A Linear Equation in one variable is an equation that can be written in the form ax+b=0.
It must be equal to 1 (variable). It can not be squared or cubed unless they cancel out. Some may not even have variables, which it then is undefined.
What are the Properties of Equality?
How do you do Intersection-of-Graph method on a Calculator?
Type in your results using Y=, Graph and then hit 2nd>Calc and Intersect to find the Intersection. Can also be done by hand.
How do you Solve Application Problems?
What is the Formula when dealing with Distance, Speed, Time, etc.?
What are the Inequality Symbols?
What is Set Builder Notation?
What is a Compound Inequality?
Make sure to isolate the variable in between.
The solution set (in interval notation) of a compound inequality is always an interval of these: (a,b), [a,b], (a,b], [a,b).
What is the Absolute Value Function?
It is V-Shaped and is represented by y=|x|. It cannot be represented by single linear function.
What is the basic Absolute Value Equation?
You can also have a number inside, such as |x-3| and the equal sign can be changed to something such as < or >.
Absolute Value Function
a function written in the form y = /x/, and the graph is always in the shape of a v
Constant of Variation
the number k in equations of the form y=kx
Dependent Variable
a factor that can change in an experiment in response to changes in the independent variable
Direct Variation
y=kx
Domain
The set of x-coordinates of the set of points on a graph; the set of x-coordinates of a given set of ordered pairs. The value that is the input in a function or relation.
Function
A relation which one element from the domain is paired with range
Function Notation
an equation in the form of 'f(x)=' to show the output value of a function, f, for an input value x
Independent Variable
variable that is changed in an experiment
Linear Equation
an equation whose graph is a straight line
Linear Function
a function in which the graph of the solutions forms a line
Linear Inequality
an inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line
Mapping Diagram
a way to show a relation that links elements of the domain with cooresponding elements of the range
Parameter
a determining or characteristic element; a factor that shapes the total outcome; a limit, boundary
Parent Function
The simplest function in a family; all functions in the family are transformations of it
Point slope form
Y-Y1=m(x-x1)
Range
The y-coordinates of the set of points on a graph. Also, the y-coordinates of a given set of ordered pairs.
Reflection
A transformation that "flips" a figure over a mirror or reflection line.
Relation
A set of ordered pairs
Scatter Plot
a graph with points plotted to show a possible relationship between two sets of data.
Shrink
Reduces y values by a factor between 1-0
Slope
the steepness of a line, equal to the ratio of a vertical change to the corresponding horizontal change
Slope Intersept Form
the equation of a line in the form of y=mx+b, where m is the slope and b is the y intersept
Standerd Form
ax+by=c
Stretch
Multiplys all y values by the same factor greater than 1
Transformation
An operation that moves or changes a geometric figure in some way to produce a new figure
Translation
A transformation that "slides" each point of a figure the same distance in the same direction.
Trend Line
a line that approximates the relationship between the data sets of a scatter plot
Vertex
The point in a function where the function reaches a min or a max
Vertical Line Test
a method to determine if a graph is a function or not
x-intercept
The x-coordinate of the point where a line crosses the x-axis.
y-intercept
the y-coordinate of the point where the line crosses the y-axis.
What can a Linear Function Equation be written as?
f(x) = ax + b or f(x) = mx + b.
The formula for a quadratic function is different from that of a linear function because it contains an x^2 term.
Examples:
f(x) = 3x^2 + 3x + 5 and g(x) = 5 - x^2 are not linear equations because of the ^2's.
What is the Quadratic Function Equation?
The "a" in f(x) is called the Leading Coefficient.
What is a Parabola?
The graph f(x) is called a Parabola.
Examples:
f(x) = 2x^2 - x + 1 (Opens Upward because positive. a>0.)
g(x) = -x^2 + 3x -5 (Opens Downward because negitive. a<0.)
The highest (Absolute Max)/lowest (Absolute Min) point is called the Vertex.
What is the Vertex Form?
Parabola graph of f(x) = a (x-h)^2 + k with a not equal to 0; vertex (h,k). For the Vertex, h is always the opposite and k is what is displayed.
This formula is known as the Standard Form of the Parabola.
How do you make a Perfect Square?
What is Completing the Square?
As a Quadratic Expression: x^2 + kx + (k/2)2.
As a Perfect Square Trinomial: x^2 + kx + (k/2)^2 = (x+k/2)^2.
Completing the Square Example...
x^2 + 4x + 5
1. x^2 + 4x ____ + 5
2. 4/2. ALWAYS divide by 2! Then square the answer; 2^2 = 4. Now add AND THEN ALSO subtract it.
3. (x^2 + 4x + 4) + 5 - 4
4. Now make it a perfect square!
(x + 2)^2 + 1.
This is the Standard Form - f(x) = a(x - h)^2 + k.
1(x+2)^2+1; 1 is a, 2 is h and 1 is k. Vertex is (-2,1) and it is going Upwards.
How can you Graph Quadratic Functions?
By hand or by using a calculator. A calculator is a good tool to check your answer by hand!
How do you find the Area of a Rectangle?
What is an easier alternate formula to find the vertex?
A Projectile is...
... anything you can throw.
What is the Square Root Property?
How do you do a Quadratic Equation by Calculator?
Use Y^1 & Y^2 in your Calc. Hit Graph, and then use the Intersection.
What is the Quadratic Formula?
Factoring can be done by...
... trial and error.
What is the Zero-Product Property?
ab=0 if and only if a=0 or b=0.
What does Area equal?
Two important things to remember in math are...
PEMDAS and to be careful with your signs!
The Discriminate Formula is...
What are Complex Numbers?
A Complex Number is a number than can be written as a + bi. A is Real and B is Imaginary.
>0 (is postive) - 2 distinct real sol.
>| (bottom line) ( is zero) - Two solutons, but same. So 1.
<0 (is negative) - No real solutions.
What are some Complex Numbers?
1 + 2i, -3i, 4, 3 - 11i, etc.
Every Real Number is a Complex Number (Ex. 4 --> 4+0i).
What is an Imaginary Unit?
There is also more, but the most commonly used is the one above and i^2 = -1.
What is the Expression of sqrt(-a)?
If a > 0, then sqrt(-a) = i sqrt(a).
What is a Complex Conjugate?
If a + bi is given, then the Complex Conjugate is a - bi.
Fact: (a + bi)(a - bi) = a^2 + b^2. The i disappears.
Fact: (Conjugate Zeros Therom) - If a + bi is a solution at some polynomial eqn., then a - bi is also a solution.
What is a Transformation?
A Transformation is a shift or translations in the xy-plane.
What types of Transformations are there?
There are many types of Transformations. All moving either left, right, up or down.
Some are...
y = f(x) + c --> Upward.
y = f(x) - c --> Downward.
y = f(x - c) --> Right.
y = f(x + c) --> Left.
If a number is in front of the X, this number makes it go slower (Ex. 1/2) or faster (Ex. 2).
These (dealing with i & Transformations) can be done by...
Hand and/or Calc.
Two rules when dealing with reflections and negatives are...
1. When negative is outside, it will make it look down. (Also the opposite.)
2. When negative is inside, it will make it look left. (Also the opposite.)
* When dealing with square roots, when the negative is inside, the only values we can then add are those that are negative... which then make it a positive and is valid.
Combining Transformations...
Ex. y = -2(x - 1)^2 + 3
- Reflects across the x-axis.
2 Stretches vertically by a factor of 2.
- 1 Shifts to the right 1 unit.
+ 3 Shifts upward 3 units.
What is a Polynomial Function?
For example, 2x^2 - 3x + 1 is a Polynomial; C(X) = 4 is a Polynomial... you just don't see the "0's."
A Function is Even if...
f(-x) = f(x)
(+ - +)
Always Symmetrical with respect to the y-axis.
A Function is Odd if...
f(-x) = -f(x)
(- + -)
Always Symmetrical with respect to the origin.
A Function is Neither if...
A Function is "Neither" if it is not odd nor even. Such as...
+ + +.
Local/Relative Maximum (Minimum) & Absolute/Global Maximum (Minimum) is...
Basics of division...
When Dividing Polynomials, you must make sure...
Make sure that everything is written in decreasing order of powers!
Ex. 3x^3 - x^2 + 5 --> 3x^3 - x^2 + 0x + 5
Two ways to check when Dividing by Polynomials...
Dividend/Divisor = Quotient + Remainder/Divisor
or
Dividend = (Divisor)(Quotient) + (Remainder)
Long Division...
Make sure that everything is written in decreasing order of powers!
Synthetic Division...
A shortcut that can be used to divide h-k into a polynomial.
Make sure that everything is written in decreasing order of powers!
The Remainder Theroem...
The Remainder Theroem is when the remainder of dividing P(X) by X-K is P(K). You can also use Linear or Synthetic of course.
Ex. f(2) = (2)^4 - 3(2)^2 - 4(2)^2 + 12(2)...
Note: P(X) = (X-K) Q(X) + R
If the remainder is 0, P(X) = (X-K) Q(X). Thus, (X-K) (The divisor) is a factor of P(X).
The Factor Theroem...
In the Factor Theroem, X-K is a factor of P(X) if and only if the remainder is P(K) = 0. (The remainder is 0).
Ex. Decide if x-3 is a factor of x^3 - 2x +1
Use Synthetic, Linear or Remainder Theorem.
P(X) = x^3 - 2x + 1
P(3) = 3^3 - 2(3) + 1 = 22
It's not 0, so x-3 is not a factor.
Complete Factored Form...
f(x) = a^n (x-c)(x-b)(x-n)...
Such as...
7x^3 - 21x^2 + 7x + 21 = 7(x+1)(x-1)(x-3)
Rational Zero's Test is...
The possible Rational Zero's of f(x) are: factors of a^o / factors of a^n.
Steps:
1. Consider the possible rational zeros.
2. Find the zero's of the quotient.
3. Write f(x) in factoral form.
The Fundamentals Theorem of Algebra is...
The Fundamentals Theorem of Algebra is a Polynomail of degree n with complex coefficients has a complex zero.
Fact: A Polynomial of degree n has n zeros (counting multiplicity.)
Conjugate Zeros Theorem is...
The Conjugate Zeros Theorem is if a Polynomial f(x) has only real coefficients and if a + bi is a zero of f(x), then the conjugate a - bi is also a zero f(x).
Ex. f(x) = 1 + 2i is a zero.
f(x) = 1 - 2i is also a zero.
When dealing with shifting of graphs (ex.)...
Ex. from Exam 2...
1. f(x) = x^2 + 2x - 7; left 8 unit, up 11 units.
2. f(x+8) + 11
3. f(x+8) = (x+8)^2 + 2(x+8) - 7 + 11.
Clean up and done!
When doing Quadratic Equations...
When doing Quadratic Equations, you can separate the denominator. Such as...
(-3/2) + or - (sqrt3/2)
or
(-3 + or - sqrt3)/(2)
Real Zero's of a Polynomial facts...
Real Zero's of a Polynomial = x-intercept.
No x-intercept means no real zeros.
Factor
Number multiplied by another factor to get a product in a multiplication problem.
Prime
A number that has no other factors but itself and 1.
Composite
A number that has three or more whole number factors.
Base
Number getting multiplied by itself.
Exponent
Indicates how many times the base gets multiplied by itself.
Power
Indicates the number of times a number is multiplied by itself.
Order of Operations
Order to solve a problem.
PEMDAS.
Parentheses
Used to group things to be done first, or show multiplication.
Brackets
used to group things to be done first, or show multiplication.
Property
The actions of numbers when combined.
Commutative Property
Change the order without changing the outcome.
Associative Property
Change the grouping without changing the outcome.
Distributive Property
Distribute the the number to the other ones.
Addition Property of Zero
Adding zero to a number is equal to the same number.
Additive Identity Property
Addition Propery of Zero
Multiplication Property of 1
Multiplying 1 by a number is equal to the same number.
Multiplicative Inverses (Reciprocals)
Multiplication Property of 1
Multiplication Property of Zero
Multiplying 0 by a number is equal to 0.
Universal Set
All the elements you can use.
Intersection
All the elements that belong to both sets.
Disjoint Sets
Empty Sets { } or null.
Empty Sets
Disjoint Sets { } or null.
Union
Everything that belongs to BOTH sets.
Complement
Everything that's NOT in that set.
X- Axis
Horizontal line
Y- Axis
Vertical line
Coordinate axes
lines that have the same scale and are drawn perpendicular to eachother.
Coordinate Plane
Plane determined by the axes
Origin
Intersection of the two axes.
Quadrants
4 regions, roman numerals 1, 2, 3 and 4, in a counterclockwise order.
Ordered Pair
two numbers to be plotted.
x- coordinate
(Abscissa) 1st number in the pair
y- coordinate
(Ordinate) 2nd number in the pair.
Graph of the Ordered Pair
Graph that order pairs get plotted onto.
Polygon
Closed figure whose sides are line segments.
Quadrilateral
polygon with 4 sides.
Verticies
Endpoints of the sides.
Natural Numbers (Counting Numbers)
1, 2, 3, 4, 5, 6, 7, 8, 9, etc.
Whole numbers that are not negative.
Integers
-3, -2, -1, 0, 1, 2, 3, etc.
These are the natural numbers, their additive inverses (negatives), and 0.
Rational Numbers
2/1, 1/3, (-1/4), 22/7, 0, 1.2, etc.
Every integer is a rational number. Rational numbers can be expressed as decimals that either terminate (end) or repeat a sequence of digits.
Irrational Numbers
Irrational Numbers are numbers which are no rational numbers. They cannot be expressed as the ratio of two integers and has a decimal representation that does not terminate or repeat.
Real Numbers
2, -10, -131.3337, 1/3, etc.
Real Numbers can be represented by decimal numbers. Real numbers include both the rational and irrational numbers.
Classify: 5
Natural Number, Integer, Rational Number.
Classify: -1.2
Rational Number.
Classify: 13/7
Rational Number
Order of Operations is...
PEMDAS
Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction.
Example: a/0
Not defined.
Example: 0/a
0.
What is Mean?
The quotient of the sum of several quantities and their number; an average.
What is Median?
The median (middle) value of a range of values.
What is a Cartesian (rectangular) coordinate plane?
What does Relation mean?
A relation is a set of ordered pairs.
What is an Ordered Pair?
(x, y)
What is the Domain?
x.
What is the Range?
y.
What is the Distance Formula?
What is the Midpoint Formula?
What is the Standard Equation of a Circle?
What is a Circle?
A Circle is a collection/set of points that are on a fixed distance (radius) from a fixed point (center).
What must you ALWAYS be careful with?
Your signs!
Absolute Value is...
Absolute Value is the distance from zero; always positive.
What is Function Notation?
f(x)=y
What is a Function?
A Function is a relation in which every element in a first set (domain) is paired to a unique element on a second set (range).
What is the Domain of a Function?
The Domain of a Function is the set of values that makes the function well defined. If the function is not well defined at a point, then that point is not in the domain.
What is the Range of a Function?
The Range of a Function f(x) is the set {y=f(x)}.
What is the Vertical Line Test?
If every vertical line intersects a graph at no more than one point, then the graph represents a function.
Increasing & Decreasing Functions
f increases on l if, whenever x^1<x^2, f(x^1)<f(x^2).
f decreases on l if, whenever x^1<x^2, f(x^1)>f(x^2).
What are the "Library of Functions?"
How do we read? And how should I read a graph?
From left to right. Read a graph the same way.
What is Interval Notation and how do you use it?
What is the U (Union) Symbol?
Joins two pairs. You write this in between the inequality of the interval notation.
What is the Average Rate of Change?
The Average Rate of Change of f from x^1 to x^2 is...
y^2-y^1/x^2-x^1
What is the Difference Quotient?
What is the Point-Slope Form of an Equation of a Line?
What is the Slope Formula?
What is the Slope-Intercept Form of an Equation of a Line?
Also known as a Linear Function.
What is the Standard Form?
With Verticle Lines...
The Slope is undefined!
The General Form of the Equation of a Line is...
What is an Equation?
An Equation is a statement that two mathematical expressions are equal.
What are the three types of Equations?
1. Contradiction - No solution.
2. Identity - There is a solution.
3. Conditional - Satisfied by some, but not by all values of the variable.
What is a Linear Equation?
A Linear Equation in one variable is an equation that can be written in the form ax+b=0.
It must be equal to 1 (variable). It can not be squared or cubed unless they cancel out. Some may not even have variables, which it then is undefined.
What are the Properties of Equality?
How do you do Intersection-of-Graph method on a Calculator?
Type in your results using Y=, Graph and then hit 2nd>Calc and Intersect to find the Intersection. Can also be done by hand.
How do you Solve Application Problems?
What is the Formula when dealing with Distance, Speed, Time, etc.?
What are the Inequality Symbols?
What is Set Builder Notation?
What is a Compound Inequality?
Make sure to isolate the variable in between.
The solution set (in interval notation) of a compound inequality is always an interval of these: (a,b), [a,b], (a,b], [a,b).
What is the Absolute Value Function?
It is V-Shaped and is represented by y=|x|. It cannot be represented by single linear function.
What is the basic Absolute Value Equation?
You can also have a number inside, such as |x-3| and the equal sign can be changed to something such as < or >.
Quadratic Equation
Distance
Midpoint Formula
Slope
Slope Intercept
Stnd Form of a Circle
General Form of a Circle
Vertex Formula
Testing Symmetry
Difference Quotient
Compound Interest Formula
Compound Interest Formula solved for P
Continuously Compound Interest Formula
Radioactive Decay Formula
Af=Ai(2)-t/h Where Af is Amount Final(at present) is equal to Amount Initial times two raised to the negative of the amount of time passed divided by the Halflife.
Malthusian Population Growth
P = Pµeⁿ°
Where P is current population, Pµ is initial population, e is raised to the ()number in years times the °growth rate (birth rate - death rate)
Logarithmic Relationships
Pascals Triangle
Permutations Formula
Combinations Formula
nCr = n!/r!(n-r)!
Combinations Formula
P(n,r) = n!/(n-r)!
Permutations Formula
Log Power Rule
logαMⁿ =nlogαM
Log Change of Base Formula
Binomial Expansion
Factorial Notation
n! = n × (n - 1) × (n - 2) × (n - 3) × ... × 3 × 2 × 1
Binomial Expansion By Pascals Triangle
Definition of Term in Binomial Expansion
Summation Notation
Point Slope Form
Standard Form of a Line
numerical expression
consists of numbers and operations
evaluate
to find the value of an expression
order of operations
the order in which operations in an expression to be evaluated are carried out. 1. parentheses 2. exponets 3. multiplication and divison 4. addition and subtraction
variable
a letter used to represent one or more numbers
variable expression
consists of numbers, variables, and operations
power
is a number made of repeated factors
exponent
a mathematical notation indicating the number of times a quantity is multiplied by itself
base
is the number that is repeatedly multiplied in a power
equation
a mathematical sentence with an equal sign that shows that two expressions are equivalent
solving an equation
finding all the solutions of an equation
perimeter
The sum of the lengths of the sides of a polygon
area
the number of square units needed to cover a flat surface
integers
Positive and Negative Whole Numbers and 0. Examples: .... -3, -2, -1, 0, 1, 2, 3.....(Does not include fractions or decimals)
additive identity property
The sum of a number and zero is always that number.
additive inverse property
The sum of a number and its opposite is zero.
multiplication identity property
The product of any number and one is that number.
commutative property of addition
In a sum, you can add terms in any order, a + b = b + a
commutative property of multiplication
the order of the factors does not change the product a x b = b x a
associative property of addition
changing the grouping of terms will not change the sum, (a + b) + c = a + (b + c)
associative property of multiplication
changing the grouping of factors will not change the product, (ab)c = a(bc)
distributive property
a(b + c) = ab + ac
an + ac = a(b+ c)
terms
in an expression are separated by addition and subtraction signs
like terms
terms that have identical variable parts raised to the same power
coefficient
number in front of a variable
constant
a term that has no variable and does not change
coordinate plane
A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis.
aka "the Cartesian plane" after René Descartes
ordered pair
A pair of numbers, (x, y), that indicate the position of a point on a coordinate plane.
quadrant
one of four sections into which the coordinate plane is divided
Inverse Operation
operations that undo each other, such as addition and subtraction
Solve for a variable
To get a variable alone on one side of an equation or inequality in order to solve the equation or inequality
Equivalent Equations
equations that have the same solution
Inequality
A statement that compares two quantities using <, >, ≤,≥, or ≠
reciprocal
one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
rational number
A number that can be written as a/b where a and b are integers, but b is not equal to 0.
ratio
a comparison of two numbers by division
rate
a ratio that compares two quantities measured in different units
irrational number
a number that can not be written a/b
input
the x-value in a function
output
the y-value in a function
function
a relation that assigns exactly one output value for each input value
domain
the set of all the input (x-values) for a function
range
the set of all the output (y-values) for a function
linear equation
an equation whose answers are ordered pairs (x,y) and whose answers graph a straight line
x-intercept
the point where a graph crosses the x-axis
y-intercept
the point where a graph crosses the y-axis
slope
the steepness of a line on a graph, rise over run
Slope Intercept Form
y= mx + b
where "m=slope" and "b=y-intercept"
Slope Formula
Slope=m= Y2 - Y1 / X2 - X1
additional
existing or coming by way of addition
annual
occurring or payable every year
approximately
Almost, but not exactly; more or less
category
a classification or grouping
chemical compound
a substance formed by the chemical combination of two or more elements in definite proportions
common
to be expected
composed of
made up of
conclusion
the act of making up your mind about something
corresponding
Angles or lines of 2 different polygons that are in the same position
counterexample
refutation by example
creates
to make, form.
determine
shape or influence
effect
an outward appearance
elapsed
passed
exceeds
to go beyond
formed
clearly defined
identical
exactly alike
international
concerning or belonging to all or at least two or more nations
joined
joined
non-included
Not between
per
by or through
population
statistics the entire aggregation of items from which samples can be drawn
prove
prove formally
regarding
concerning about or with respect to
regardless
in spite of everything
region
the approximate amount of something usually used prepositionally as in 'in the region of'
represents
under or below
routine
found in the ordinary course of events
satisfy
fill or meet a want or need
shown
Revealed
sufficent
all that is needed, enough
systems
groups of organs working together to perform complex functions
twice
two times
shown
to reveal
view
the act of looking or seeing or observing
promoting
calling attention to, advertising or publicizing
best represents
closest to; most similar to
closets to
something that is close to another object
in terms of
with regard to,with respect to
reasonable conclusion
a position or opinion or judgment reached after considering facts and observations
Absolute Value
a number's distance from zero
Coefficient
the number in front of the variable (ie: 5 in 5a)
Irrational Numbers
numbers that do not repeat or terminate
Rational Numbers
numbers that stop and can be written as fractions
Domain
the first term in an ordered pair
Range
the second term in an ordered pair
Function
one domain paired with exactly one range
Independent Variable
the variable that makes up the domain
Feasible Region
the area between intersecting lines (shaded area)
Systems of equations
2 or more equations with the same variable
Elimination Method
eliminate a variable by adding or subtracting the equations
Substitution Method
solve an equation and substitute it into another equation
Vertex
the point of intersection between lines
Scatter Plot
a graph of ordered pairs
x-intercept
where a graph crosses the x-axis
Empty Set
when an equation has no solution
Monomial
a number, variable, or a product of a number and variable
Slope
change in y over the change in x (rise/run)
Standard Form
Ax+By=c
Determinant
numbers in a square array enclosed by parallel lines
Dilation
enlargement or reduction of an image
Element
each number in a matrix
Matrix
a rectangular array of numbers inside square brackets
Scalar Multiplication
multiplying a matrix by a scalar
Translation
movement of a figure
Complex Number
a number in the form a+bi
Imaginary Unit
i = √-1
Parabola
a "u" shaped graph y=x²
Discriminant
b²-4ac
Quadratic Equation
x=(-b±√b²-4ac)/2a
Quadratic Function
an equation of the form x²
Root
solutions to a quadratic equation
Degree of a Polynomial
greatest degree of any term
Leading Coefficient
the coefficient of the term with the highest degree
Polynomial Function
function represented by a monomial or a sum of monomials
Relative Maximum
highest point on a graph
Relative Minimum
lowest point on a graph
Synthetic Division
method used to divide polynomials by monomials
Synthetic Substitution
the use of synthetic division to evaluate a polynomial
additional
existing or coming by way of addition
annual
occurring or payable every year
approximately
Almost, but not exactly; more or less
category
a classification or grouping
chemical compound
a substance formed by the chemical combination of two or more elements in definite proportions
common
to be expected
composed of
made up of
conclusion
the act of making up your mind about something
corresponding
Angles or lines of 2 different polygons that are in the same position
counterexample
refutation by example
creates
to make, form.
determine
shape or influence
effect
an outward appearance
elapsed
passed
exceeds
to go beyond
formed
clearly defined
identical
exactly alike
international
concerning or belonging to all or at least two or more nations
joined
joined
non-included
Not between
per
by or through
population
statistics the entire aggregation of items from which samples can be drawn
prove
prove formally
regarding
concerning about or with respect to
regardless
in spite of everything
region
the approximate amount of something usually used prepositionally as in 'in the region of'
represents
under or below
routine
found in the ordinary course of events
satisfy
fill or meet a want or need
shown
Revealed
sufficent
all that is needed, enough
systems
groups of organs working together to perform complex functions
twice
two times
shown
to reveal
view
the act of looking or seeing or observing
promoting
calling attention to, advertising or publicizing
best represents
closest to; most similar to
closets to
something that is close to another object
in terms of
with regard to,with respect to
reasonable conclusion
a position or opinion or judgment reached after considering facts and observations
numerical expression
consists of numbers and operations
evaluate
to find the value of an expression
order of operations
the order in which operations in an expression to be evaluated are carried out. 1. parentheses 2. exponets 3. multiplication and divison 4. addition and subtraction
variable
a letter used to represent one or more numbers
variable expression
consists of numbers, variables, and operations
power
is a number made of repeated factors
exponent
a mathematical notation indicating the number of times a quantity is multiplied by itself
base
is the number that is repeatedly multiplied in a power
equation
a mathematical sentence with an equal sign that shows that two expressions are equivalent
solving an equation
finding all the solutions of an equation
perimeter
The sum of the lengths of the sides of a polygon
area
the number of square units needed to cover a flat surface
integers
Positive and Negative Whole Numbers and 0. Examples: .... -3, -2, -1, 0, 1, 2, 3.....(Does not include fractions or decimals)
additive identity property
The sum of a number and zero is always that number.
additive inverse property
The sum of a number and its opposite is zero.
multiplication identity property
The product of any number and one is that number.
commutative property of addition
In a sum, you can add terms in any order, a + b = b + a
commutative property of multiplication
the order of the factors does not change the product a x b = b x a
associative property of addition
changing the grouping of terms will not change the sum, (a + b) + c = a + (b + c)
associative property of multiplication
changing the grouping of factors will not change the product, (ab)c = a(bc)
distributive property
a(b + c) = ab + ac
an + ac = a(b+ c)
terms
in an expression are separated by addition and subtraction signs
like terms
terms that have identical variable parts raised to the same power
coefficient
number in front of a variable
constant
a term that has no variable and does not change
coordinate plane
A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis.
aka "the Cartesian plane" after René Descartes
ordered pair
A pair of numbers, (x, y), that indicate the position of a point on a coordinate plane.
quadrant
one of four sections into which the coordinate plane is divided
Inverse Operation
operations that undo each other, such as addition and subtraction
Solve for a variable
To get a variable alone on one side of an equation or inequality in order to solve the equation or inequality
Equivalent Equations
equations that have the same solution
Inequality
A statement that compares two quantities using <, >, ≤,≥, or ≠
reciprocal
one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
rational number
A number that can be written as a/b where a and b are integers, but b is not equal to 0.
ratio
a comparison of two numbers by division
rate
a ratio that compares two quantities measured in different units
irrational number
a number that can not be written a/b
input
the x-value in a function
output
the y-value in a function
function
a relation that assigns exactly one output value for each input value
domain
the set of all the input (x-values) for a function
range
the set of all the output (y-values) for a function
linear equation
an equation whose answers are ordered pairs (x,y) and whose answers graph a straight line
x-intercept
the point where a graph crosses the x-axis
y-intercept
the point where a graph crosses the y-axis
slope
the steepness of a line on a graph, rise over run
Slope Intercept Form
y= mx + b
where "m=slope" and "b=y-intercept"
Slope Formula
Slope=m= Y2 - Y1 / X2 - X1
5 ∙ 0 = 0
n ∙ 0 = 0
0 ∙ 3 = 0
n ∙ 0 = 0
(¾)(0) = 0
n ∙ 0 = 0
(.2)(0) = 0
n ∙ 0 = 0
0 ∙ 1 = 0
n ∙ 0 = 0
9 ∙ 1 = 9
n ∙ 1 = n
7/8 ∙ 1 = 7/8
n ∙ 1 = n
(.6)(1) = 1
n ∙ 1 = n
(1)(1) = 1
n ∙ 1 = n
(-5)1 = -5
nm = mn
2 ∙ 2 = 22
n ∙ n = n[sq]
(-6)(-6) = (6)2
n ∙ n = n[sq]
3 ∙ 3 = 32
n ∙ n = n[sq]
(½)(½) = (½)2
n ∙ n = n[sq]
(.9)(.9) = (.9)2
n ∙ n = n[sq]
4 ∙ 7 = 7 ∙ 4
nm = mn
(.4)(.6) = (.6)(.4)
nm = mn
(-5)(2) = (2)(-5)
nm = mn
6 + 0 = 6
n + 0 = n
2/3 + 0 = 2/3
n + 0 = n
.25 + 0 = .25
n + 0 = n
72 + 0 = 72
n + 0 = n
34 - 34 = 0
n - n = 0
9.4 - 9.4 = 0
n - n = 0
¼ - ¼ = 0
n - n = 0
5 - 5 = 0
n - n = 0
5 + 4 = 4 + 5
n + m = m + n
2/5 + ¼ = ¼ + 2/5
n + m = m + n
6 + 1.3 = 1.3 + .6
n + m = m + n
3 ∙ 1/3 = 1
n ∙ 1/n = 1
7/8 ∙ 8/7 = 1
n ∙ 1/n = 1
7 ∙ 1/7 = 1
n ∙ 1/n = 1
½ ∙ 2 = 1
n ∙ 1/n = 1
14/14 = 1
n/n = 1
9/9 = 1
n/n = 1
.3/.3 = 1
n/n = 1
5/8 ÷ 5/8 = 1
n/n = 1
Eigenvalue
a scalar, nonzero solution to Ax=λx;Basically it is when you have a vector multiplied by a given matrix that produces another vector, which can be represented as a scalar (eigenvalue) and the original vector; Found by solving the characteristic polynomial (they are the roots)
Eigenvector
a nonzero vector (x) where Ax=λx;basically its a scaled vector, where the linear transformation doesn't change; It's also a nonzero in the nullspace of a given matrix [think(A-I)x=0
Spectrum (of a matrix)
the range or set of all eigenvalues of a square
Eigenspace
the set of all solutions for Ax=λx for a specific λ; the null-space of (A-λI)
Algebraic Multiplicity
number of times an eigenvalue repeats. Or the number of times that a root appears in the characteristic polynomial.
Geometric Multiplicity
dimension of eigenspace (number of free variables)
Trace (of a square matrix)
Sum of the diagonal entries in a square matrix. Also important: It is the sum of the eigenvalues of A
Characteristic (polynomials of square matrices)
Found by setting det(A-I)=0, a scalar equation that gives us roots of eigenvalues
symmetric matrix
A matrix such that it equals its transpose; any two of its eigevectors from different eigenspaces ( made from different eigenvalues) are orthogonal; counting multiplicities, has as many eigenvalues as rows or columns;orthogonally diagonizable.
similar matrix
When two matrices A, B and another (invertible) matrix P satisfy A=P⁻¹BP
diagonalizable matrix
the factorization of a matrix into three other matrices, PDP⁻¹, where D is a matrix containing the eigenvalues
orthogonal diagonalization (of a real symmetric matrix)
When an orthogonal matrix P with a diagonalized matrix D where A=PDP⁻¹=PDP^T (P⁻¹=P^T)
complex number
z=a+bi where i=√-1
conjugate (of complex numbers)
z=a-bi, etc.
modulus
(length of a complex number)
z=√(a²+b²)=‖a,b‖
Length (complex and real vectors)
The square root of all the entries of the vector squared, if it is a vector with complex entries than you must use the absolute values of the complex numbers.
unit vector
A vector with a length of one (unit, hence the name)
angles between real vectors
U·V=‖‖U‖‖ ‖‖V‖‖ cos (ϴ); ϴ=cos-1(U·V)/U·V
Cauchy-Schwarz inequality
We use this to find the angle between two real vectors in terms of approximations; (find out more!)
orthogonality
Subspaces and orthogonality: A vector is only in a space (that is transposed) if it is orthogonal to every other vector in that space.
WT is a subspace of Rn
orthogonal vector
a set of non-zero vectors;
angle between vectors is 90;
the dot product is zero
Orthonormal Vector
orthogonal vectors which have a length of 1(are unit vectors)
Orthogonal Basis
Where all the vectors in the set are orthogonal to each other that forms a subspace; a.k.a where an orthogonal set that is also a subspace
Orthonormal Basis
a subspace spanned by an orthogonal set of unit vectors
Orthogonal Matrix
a square invertible matrix such that U-1=UT; has orthonormal columns and rows
Orthogonal Projection
turning a vector into two other vectors r that sum up to it requires an orthogonal projection; Finding the weights for the linear combination: C1 =Y·U1/U1·U1 , given that y=c1v1+c2v2...
Gram-Schmidt Process
used for making orthonormal/orthogonal bases;
The process, which makes a set of vectors {x1..xn} into an orthogonal basis {v1...vn}
1. v1 = x1
2. v2 =x1- (x2·v1/v1·v1)v1
3. v3 = x3- (x3·v1/v1·v1)v1-x3·v2/v2·v2)v2 (etc.)
Least Square fit
approximating inconsistent systems of Ax = b; the smaller difference of ||b-Ax||, the better the approximation;
ATAx(approx)=ATb(approx)
spectral theorem
if A is symmetric, then it has real eigenvalues and can be orthogonally diagonalized
spectral decomposition
A=1u1uT1 +2u2uT2....... where u are the columns of P in A=PDPTwhere PT=P-1; this is the equation for matrix A where the spectrum of eigenvalues determine each piece
quadratic form
A function (Q()) where the vector inputed (x) = x^T Ax, where A is a symmetric matrix
Positive definite
Q(x)0 for all x≠0
All eigenvalues are all positive
Negative definite
Q(x)0 for all x≠0
All eigenvalues are all negative
Indefinite
Q(x) assumes both positive and negative values
Eigenvalues are both positive and negative
Linear Substitution
find out about linear substitution
Principle Axes Theorem
Let A be a symmetric matrix; there is an orthogonal change of variable, x = Py, that transforms xTAx into yTDy with no cross-products
Transpose Properties (of a matrix)
1. (A^t)^t = A
2. (AB)^t = Bt At
3. (cA)t = c (A)t
Positive Semidefinite
Q(x)0 for all x
Eigenvalues are all non-negative
Change of Variable
To change:
1. orthogonally diagonalize matrix A
2. Make D a diagonal matrix with eigenvalues
3.substitute x^T Ax with (Py)^T A (Py) where x = Py
4.simplify to get λy² +λy²...
y = x²
y = x² − 3
y = x² + 3
y = x² + 10
y = x² − 4
y = x² −25
y = x² − 9
y = (x+2)²
y = (x+1)²
y = (x − 3)²
y = (x+5)²
y = (x−2)²
y = −x²
y = −5x²
y = − (1/5)x²
y = (x−1)² +3
y = (x−2)² +5
y = (x+3)² −5
y = ±√x
y = ±√x + 3
y = ±√x − 5
y = ±√(x+3)
y = ± √(x − 5)
y = x² + 3x
y = x² −3x
y = x² −5x
y = −x² −2x
y = ±√−x
y = ±√−x + 2
y = ±√(−x+3)
b = 4
This graph does NOT represent ax² -1/4, where a would be a fraction less than 1.
This graph is of the form (x⁴ - a)/(x² + b)
Find the integer, b.
HINT
Type "b= ..."
c) (x⁴−9)/(x²+1)
Which function is shown in blue?
a) (x⁴−9)/(x²+9)
b) (x⁴−9)/(x²+2)
c) (x⁴−9)/(x²+1)
numerical expression
consists of numbers and operations
evaluate
to find the value of an expression
order of operations
the order in which operations in an expression to be evaluated are carried out. 1. parentheses 2. exponets 3. multiplication and divison 4. addition and subtraction
variable
a letter used to represent one or more numbers
variable expression
consists of numbers, variables, and operations
power
is a number made of repeated factors
exponent
a mathematical notation indicating the number of times a quantity is multiplied by itself
base
is the number that is repeatedly multiplied in a power
equation
a mathematical sentence with an equal sign that shows that two expressions are equivalent
solving an equation
finding all the solutions of an equation
perimeter
The sum of the lengths of the sides of a polygon
area
the number of square units needed to cover a flat surface
integers
Positive and Negative Whole Numbers and 0. Examples: .... -3, -2, -1, 0, 1, 2, 3.....(Does not include fractions or decimals)
additive identity property
The sum of a number and zero is always that number.
additive inverse property
The sum of a number and its opposite is zero.
multiplication identity property
The product of any number and one is that number.
commutative property of addition
In a sum, you can add terms in any order, a + b = b + a
commutative property of multiplication
the order of the factors does not change the product a x b = b x a
associative property of addition
changing the grouping of terms will not change the sum, (a + b) + c = a + (b + c)
associative property of multiplication
changing the grouping of factors will not change the product, (ab)c = a(bc)
distributive property
a(b + c) = ab + ac
an + ac = a(b+ c)
terms
in an expression are separated by addition and subtraction signs
like terms
terms that have identical variable parts raised to the same power
coefficient
number in front of a variable
constant
a term that has no variable and does not change
coordinate plane
A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis.
aka "the Cartesian plane" after René Descartes
ordered pair
A pair of numbers, (x, y), that indicate the position of a point on a coordinate plane.
quadrant
one of four sections into which the coordinate plane is divided
Inverse Operation
operations that undo each other, such as addition and subtraction
Solve for a variable
To get a variable alone on one side of an equation or inequality in order to solve the equation or inequality
Equivalent Equations
equations that have the same solution
Inequality
A statement that compares two quantities using <, >, ≤,≥, or ≠
reciprocal
one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
rational number
A number that can be written as a/b where a and b are integers, but b is not equal to 0.
ratio
a comparison of two numbers by division
rate
a ratio that compares two quantities measured in different units
irrational number
a number that can not be written a/b
input
the x-value in a function
output
the y-value in a function
function
a relation that assigns exactly one output value for each input value
domain
the set of all the input (x-values) for a function
range
the set of all the output (y-values) for a function
linear equation
an equation whose answers are ordered pairs (x,y) and whose answers graph a straight line
x-intercept
the point where a graph crosses the x-axis
y-intercept
the point where a graph crosses the y-axis
slope
the steepness of a line on a graph, rise over run
Slope Intercept Form
y= mx + b
where "m=slope" and "b=y-intercept"
Slope Formula
Slope=m= Y2 - Y1 / X2 - X1
numerical expression
consists of numbers and operations
evaluate
to find the value of an expression
order of operations
the order in which operations in an expression to be evaluated are carried out. 1. parentheses 2. exponets 3. multiplication and divison 4. addition and subtraction
variable
a letter used to represent one or more numbers
variable expression
consists of numbers, variables, and operations
power
is a number made of repeated factors
exponent
a mathematical notation indicating the number of times a quantity is multiplied by itself
base
is the number that is repeatedly multiplied in a power
equation
a mathematical sentence with an equal sign that shows that two expressions are equivalent
solving an equation
finding all the solutions of an equation
perimeter
The sum of the lengths of the sides of a polygon
area
the number of square units needed to cover a flat surface
integers
Positive and Negative Whole Numbers and 0. Examples: .... -3, -2, -1, 0, 1, 2, 3.....(Does not include fractions or decimals)
additive identity property
The sum of a number and zero is always that number.
additive inverse property
The sum of a number and its opposite is zero.
multiplication identity property
The product of any number and one is that number.
commutative property of addition
In a sum, you can add terms in any order, a + b = b + a
commutative property of multiplication
the order of the factors does not change the product a x b = b x a
associative property of addition
changing the grouping of terms will not change the sum, (a + b) + c = a + (b + c)
associative property of multiplication
changing the grouping of factors will not change the product, (ab)c = a(bc)
distributive property
a(b + c) = ab + ac
an + ac = a(b+ c)
terms
in an expression are separated by addition and subtraction signs
like terms
terms that have identical variable parts raised to the same power
coefficient
number in front of a variable
constant
a term that has no variable and does not change
coordinate plane
A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis.
aka "the Cartesian plane" after René Descartes
ordered pair
A pair of numbers, (x, y), that indicate the position of a point on a coordinate plane.
quadrant
one of four sections into which the coordinate plane is divided
Inverse Operation
operations that undo each other, such as addition and subtraction
Solve for a variable
To get a variable alone on one side of an equation or inequality in order to solve the equation or inequality
Equivalent Equations
equations that have the same solution
Inequality
A statement that compares two quantities using <, >, ≤,≥, or ≠
reciprocal
one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
rational number
A number that can be written as a/b where a and b are integers, but b is not equal to 0.
ratio
a comparison of two numbers by division
rate
a ratio that compares two quantities measured in different units
irrational number
a number that can not be written a/b
input
the x-value in a function
output
the y-value in a function
function
a relation that assigns exactly one output value for each input value
domain
the set of all the input (x-values) for a function
range
the set of all the output (y-values) for a function
linear equation
an equation whose answers are ordered pairs (x,y) and whose answers graph a straight line
x-intercept
the point where a graph crosses the x-axis
y-intercept
the point where a graph crosses the y-axis
slope
the steepness of a line on a graph, rise over run
Slope Intercept Form
y= mx + b
where "m=slope" and "b=y-intercept"
Slope Formula
Slope=m= Y2 - Y1 / X2 - X1
y = x²
y = x² − 3
y = x² + 3
y = x² + 10
y = x² − 4
y = x² −25
y = x² − 9
y = (x+2)²
y = (x+1)²
y = (x − 3)²
y = (x+5)²
y = (x−2)²
y = −x²
y = −5x²
y = − (1/5)x²
y = (x−1)² +3
y = (x−2)² +5
y = (x+3)² −5
y = ±√x
y = ±√x + 3
y = ±√x − 5
y = ±√(x+3)
y = ± √(x − 5)
y = x² + 3x
y = x² −3x
y = x² −5x
y = −x² −2x
y = ±√−x
y = ±√−x + 2
y = ±√(−x+3)
b = 4
This graph does NOT represent ax² -1/4, where a would be a fraction less than 1.
This graph is of the form (x⁴ - a)/(x² + b)
Find the integer, b.
HINT
Type "b= ..."
c) (x⁴−9)/(x²+1)
Which function is shown in blue?
a) (x⁴−9)/(x²+9)
b) (x⁴−9)/(x²+2)
c) (x⁴−9)/(x²+1)
numerical expression
consists of numbers and operations
evaluate
to find the value of an expression
order of operations
the order in which operations in an expression to be evaluated are carried out. 1. parentheses 2. exponets 3. multiplication and divison 4. addition and subtraction
variable
a letter used to represent one or more numbers
variable expression
consists of numbers, variables, and operations
power
is a number made of repeated factors
exponent
a mathematical notation indicating the number of times a quantity is multiplied by itself
base
is the number that is repeatedly multiplied in a power
equation
a mathematical sentence with an equal sign that shows that two expressions are equivalent
solving an equation
finding all the solutions of an equation
formula
An equation that shows a relationship among two or more quantities
perimeter
The sum of the lengths of the sides of a polygon
area
the number of square units needed to cover a flat surface
integers
Positive and Negative Whole Numbers and 0. Examples: .... -3, -2, -1, 0, 1, 2, 3.....(Does not include fractions or decimals)
absolute value
the distance a number is from 0 on the number line, value of n is written l n l
additive identity property
The sum of a number and zero is always that number.
additive inverse property
The sum of a number and its opposite is zero.
multiplication identity property
The product of any number and one is that number.
mean
the sum of the values in a data set divided by the number of values in the set
commutative property of addition
In a sum, you can add terms in any order, a + b = b + a
commutative property of multiplication
the order of the factors does not change the product a x b = b x a
associative property of addition
changing the grouping of terms will not change the sum, (a + b) + c = a + (b + c)
associative property of multiplication
changing the grouping of factors will not change the product, (ab)c = a(bc)
distributive property
a(b + c) = ab + ac
an + ac = a(b+ c)
terms
in an expression are separated by addition and subtraction signs
like terms
terms that have identical variable parts raised to the same power
coefficient
number in front of a variable
constant
a term that has no variable and does not change
coordinate plane
A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis.
aka "the Cartesian plane" after René Descartes
ordered pair
A pair of numbers, (x, y), that indicate the position of a point on a coordinate plane.
quadrant
one of four sections into which the coordinate plane is divided
Inverse Operation
operations that undo each other, such as addition and subtraction
Isolate the variable
To get a variable alone on one side of an equation or inequality in order to solve the equation or inequality
Equivalent Equations
equations that have the same solution
base
the bottom of a triangle
height
the perpendicular (90 degrees) distance from the base of a triangle to the opposite vertex
Inequality
A statement that compares two quantities using <, >, ≤,≥, or ≠
Equivalent Inequalities
inequalities that have the same solution
prime number
A whole number that has exactly two factors, 1 and itself.
composite
a number that has more than two factors
monomial
a single term made up of numbers and variables
prime factorization
a number written as the product of its prime factors
greatest common factor
The largest factor that two or more numbers have in common.
simplest form
when the GCF of the numerator and denominator is 1
equivalent fractions
fractions that have the same value and the same simplest form
multiple
skip-counting by any given number Ex: 4, 8, 12, 16...
least common multiple
the smallest multiple that two or more numbers have in common
least common denominator
The least common multiple of the denominators of two or more fractions.
scientific notation
a method of writing very large or very small numbers by using powers of 10
reciprocal
one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
rational number
A number that can be written as a/b where a and b are integers, but b is not equal to 0.
repeating decimal
a decimal in which one or more digits repeat infinitely
median
is the middle value in a data set where the numbers are ordered least to greatest
mode
is the value in a data set that occurs most often. If all values occur the same amount of times there is no mode for that set.
range
is the difference of the greatest value and the least value in a set of data.
circle
is the set of all points that are an equal distance from a point called the center
center
the point in the exact middle of a circle
radius
a line segment from the center of a circle to any point on the circle (is also half the diameter)
diameter
the distance across a circle through its center (is also twice the radius)
circumference
The distance around a circle.
ratio
a comparison of two numbers by division
rate
a ratio that compares two quantities measured in different units
unit rate
a rate that has a denominator of 1
equivalent ratios
Ratios that have the same value.
proportion
an equation that states that two ratios are equal
Ex: 1/2 = x/10
scale model
a model of an object in which the dimensions are in proportion to the actual dimensions of the object.
percent
a ratio whose denominator is 100
interest
is the amount paid for borrowing or lending money
principal
the original amount of money loaned or borrowed
annual interest rate
apr, the percent of the principal you pay or earn a year
probability of an event
number of favorable outcomes divided by total number of possible outcomes
theoretical probability
what should occur in a probability experiment...an experiment is not actually done
experimental probability
probability based on what happens when an experiment is actually done
opposite
the number that is on the other side of 0 and is exactly the same distance away from 0
radical expression
an expression that contains a square root
perfect square
a number that has a whole number for a square root ex: 9, 16, 121
irrational number
a number that can not be written a/b
Pythagorean Theorem
a² + b² = c²
Pythagorean triple
a set of three positive integers that work in the pythagorean theorem
Pi
3.1415.... is an irrational number resulting from the ratio of a circle's circumference to its diameter
cube
special polyhedron with faces that are all squares
surface area
The total area of the 2-dimensional surfaces that make up a 3-dimensional object.
volume
the amount of 3-dimensional space occupied by an object
input
the x-value in a function
output
the y-value in a function
function
a relation that assigns exactly one output value for each input value
domain
the set of all the input (x-values) for a function
range
the set of all the output (y-values) for a function
linear equation
an equation whose answers are ordered pairs (x,y) and whose answers graph a straight line
x-intercept
the point where a graph crosses the x-axis
y-intercept
the point where a graph crosses the y-axis
slope
the steepness of a line on a graph, rise over run
slope intercept form
y = mx + b
tree diagram (probability)
a diagram used to show the total number of possible outcomes in an experiment
odds
the ratio of the number of ways the event can occur to the number of ways the event cannot occur. Favorable over Unfavorable.
independent event (probability)
an event that is not affected by another event.
dependent event (probability)
an event who's outcome does depend on the outcome of a previous event
fundamental counting principle
...
combinations
...
permutations
...
In the equation, y = mx + b, the m stands for _____________
Slope
In the equation, y = mx + b, the b stands for _____________
y-intercept
In the equation y = 3x + 7, the slope is _______________
3
In the equation y = 3x + 7, the y intercept is _______________
7
In the equation y = −2x − 6, the slope is ________________
−2
In the equation, y = −2x − 6, the y intercept is _____________
−6
Look at the graph for problem #12 on page 218. What is the equation in slope intercept form?
y = −1/5 x + 1
Look at the graph for problem #14 on page 218. What is the equation in slope intercept form?
y = −2 x + 3
Look at the graph for problem #34 on page 219. What is the equation in slope intercept form?
y = −4/7 x − 2
Given the point (1, 9) and the slope 4, write the equation in slope intercept form
y = 4x + 5
Given the point (1, 3) and (−3, −5), write the equation in slope intercept form
y = 2x + 1
Write the equation in slope intercept form: 2x + 4y = 12
y = −1/2x + 3
Look at the graph for problem #40 on page 235. Write the equation in point slope form
y − 3 = 4(x − 1)
Look at the graph for problem #42 on page 235. Write the equation in point slope form
y − 7 = −4/3(x + 3)
Given the point (2, 2) and m = −3, write the equation in point slope form
y − 2 = −3(x − 2)
Given the point (−8, 5) and m = −2/5, write the equation in point slope form
y − 5 = −2/5(x + 8)
The point slope form of a linear equation is ________
y - y1 = m(x − x1)
The slope intercept form of a linear equation is ____
y = mx + b
Write the equation in standard form: y − 11 = 3(x − 2)
3x − y = −5
Write the equation in standard form: y − 10 = −(x − 2)
x + y = 12
Write the equation in slope intercept form:. . . y + 2 = 4(x + 2)
y = 4x + 6
Determine whether the lines are parallel, perpendicular, or neither: . . . . . . . . . . . . . . . . . y = −2x and 2x + y = 3
Parallel
Determine whether the lines are parallel, perpendicular, or neither: . . . . . . . . . . . . . . . . . 3x + 5y = 10 and 5x − 3y = −6
Perpendicular
Determine whether the lines are parallel, perpendicular, or neither: . . . . . . . . . . . . . . . . . 2x + 5y = 15 and 3x + 5y = 15
Neither
What form should you put lines in to determine if they are parallel, perpendicular, or neither?
Slope intercept form
What is true about the slopes of parallel lines?
The slopes of parallel lines are the same
What is true about the slopes of perpendicular lines?
The product of slopes of perpendicular lines is −1. (Slopes of perpendicular lines are opposite reciprocals)
Write the equations for two lines that are parallel.
There are many answers. An example would by y = 2x and y = 2x + 7.
Write the equations for two lines that are perpendicular.
There are many answers. An example would by y = 2x and y = −1/2 x + 6. The product of the two slopes has to be −1
What is true about any horizontal line and a vertical line?
They are perpendicular
Write an equation in slope intercept form for the line that passes through (−2, 2) and is perpendicular to y = −1/3 x + 9.
y = 3x + 8
Write an equation in slope intercept form for the line that passes through (3, 2) and is parallel r to y = x + 5.
y = x − 1
Write an equation in slope intercept form for the line that passes through (10, 5) and is perpendicular to 5x + 4y = 8.
y = 4/5x − 3
Write an equation in slope intercept form for the line that passes through (−1, −2) and is parallel to 3x − y = 5.
y = 3x + 1
Describe how you would graph y = 1/3 x + 2 using the slope and the y interept.
Put a dot on the y-axis on the number 2. Count a slope of 1/3 by going up 1 and right 3 or down 1 and left 3. Count the slope two or three times and then draw the line.
Title
Algebra
Structure and Method
Book 1
Chapter
1
Introduction to Algebra
Chapter 1
Section 1
Variables and Equations
Lesson
1-1
Variables
Objective
1-1
To simplify numerical expressions and
evaluate algebraic expressions.
variable
A symbol
used to represent
one or more numbers.
values
of
a
variable
The numbers
that can be represented by
the variable.
variable
expression
An expression
that contains
a variable.
numerical
expression
An expression
that names
a particular number;
a
numeral.
numeral
An expression
that names
a particular number;
a
numerical expression.
value
of
a
numerical
expression
The number
named by
the expression.
simplifying
a
numerical
expression
Replacing the expression by
the simplest name
for its value.
substitution
principle
An expression
may be replaced by
another expression
that has the same value.
evaluating
a
variable
expression
Replacing each variable
in the expression by
a given value
and simplifying the result.
Lesson
1-2
Grouping Symbols
Objective
1-2
To simplify expressions with
and without grouping symbols.
grouping symbol
A device
used to enclose an expression
that should be simplified
before other operations are performed.
Examples:
parentheses, brackets, fraction bar.
When there are no grouping symbols,
simplify in the following order:
1. Do all multiplications and divisions
in order from left to right.
2. Do all additions and subtractions
in order from left to right.
Lesson
1-3
Equations
Objective
1-3
To find solution sets
of equations
over a given domain.
equation
A statement
formed by
placing an equals sign
between
two numerical or variable expressions.
sides
of an equation
The two expressions
joined by
the equals sign.
open sentence
A sentence
containing
one or more variables.
domain
of a variable
The given set
of numbers
that the variable
may represent.
solution
of a sentence
Any value
of a variable
that turns an open sentence
into a true statement.
satisfy
an open sentence
Any solution
of the sentence
satisfies
the sentence.
solution set
of an open sentence
The set
of all solutions
of the sentence.
solve
an open sentence
To find
the solution set
of the sentence.
Chapter 1
Section 2
Applications and Problem Solving
Lesson
1-4
Translating Words into Symbols
Objective
1-4
To translate phrases into variable expressions.
formula
An equation
that states a rule
about a relationship.
A = lw
Area of rectangle
= length of rectangle × width of rectangle
P = 2l + 2w
Perimeter of rectangle
= ( 2 × length ) + ( 2 × width )
D = rt
Distance traveled
= rate × time traveled
C = np
Cost
= number of items × price per item
area
The area of a region is
the number of square units
it contains.
perimeter
The perimeter of a plane figure is
the distance around it.
Lesson
1-5
Translating Sentences into Equations
Objective
1-5
To translate word sentences
into equations.
Lesson
1-6
Translating Problems into Equations
Objective
1-6
To translate simple word problems
into equations.
Step 1
Read the problem carefully.
Step 2
Choose a variable and
represent the unknows.
Step 3
Reread the problem and
write an equation.
Lesson
1-7
A Problem Solving Plan
Objective
1-7
To use
the five-step plan
to solve word problems
over a given domain.
Plan for Solving a Word Problem
Step 1
Read the problem carefully.
Decide what unknown numbers are asked for
and
what facts are known.
Making a sketch may help.
Plan for Solving a Word Problem
Step 2
Choose a variable and
use it with the given facts to
represent the unknowns
described in the problem.
Plan for Solving a Word Problem
Step 3
Reread the problem and
write an equation
that represents
relationships among the numbers
in the problem.
Plan for Solving a Word Problem
Step 4
Solve the equation and
find the unknowns
asked for.
Plan for Solving a Word Problem
Step 5
Check your results
with the words
of the problem.
Give the answer.
Chapter 1
Section 3
Numbers on a Line
Lesson
1-8
Number Lines
Objective
1-8
To graph real numbers on a number line and to compare real numbers.
origin
The zero point
on a number line.
The intersection
of the axes
on a coordinate plane.
positive side
On a horizontal number line,
the side to the right
of the origin.
negative side
On a horizontal number line,
the side to the left
of the origin.
Zero is niether
positive nor negative.
positive integers
The numbers
1, 2, 3, 4, and so on.
negative integers
The numbers
−1, −2, −3, −4, and so on.
integers
The set consisting of
the positive integers,
the negative integers,
and zero.
whole numbers
The set consisting of
zero and
all the positive integers.
positive number
A number paired with
a point on the positive side
of a number line.
negative number
A number paired with
a point on the negative side
of a number line.
graph
of a number
The point
on a number line
that is paired with
the number.
coordinate
of a point
The number
paired with
that point
on a number line.
real number
Any number
that is either
positive, negative, or zero.
inequality symbols
Symbols used to show
the order
of two real numbers.
The symbol ≠
means "is not equal to."
Lesson
1-9
Opposites and Absolute Values
Objective
1-9
To use opposites and absolute values.
opposite
of a number
Each
of the numbers
in a pair
such as 6 and −6 or
−2.5 and 2.5.
Also called additive inverse.
1. If a is positive,
then −a is negative.
2. If a is negative,
then −a is positive.
3. If a = 0,
then −a = 0.
4. The opposite of −a is
a; that is, −(−a) = a.
absolute value
The positive number
of any pair
of opposite nonzero real numbers
is the absolute value
of each number in the pair.
The absolute value of 0 is 0.
The absolute value of a number a
is denoted by |a|.
1. If a is positive,
|a| = a.
2. If a is negative,
|a| = −a.
3. If a is zero,
|a| = 0.
axis
a line about which a three-dimensional body or figure is symmetrical.
base
in an expression of the form x m the base is x
bisect
to cut or divide into two equal parts: to bisect an angle.
bisector
a point, ray, line, line segment, or plane that intersects the segment at its midpoint
centimeter
a metric unit of length equal to one hundredth of a meter
congruent
Having the same size and shape
constant
a number representing a quantity assumed to have a fixed value in a specified mathematical context
coordinate
a number that identifies a position relative to an axis
cubes
a number that is a whole number raised to the third power Ex. 8, 27, 64, 125, etc.
cylinders
a three dimensional figure with two parallell, congruent circular basis connected by a curved lateral surface
dependent
the variable in the relation whos value depends on the value of the independent variable
diagram
A visual representation of data to help readers better understand relationships among data
diameter
the length of a straight line passing through the center of a circle and connecting two points on the circumference
dilation
A transformation that changes the size of an object, but not the shape.
dimensions
The length, width, and height of an object being measured.
domain
the set of values of the independent variable for which a function is defined
end point
the point in a titration at which a marked color change takes place
estimate
an approximate calculation of quantity or degree or worth
expression
a group of symbols that make a mathematical statement
formula
mathematics a standard procedure for solving a class of mathematical problems
function
a mathematical relation such that each element of one set is associated with at least one element of another set
graph
a drawing illustrating the relations between certain quantities plotted with reference to a set of axes
grid
a network of horizontal and vertical lines that provide coordinates for locating points on an image
height
the vertical dimension of extension
hexagon
a six-sided polygon
increase
the amount by which something increases
decrease
the amount by which something decreases
intercept
the point at which a line intersects a coordinate axis
intersection
the act of intersecting as joining by causing your path to intersect your target's path
interval
a set containing all points or all real numbers between two given endpoints
inverse
opposite in nature or effect or relation to another quantity
isosceles triangle
a triangle with at least two congruent sides
length
the linear extent in space from one end to the other
level
having a horizontal surface in which no part is higher or lower than another
linear
designating or involving an equation whose terms are of the first degree
matrix
a rectangular array of elements or entries set out by rows and columns
maximum
the point on a curve where the tangent changes from positive on the left to negative on the right
minimum
the point on a curve where the tangent changes from negative on the left to positive on the right
meter
any of various measuring instruments for measuring a quantity
milligrams
used to measure the mass of very small objects one thousandths
modeled
resembling sculpture
narrowest
the field of vision is a cone-shaped area with its narrowest/widest end near the driver
non-collinear
points that do not lie on the same line
ordered pair
A pair of numbers, x, y, that indicate the position of a point on a Cartesian plane.
parallelogram
a quadrilateral whose opposite sides are both parallel and equal in length
pendulum
a weight, hanging from a point, that swings an equal distance from side to side; often used in clocks
perimeter
the size of something as given by the distance around it
perpendicular
intersecting at or forming right angles
plane
an unbounded two-dimensional shape
point
a geometric element that has position but no extension
polygon
closed plane figure having, literally, many angles and therefore many sides
pyramid
a polyhedron having a polygonal base and triangular sides with a common vertex
quadratic equations
a function that has the variable raised to the second power
quadrilaterals
4 sided polygon
quantity
something that has a magnitude and can be represented in mathematical expressions by a constant or a variable
range
the difference between the highest and lowest scores in a distribution
rectangle
a parallelogram with four right angles
reduced
made less in size or amount or degree
reflections
flips a figure over a line
relations
a set of ordered pairs
rotation
a single complete turn axial or orbital
scale factor
The ratio of the lengths of two corresponding sides of two similar polygons
secant
ratio of the hypotenuse to the adjacent side of a right-angled triangle
segments
set of points on a line that consist of two points called endpoints, and all the points between them
semi circle
half of a circle
slope
the steepness of a line on a graph, equal to its vertical change divided by its horizontal change
solutions
ways to solve problems
squares
all sides are equal, opposite sides are parallel, all angles are 90 degrees
symmetry
an attribute of a shape or relation
tables
Used to arrange text in columns and rows
tax rate
the amount of tax people are required to pay per unit of whatever is being taxed
three dimensional
Having the dimensions of height, width, and depth.
translation
the act of changing in form or shape or appearance
value
a numerical quantity measured or assigned or computed
volume
the amount of 3-dimensional space occupied by an object
widest
The widest/narrowest part of the nuchal translucency should be measured.
Absolute Value
The distance a number is from the 0 on the number line
Algebraic Expression
An expression that contains numbers, operations and variables.
Associative Property
A property that states that numbers in addition of multiplication expressions can be grouped without affecting the value of the expression.
Axes
A horixontal and vertical number line on a coordinate plane.
Base of a Power
The repeated factor in a power.
Coefficient
The number mulltiplied by a variable in a term.
Commutative Property
A property that states numbers can be added or multiplied in any order.
Constant
A term that has no variable.
Unit Conversion
The process of renaming a measurement using different units.
Coordinate Plane
A coordinate system formed by the intersection of a horizontal number line, called the x-axis, and a vertical number line, called the y-axis.
Decimal
a number with one or more digits to the right of the decimal point
Distributive Property
a(b+c)=ab+ac, Multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.
Dividend
a number to be divided by another number
Divisor
the number by which a dividend is divided.
Elimination Method
to solve a system of equations by adding equations to get rid of one variable. sometimes needing to multiply one equation by a # to make terms opposites
Equation
a mathematical statement in which two expressions are equal
Equivalent expressions
algebraic expressions that have the same values for all values of variables
Equivalent fractions
fractions that have the same value
Evaluate
to find the value of a numerical or algebraic EXPRESSION.
Exponent
The number of times a factor in repeated in a power.
Factors
Whole numbers that can be multiplied together to hind a product.
Formula
An algebraic equation that shows the relationship among specific quantities.
Function
A pairing of input and output values according to a specific rule.
Greatest Common Factor
The greatest factor that is common to two or more numbers.
Grouping Symbols
parentheses ( ), brackets [ ], and braces { } that group parts of an expression.
Improper Fraction
A fraction whose numerator is greater than or equal to its denominator.
Integers
The set of all whole numbers, their opposites, and 0.
Inverse Operations
Operations that undo each other.
Irrational numbers
numbers that cannot be expressed in the form a/b, where a and b are integers and b =0.
Least Common Denominator
The least common multiple of the denominators of two or more fractions.
What will the solution of this absolute value equation look like |exp.| > a ?
exp. < -a or exp. > a
What will the solution of this absolute value equation look like |exp.| = a ?
exp = a or exp = -a
What will the solution of this absolute value equation look like |exp.| < a ?
-a < exp < a
What is the first step in this equation : 2|2x+1| -4 = 16 ?
Get rid of the variables outside of the absolute value.
What do you do when you have a negative denominator? ex b-2/-5
Move it up top. answer: 2-b/5 (Make sure to change signs)
How would you describe this inequality in a solution set: x < 5 ?
{x|x<5}
How would you describe this inequality in interval notation: x ≥ 2 ?
[ 2 , ∞ )
How would you express this inequality in a solution set: -4 < t ≤ 5/3 ?
{t| -4 < t ≤ 5/3}
Describe this inequality in interval notation: x < 5/3 ?
( -∞ , 5/3)
What do you do when you divide a negative number in an inequality?
Switch the signs
For what inequality symbols do you use brackets?
≤ and ≥
For what inequality symbols do you use parenthesis?
< and >
What is the solution for |x - 5| < 0?
∅ The absolute value of a number can never be < 0. No solution.
What is the solution for |x - 5| ≤ 0?
{5} Absolute value of a number can never be < 0 but it can be 0 when x =5. |5 - 5| = 0
What is the solution for |x - 5| ≥ 0?
R. Any real number, because the absolute value of a number is always ≥ 0.
How would you word the function (front) part of this equation? P(x) = 7.25x
P is a function of x
Is this a linear equation or a linear function: y = 2x - 1 ?
A linear equation.
Is this a linear equation or a linear function: f(x) = 2x - 1 ?
A linear function
What is the standard form for a linear equation?
ax + by = c Both x and y are on the right side of the equation.
How do you find the y-intercept in a linear equation? ax + by = c
Set x = 0 to isolate the y.
How do you find the x-intercept in a linear equation? ax + by = c
Set y = 0 to isolate the x.
The graph of y = b is?
A horizontal line through b on the y axis. (Slope = 0)
The graph of x = b is?
A vertical line through b on the x axis. (Slope undefined)
How is slope defined?
The slope of a line measures the tilt of the line.
How does slope tell us the tilt? y/x
Because y/x = rise/run
How do we view lines on the graph to get the correct slope?
Left to right
How is a positive line viewed?
It runs upwards. m>0
How is a negative line viewed?
It runs downwards. m<0
If m=0 what type of line would be viewed on the graph?
A horizontal line.
How do we calculate slope from two given points? (x¹,y¹) (x²,y²)
y² - y¹/x² - x¹ to get the slope. (rise/run)
What is the formula for slope-intercept form?
y = mx + b
What is the point slope equation? Used when you have a slope and a random point.
y - y¹ = m(x - x¹)
How do you find the equation of a line if you are given two points, but no y-intercept or slope? Such as: (4 , 3) (6 , -2)
You would have to find the slope first, x² - x¹/y² - y¹, and then use the point slope equation, y - y¹ = m(x - x¹)
What do parallel lines have in common?
Equal slopes. m¹ = m²
What is the difference in slopes for perpendicular lines?
They have negative reciprocals. ex. Line¹: m= 2/4, would mean Line²: m= -4/2
What is the equation of the line through (0 , 2) that is perpendicular to the graph of the line: y = 1/2x - 4 ? (Think of the perpendicular rule)
y = -2/1x + 2 because (0 , 2) gives us the y intercept and -2/1 is the reciprocal of the line in the question.
When graphing linear inequalities, for what symbols would you use a solid line in the graph?
≤ and ≥
When graphing linear inequalities, for what symbols would you use a dotted line in the graph?
< and >
What is the first step in graphing a linear inequality? y ≥ 1/2x - 3
Decide what type of line it will be. In this case it's a solid line because of the ≥ sign.
How do you graph the line of a linear inequality? (Also the 2nd step) y ≥ 1/2x - 3
Set the inequality as an equation y ≥ 1/2x - 3 turns into y = 1/2x - 3, and plot the line.
How should you decide what side will be shaded in while graphing a linear inequality? y ≥ 1/2x - 3
Test it by putting in a point, typically (0 , 0) and see if it's true. 0 ≥ 1/2(0) -3 is 0 ≥ 3, which means true. Shade on the side of the point. If false shade on opposite side.
What three methods could be used to solve systems of linear equations? Equation¹: 2x + y = 6 Equation²: 3x - 2y = 16
By graphing, the substitution method, and the addition method.
These two equations would be easiest to solve by what method Equation¹: y = -2x + 6 Equation²: 3x - 2y = 16?
The substitution method, substitue first equation in place of y in the second: 3x - 2(-2x + 6) = 16 and solve.
These two equations would be easiest to solve by what method Equation¹: 4x + 2y = 6 Equation²: 3x - 2y = 16?
The addition method. If set up as if adding them together the y's would cancel. Then you could solve for x and then solve for y.
A linear system where two lines cross (one answer) would be what type of system?
A consistent system.
A linear system where two lines are parallel (no answer) would be what type of system?
An inconsistent system.
A linear system where two lines are directly on top of each other (infinite answers) would be what type of system?
A dependent system.
When solving a linear system of equations, if the substitution or addition method resulted in 2 = 2, what would you have?
A dependent system, two lines on top of each other. ∞ Solutions
When solving a linear system of equations, if the substitution or addition method resulted in 2 ≠ 4, what system would you have?
An inconsistent system, parallel lines. No solution.
Title
Algebra
Structure and Method
Book 1
Chapter
1
Introduction to Algebra
Chapter 1
Section 1
Variables and Equations
Lesson
1-1
Variables
Objective
1-1
To simplyfy numerical expressions and
evaluate algebraic expressions.
variable
A symbol
used to represent
one or more numbers.
values
of
a
variable
The numbers
that can be represented by
the variable.
variable
expression
An expression
that contains
a variable.
numerical
expression
An expression
that names
a particular number;
a
numeral.
numeral
An expression
that names
a particular number;
a
numerical expression.
value
of
a
numerical
expression
The number
named by
the expression.
simplifying
a
numerical
expression
Replacing the expression by
the simplest name
for its value.
substitution
principle
An expression
may be replaced by
another expression
that has the same value.
evaluating
a
variable
expression
Replacing each variable
in the expression by
a given value
and simplifying the result.
Lesson
1-2
Grouping Symbols
Objective
1-2
To simplify expressions with
and without grouping symbols.
grouping symbol
A device
used to enclose an expression
that should be simplified
before other operations are performed.
Examples:
parentheses, brackets, fraction bar.
When there are no grouping symbols,
simplify in the following order:
1. Do all multiplications and divisions
in order from left to right.
2. Do all additions and subtractions
in order from left to right.
Lesson
1-3
Equations
Objective
1-3
To find solution sets
of equations
over a given domain.
equation
A statement
formed by
placing an equals sign
between
two numerical or variable expressions.
sides
of an equation
The two expressions
joined by
the equals sign.
open sentence
A sentence
containing
one or more variables.
domain
of a variable
The given set
of numbers
that the variable
may represent.
solution
of a sentence
Any value
of a variable
that turns an open sentence
into a true statement.
satisfy
an open sentence
Any solution
of the sentence
satisfies
the sentence.
solution set
of an open sentence
The set
of all solutions
of the sentence.
solve
an open sentence
To find
the solution set
of the sentence.
Chapter 1
Section 2
Applications and Problem Solving
Lesson
1-4
Translating Words into Symbols
Objective
1-4
To translate phrases into variable expressions.
formula
An equation
that states a rule
about a relationship.
A = lw
Area of rectangle
= length of rectangle × width of rectangle
P = 2l + 2w
Perimeter of rectangle
= ( 2 × length ) + ( 2 × width )
D = rt
Distance traveled
= rate × time traveled
C = np
Cost
= number of items × price per item
area
The area of a region is
the number of square units
it contains.
perimeter
The perimeter of a plane figure is
the distance around it.
Lesson
1-5
Translating Sentences into Equations
Objective
1-5
To translate word sentences
into equations.
Lesson
1-6
Translating Problems into Equations
Objective
1-6
To translate simple word problems
into equations.
Step 1
Read the problem carefully.
Step 2
Choose a variable and
represent the unknows.
Step 3
Reread the problem and
write an equation.
Lesson
1-7
A Problem Solving Plan
Objective
1-7
To use
the five-step plan
to solve word problems
over a given domain.
Plan for Solving a Word Problem
Step 1
Read the problem carefully.
Decide what unknown numbers are asked for
and
what facts are known.
Making a sketch may help.
Plan for Solving a Word Problem
Step 2
Choose a variable and
use it with the given facts to
represent the unknowns
described in the problem.
Plan for Solving a Word Problem
Step 3
Reread the problem and
write an equation
that represents
relationships among the numbers
in the problem.
Plan for Solving a Word Problem
Step 4
Solve the equation and
find the unknowns
asked for.
Plan for Solving a Word Problem
Step 5
Check your results
with the words
of the problem.
Give the answer.
Chapter 1
Section 3
Numbers on a Line
Lesson
1-8
Number Lines
Objective
1-8
To graph real numbers on a number line and to compare real numbers.
origin
The zero point
on a number line.
The intersection
of the axes
on a coordinate plane.
positive side
On a horizontal number line,
the side to the right
of the origin.
negative side
On a horizontal number line,
the side to the left
of the origin.
Zero is niether
positive nor negative.
positive integers
The numbers
1, 2, 3, 4, and so on.
negative integers
The numbers
−1, −2, −3, −4, and so on.
integers
The set consisting of
the positive integers,
the negative integers,
and zero.
whole numbers
The set consisting of
zero and
all the positive integers.
positive number
A number paired with
a point on the positive side
of a number line.
negative number
A number paired with
a point on the negative side
of a number line.
graph
of a number
The point
on a number line
that is paired with
the number.
coordinate
of a point
The number
paired with
that point
on a number line.
real number
Any number
that is either
positive, negative, or zero.
inequality symbols
Symbols used to show
the order
of two real numbers.
The symbol ≠
means "is not equal to."
Lesson
1-9
Opposites and Absolute Values
Objective
1-9
To use opposites and absolute values.
opposite
of a number
Each
of the numbers
in a pair
such as 6 and −6 or
−2.5 and 2.5.
Also called additive inverse.
1. If a is positive,
then −a is negative.
2. If a is negative,
then −a is positive.
3. If a = 0,
then −a = 0.
4. The opposite of −a is
a; that is, −(−a) = a.
absolute value
The positive number
of any pair
of opposite nonzero real numbers
is the absolute value
of each number in the pair.
The absolute value of 0 is 0.
The absolute value of a number a
is denoted by |a|.
1. If a is positive,
|a| = a.
2. If a is negative,
|a| = −a.
3. If a is zero,
|a| = 0.
y = x²
y = x² − 3
y = x² + 3
y = x² + 10
y = x² − 4
y = x² −25
y = x² − 9
y = (x+2)²
y = (x+1)²
y = (x − 3)²
y = (x+5)²
y = (x−2)²
y = −x²
y = −5x²
y = − (1/5)x²
y = (x−1)² +3
y = (x−2)² +5
y = (x+3)² −5
y = ±√x
y = ±√x + 3
y = ±√x − 5
y = ±√(x+3)
y = ± √(x − 5)
y = x² + 3x
y = x² −3x
y = x² −5x
y = −x² −2x
y = ±√−x
y = ±√−x + 2
y = ±√(−x+3)
b = 4
This graph does NOT represent ax² -1/4, where a would be a fraction less than 1.
This graph is of the form (x⁴ - a)/(x² + b)
Find the integer, b.
HINT
Type "b= ..."
c) (x⁴−9)/(x²+1)
Which function is shown in blue?
a) (x⁴−9)/(x²+9)
b) (x⁴−9)/(x²+2)
c) (x⁴−9)/(x²+1)
What is (3.2 x 10^5)(1.4 x 10^-2) written in
scientific notation?
4.48 x 10³
What is the end behavior for f(x) = -2x³ + x² - 2?
Left side up, right side down.
What is the end behavior for f(x) = 3x⁴ - x² +1 ?
Both sides up.
What is the complete factorization of 3x⁴ - 3x² ?
3x²(x-1)(x+1)
If x + 3 is a factor of x³ − x² − 17x − 15,
what are the other factors?
(x + 1) and (x - 5)
When multiplying two terms with the same base, you should ____ the exponents.
add
When dividing two terms with the same base, you should _____ the exponents.
subtract
What is the degree of the polynomial h(t) = -8t² + 5 - 3t³?
The degree is 3
What is the greatest common mononial factor of 9x³y² + 15x²y - 6xy² ?
3xy
If x-2 is a factor of a polynomial f(x), is it true that f(2) = 0 ?
True.
Is ± ½ a possible rational solution of f(x)= - 3x³ - 11x² + 5x - 6?
No.
Which of the following, based on the Descartes Rule of Signs, is the only possible classification of the roots of the function f(k) = -3k³ + 5k² - k + 4?
3, 1 positives; 0 negatives; 0, 2 imaginary
At which value of x does f(x) = 2x³ - x² + 1 have a local minimum value?
(0,1)
Simplify (¾)^-3
64/27
Perform the indicated operation: (7x - 3)²
49x² - 42x + 9
Perform the indicated operation: (x - 2)(x+3)(x-5)
x³ - 4x² - 11x + 30
Perform the indicated operation: (2x⁴+9x-7) - (x⁴+6x+5)
x⁴ + 3x - 12
Factor the polynomial completely: 3x³-81
3(x-3)(x²+3x+9)
Factor the polynomial completely: 3x³ + 6x² + x + 2
(3x²+1)(x+2)
Using long division: (x⁴ + 10x³ + 8x² - 59x +40) ÷ (x² + 3x -5)
x² + 7x - 8
Using synthetic division: (2x³ - 25x² + 83x - 88) ÷ (x-8)
2x² - 9x + 11
Find all real zeros of the function: f(x) = x³ - 3x² - x +3
-1, 1, 3
Find all real zeros of the function: f(x) = x³ -6x² +4x - 24
6
Find all real zeros of the function: f(x) = x⁴ - 2x³ - 8x² + 8x + 16
-2, 2, 1 ± √5
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros: -3, ± √2
f(x) = x³ + 3x² - 2x -6
Find all zeros of the polynomial function: g(x) = x³ - 2x² - x + 2
-1, 1, 2
Find all zeros of the polynomial function: h(x) = 2x⁴ - 3x³ - 27x² + 62x - 24
-4, ½, 2, 3
Determine the possible number of positive real zeros, negative real zeros, and imaginary zeros for the function: f(x) = x⁴ + 3x³ - 2x² - x + 10
positive zeros: 0 or 2, negative zeros: 0 or 2,
imaginary zeros 0, 2, or 4
Determine the possible number of positive real zeros, negative real zeros, and imaginary zeros for the function: g(x) = -x^5 + 2x⁴ + 3x² - 7x - 12
positive zeros: 0 or 2, negative zeros: 1,
imaginary zeros: 2 or 4
(3³)² is the power of power property. It tells us to do what to the exponents?
Multiply exponents, so 3^5.
What does FOIL stand for?
First, Outter, Inner, Last
When multiplying binomials, what method do we use?
FOIL
A zero of a function is also the _____ on a graph?
x-intercept
Absolute Value
The distance a number is from the 0 on the number line
Algebraic Expression
An expression that contains numbers, operations and variables.
Associative Property
A property that states that numbers in addition of multiplication expressions can be grouped without affecting the value of the expression.
Axes
A horixontal and vertical number line on a coordinate plane.
Base of a Power
The repeated factor in a power.
Coefficient
The number mulltiplied by a variable in a term.
Commutative Property
A property that states numbers can be added or multiplied in any order.
Constant
A term that has no variable.
Unit Conversion
The process of renaming a measurement using different units.
Coordinate Plane
A coordinate system formed by the intersection of a horizontal number line, called the x-axis, and a vertical number line, called the y-axis.
Decimal
a number with one or more digits to the right of the decimal point
Distributive Property
a(b+c)=ab+ac, Multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.
Dividend
a number to be divided by another number
Divisor
the number by which a dividend is divided.
Elimination Method
to solve a system of equations by adding equations to get rid of one variable. sometimes needing to multiply one equation by a # to make terms opposites
Equation
a mathematical statement in which two expressions are equal
Equivalent expressions
algebraic expressions that have the same values for all values of variables
Equivalent fractions
fractions that have the same value
Evaluate
to find the value of a numerical or algebraic EXPRESSION.
Exponent
The number of times a factor in repeated in a power.
Factors
Whole numbers that can be multiplied together to hind a product.
Formula
An algebraic equation that shows the relationship among specific quantities.
Function
A pairing of input and output values according to a specific rule.
Greatest Common Factor
The greatest factor that is common to two or more numbers.
Grouping Symbols
parentheses ( ), brackets [ ], and braces { } that group parts of an expression.
Improper Fraction
A fraction whose numerator is greater than or equal to its denominator.
Integers
The set of all whole numbers, their opposites, and 0.
Inverse Operations
Operations that undo each other.
Irrational numbers
numbers that cannot be expressed in the form a/b, where a and b are integers and b =0.
Least Common Denominator
The least common multiple of the denominators of two or more fractions.
Least Common Multiple
...
linear equation
An equation that can be written as a1x1 + a2x2 + ... = b; a1, a2, etc. are real or complex numbers known in advance
consistent system
Has one or infinitely many solutions
inconsistent system
Has no solution
leading entry
Leftmost non-zero entry in a non-zero row
Echelon form
1. All nonzero rows are above any all zero rows; 2. Each leading entry is in a column to the right of the previous leading entry; 3. All entries below a leading entry in its column are zeros
Reduced Echelon Form
Same as echelon form, except all leading entries are 1; each leading 1 is the only non-zero entry in its row; there is only one unique reduced echelon form for every matrix
Span
the collection of all vectors in R^n that can be written as c1v1 + c2v2 + ... (where c1, c2, etc. are constants)
Ax = b
1. For each b in R^n, Ax = b has a solution; 2. Each b is a linear combination of A; 3. The columns of A span R^n; 4. A has a pivot position in each row
pivot position
A position in the original matrix that corresponds to a leading 1 in a reduced echelon matrix
pivot column
A column that contains a pivot position
homogeneous
A system that can be written as Ax = 0; the x = 0 solution is a TRIVIAL solution
independent
If only the trivial solution exists for a linear equation; the columns of A are independent if only the trivial solution exists
dependent
If non-zero weights that satisfy the equation exist; if there are more vectors than there are entries
transformation
assigns each vector x in R^n a vector T(x) in R^m
Matrix multiplication warnings
1. AB != BA ; 2. If AB = AC, B does not necessarily equal C; 3. If AB = 0, it cannot be concluded that either A or B is equal to 0
Transposition
flips rows and columns
Properties of transposition
1. (A^T)^T = A; 2. (A+B)^T = A^T + B^T; 3. (rA)^T = r
A^T; 4. (AB)^T = B^T
A^T
Invertibility rules
1. If A is invertible, (A^-1)^-1 = A; 2. (AB)^-1 = B^-1 * A^-1; 3. (A^T)^-1 = (A^-1)^T
Invertible Matrix Theorem (either all of them are true or all are false)
A is invertible; A is row equivalent to I; A has n pivot columns; Ax = 0 has only the trivial solution; The columns of A for a linearly independent set; The transformation x --> Ax is one to one; Ax = b has at least one solution for each b in R^n; The columns of A span R^n; x --> Ax maps R^n onto each R^m; there is an n x n matrix C such that CA = I; there is a matrix such that AD = I; A^T is invertible; The columns of A form a basis of R^n; Col A = R^n; dim Col A = n; rank A = n; Nul A = [0]; dim Nul A = 0
Column Row Expansion of AB
col1Arow1B + ...
LU Factorization
1. Ly = b; Ux = y; 2. Reduce A to echelon form; 3. Place values in L that, by the same steps, would reduce it to I
Leontief input-output model
x = Cx + d
Subspaces
1. The zero vector is in H; 2. For u and v in H, u + v is also in H; 3. For u in H, cu is also in H (c is a constant)
Column space
Set of all the linear combinations of the columns of A
Null space
Set of all solution to Ax = 0
Basis
A linearly independent set in H that spans H; the pivot columns of A form a basis for A's column space
Dimension
The number of vectors in any basis of H; the zero subspace's dimension is 0
rank
The dimension of the column space
one-to-one
A transformation that assigns a vector y in R^m for each x in R^n; there's a pivot in every column
onto
consistent for any b; pivots in all rows
inner product
a matrix product u^Tv or u . v where u and v are vectors; if U . V = 0, u and v are orthogonal
orthogonal component
1. x is in W' if x is perpendicular to every vector that spans W; 2. W' is a subspace of R^n
orthogonal set
A set of vectors where Ui . Uj = 0 (and i != j); if S is an orthogonal set, S is linearly independent and a basis of the subspace spanned by S
orthonormal
An orthogonal set of unit vectors
Mixed Number
the sum of a whole number and a fraction
Line
straight line that has no width and no ends
Line Segment
part of a line
Intersect
when two lines cross
Point of Intersection
the place where two lines cross
Parallel Lines
lines in the same plane that do not intersect and the distance between the lines is always the same
Perpendicular
when two lines make square corners at the point of intersection
Right Angles
the angles made by perpendicular lines
Straight Angle
two right angles that forms a straight line back to back
Acute Angle
an angle smaller than a right angle
Obtuse Angle
an angle larger than a right angle
Polygon
simple, closed, flat geometric figures whose sides are line segments and whose lines do not cross
Triangle
a polygon with 3 sides and vertices
Quadrilateral
a polygon with 4 sides and vertices
Pentagon
a polygon with 5 sides and vertices
Hexagon
a polygon with 6 sides and vertices
Heptagon
a polygon with 7 sides and vertices