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Algebra 2 Honors Vocabulary
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Terms in this set (278)
set
a collection of objects ; sometimes named with capital letters
element/member
each object in a set ; typically use braces {} to enclose them
subset
a set that's a part of a larger set
real numbers
consists of several subsets of numbers ; set of rational numbers joined with the set of irrational numbers ; letter "R" represents them
natural numbers
sometimes referred to as counting numbers ; ellipsis is used to indicate that the elements of the set follow the same pattern ; go on infinitely in the positive direction ; subset of whole numbers ; letter "N" represents them
ellipsis
the 3 dots (...)
whole numbers
composed of the natural numbers combined with the number 0 ; subset of integers ; subset of integers ; letter "W" represents them
integers
combines the negatives of the natural numbers with the whole numbers ; go on infinitely in the positive and negative directions ; subset of rational numbers ; letter "Z" represents them
rational numbers
expressed as a quotient of 2 integers ; (fraction) ; includes all repreating and terminating decimals ; defined as Q = {p/q | p and q are integers, q doesn't equal 0} ; read as "the set of all elements p/q such that p and q are integers, q doesn't equal 0" ; letter "Q" represents them
irrational numbers
can't be expressed as a quotient of 2 integers (fraction) ; aren't rational ; defined as I = {x | x is not a rational number} ; letter "I" represents them
real number line
real numbers can be pictured as points on this line ; numbers on it increase from left to right and the point labeled 0 is the origin
graph
point on a number line that corresponds to a real number ; drawing it is called "graphing" the number or "plotting" the point ; convert to decimals before plotting ; one of an equation in 2 variables is the collection of all points (x, y) whose coordinates are solutions of the equation
coordinate
number that corresponds to a point on a number line
closure property
a + b and ab are real numbers
commutative property
a + b = b + a ; ab=ba
associative property
a + (b + c) = (a + b) + c ; a(bc) = (ab)c
distributive property
a(b + c) = ab + ac ; lets you combine like terms that have variables by adding the coefficients
identity property
0 + a = a ; 1 x a = a ; 0 is the additive identity ; 1 is the multiplicative identity
multiplication property of zero
0 x a = 0
additive inverse property
for each real number a, there's a unique real number -a such that a + (-a) = 0
multiplicative inverse property
for each nonzero real number a, there's a unique and real number 1/a such that a x 1/a = 1
opposite/additive inverse
this of any number a is (-a)
reciprocal/multiplicative inverse
this of any nonzero number a is 1/a
subtraction
adding the opposite
division
multiplying by the reciprocal ; "per" designates it
unit analysis
when you use the operations of addition, subtraction, multiplication, and division in real life, use this to check that your units make sense
set-builder notation
ex. {x | x < 10} reads "the set of all x such that x is less than 10" ; uses open circles/disks or solid circles/disks or "[ ]" and "( )" on the number line
open circle/open disk
represents that the number it's on isn't included in the set
solid circle/solid disk
represents that the number it's on is included in the set
arrow
on a number line, indicates that the set goes on indefinitely toward infinity and/or negative infinity
interval notation
ex. (-infinity, 10) ; uses "[ ]" for solid circles/disks and "( )" for open circles/disks ; don't confuse it with a point
union
joining all of the elements of 2 sets
intersection
the set of elements that are common to both sets
empty set
the set containing no elements
evaluate
substitution ; used in expressions and functions ; when the variables in an algebraic expression are replaced with numbers, you're doing this to the expression ; to do so, write the algebraic expression, substitute the values of the variables, and simply the expression
simplify
combining like terms ; used in expressions
numerical expression
consists of numbers, operations, and grouping symbols
exponentiation
raising to a power
exponent
used to represent repeated factors in multiplication ; number of times the base is used as a factor
base
number that's multiplied in exponentiation
power
expression in exponentiation that combines the base and exponent
PEmdas
shortcut for order of operations
order of operations
4 step process that helps avoid confusion when evaluating expressions:
1) always start with the innermost grouping symbols. includes parentheses ( ), brackets [ ], braces { }, and absolute value bars | | ; "P"
2) simplify exponential expressions ; "E"
3) perform multiplication and division working from left to right ; "md"
4) perform addition and subtraction working from left to right ; "as"
variable
a letter that's used to represent 1 or more numbers
value of a variable
any number that's used to replace a variable
algebraic expression
an expression involving variables
value of the expression
result of evaluating an algebraic expression
term
parts of an expression that are added together
coefficient of the power
when a term is the product of a number and a power of a variable, the number is this
like terms
terms that have the same variable part ; includes constant terms
verbal model
written-out version of an expression
math model
the actual expression in math terms
equation
a statement or sentence indicating that 2 algebraic expressions are equal ; can be true or false ; many functions can be represented by one in 2 variables
equality symbol
verb in an equation
solution/root
satisfies the equation ; makes it a true statement
open sentence
an equation that's neither true nor false until we choose a value for x
solution set
the set of all solutions to an equation
linear equation in one variable
an equation of the form ax + b = 0, where a and b are real numbers, with a not equal to 0
equivalent equations
2 equations with the same solution set ; adding the same real number to or subtracting the same real number form each side of an equation results in them ; multiplying or dividing each side of an equation by the same nonzero real number results in them
properties of equality
use when solving linear equations ; shown with the information "if A and B are algebraic expressions and C is a real number, then the following equations are equivalent to A = B"
addition property of equality
A + C = B + C
subtraction property of equality
A - C = B - C
multiplication property of equality
CA = CB (C isn't equal to 0)
division property of equality
A/C = B/C (C isn't equal to 0)
nonequivalent equations
using an algebraic expression for C in the properties of equality because the value of an algebraic expression is a real number can produce these
golden rule of algebra
if you do something to 1 side, then you must do it to the other side
least common denominator
the smallest number that's a common denominator for a given set of fractions ; multiply each side of an equation by it to clear the fractions
least common multiple
the smallest positive number that's a multiple of 2 or more numbers
identity
an equation that's satisfied by every real number for which both sides are defined
conditional equation
an equation that's satisfied by at least 1 real number, but isn't an identity
inconsistent equation
an equation that has no solution ; equivalent to a false statement
simple inequality
a statement that 2 algebraic expressions aren't equal in a particular way ; stated using less than (<), less than or equal to (≤), greater than (>), or greater than or equal to (≥)
brackets/parentheses
use these in place of open and closed disks when graphing inequalities ; "[ ]" and "( )"
linear inequality
given by replacing the equal sign in the general linear equation of ax + b = 0 by any of the symbols <, >, ≤, or ≥
equivalent inequalities
inequalities that have the same solution set ; solve them like we solve linear equations by performing operations on each side to get them
reverse it
do this to the direction of the inequality symbol when an inequality is multiplied or divided by a negative number
compound inequality
a sentence containing 2 simple inequalities connected with "and" or "or" ; solution can be a bounded interval
bounded interval
an interval of real numbers that doesn't involve infinity
"and"
tends to give us a bounded interval ; can collapse this kind of compound inequality into a simpler form than 2 separate inequalities ; solution makes both inequalities true ; written as 1 inequality
"or"
tends to give us 2 solutions going in opposite directions to positive and negative infinity ; to solve this kind of compound inequality, find all values of the variable that make at least 1 of the inequalities true ; solution makes either inequalities true ; rewrite as 2 equations
absolute value
the distance from 0 on the number line ; one of a real number "x" is written | x | ; quantity that's nonnegative ; opposites have the same one
absolute value equation
an equation that has a variable within the absolute value sign ; can have 2 solutions ; some have no solution, such as | x | = -5 ; important to check the possible solutions of it
extraneous solution
a solution derived from an original equation that isn't a solution of the original equation ; 1 or more of the possible solutions to an absolute value equation may be this
absolute value inequalities
inequalities that are closely related to absolute value equations
relation
a mapping/pairing of input values with output values
domain
the set of input values in a relation ; one of a function consists of the values for x for which the function is defined
range
the set of output values in a relation ; one of a function consists of the values of f(x) where x is the domain of f
function
a relation is this provided there's exactly 1 output for each input ; can't be one if you see the input twice or if at least 1 input has more than 1 output ; can still be one if 1 output maps to 2 inputs ; can be represented by other letters besides f, such as g or h
4 ways to represent relations
1) ordered pairs
2) mapping diagram
3) table of values
4) graph
ordered pairs
a relation can be represented by a set of them of the form (x, y) ; the first number is the x-coordinate and the second number is the y-coordinate ; a solution of an equation in 2 variables that represents a function when the values of x and y are substituted into the equation
coordinate plane
to graph a relation, plot each of its ordered pairs in this ; divided into 4 quadrants by the x-axis and the y-axis ; axes intersect at a point called the origin
x-values
if the function is listed as a group of ordered pairs, the domain (inputs) will be represented by these
y-values
if the function is listed as a group of ordered pairs, the range (outputs) will be represented by these
vertical line test
states that if a vertical line passes through more than 1 point on the graph of a relation, then the relation isn't a function ; use it to determine whether a relation is a function ; works because if the vertical line passes through a graph at more than 1 point, there's more than 1 value in the range that corresponds to 1 value in the domain
independent variable
in an equation, the input variable
dependent variable
in an equation, the output variable ; depends on the value of the input variable
graphing equations in 2 variables
steps to do it:
1) construct a table of values
2) graph enough solutions to recognize a pattern
3) connect the points with a line or a curve
linear function
function that's in the form y = mx + b where m is the slope and b is the y-intercept and both are constants ; graph of it is a line ; must have the highest exponent being an understood 1 ; multiple ones form a family of functions ; each one is a transformation of the function y = x ; graph shows all lines going straight
function notation
by naming a function f, you can write the function using this ; in the form of f(x) = mx + b
f(x)
symbol that's read as "the value of x" or simply as "f of x" ; another name for y, so y = f(x), (x,y) = (x, f(x)), and -y = -f(x)
function rule
an equation that represents an output value in terms of an input value ; can write it in function notation ; can be written using y, f(x), f(1), etc.
families
sets of functions in which each function is a transformation of a special function called the parent
parent linear function
the simplest form in a set of functions that form a family ; the function y = x
transformation
a change in the position, size, or shape of a figure or a graph ; each function in the family is this of the parent function ; can consist of a translation, reflection, compression, or stretch
translation
a transformation that shifts or slides every point of a figure or graph the same distance in the same direction ; shifts the graph of the parent function horizontally, vertically, or both without changing shape or orientation
vertical translation
for a positive constant k and a parent function f(x), f(x) +/- k ; + shifts the graph up and - shifts the graph down ; graph moves in normal directions
horizontal translation
for a positive constant k and a parent function f(x), f(x +/- h) ; + shifts the graph left and - shifts the graph right ; graph moves in opposite directions
reflection
a transformation that reflects/flips a graph or figure across a line, called the line of reflection, such that each reflected point is the same distance from the line of reflection, but is on the opposite side of the line ; makes a mirror image
reflection in the y-axis
for a function f(x), f(-x)
reflection in the x-axis
for a function f(x), -f(x) ; y-coordinates become negated/opposite ; x-coordinates stay the same
compression
a transformation that pushes the points of a graph horizontally toward the y-axis or vertically toward the x-axis ; sometimes called a dilation ; y = a x f(x) when 0 < a < 1 compresses the graph vertically and makes it flatter
stretch
a transformation that pulls the points of a graph horizontally away from the x-axis or vertically away from the x-axis ; sometimes called a dilation ; y = a x f(x) when a > 1 stretches the graph vertically and makes it steeper
vertical stretch
a transformation that multiplies all y-values of a function by the same factor greater than 1 ; for a function f(x) and a constant a, y = af(x) is one when a > 1
vertical compression
a transformation that reduces all y-values of a function by the same factor between 0 and 1 ; for a function f(x) and a constant a, y = af(x) is one when 0 < a < 1
h value
when we generically write a linear function as f(x) = a(x - h) + k, this affects the horizontal translation ; results in opposite signs
k value
when we generically write a linear function as f(x) = a(x - h) + k, this affects the vertical translation
a value
when we generically write a linear function as f(x) = a(x - h) + k, this affects reflecting if it's negative, stretching, and compressing ; depends on if it's < 1 or > 1
parent functions
basic ones are linear, quadratic, cubic, absolute value, and square root ; these 5 have basic transformations ; starting point for them for now is (0, 0) or the origin
slope
the ratio of vertical change (the rise) to horizontal change (the run) of a nonvertical line ; represented by the letter m ; just as 2 points determine a line, 2 points are all that are needed to determine one of a line ; one of a line is the same regardless of which 2 points are used ; formulas for it are y2-y1/x2-x1, change in y-coordinates/change in x-coordinates, and rise/run ; doesn't matter which point you choose as point #1 or #2, but you must stay consistent during the problem once you choose ; besides telling you whether the line rises, falls, is horizontal, or is vertical, it tells you the steepness of the line ; can be used to determine whether 2 different (nonvertical) lines are parallel or perpendicular ; for a standard form equation, can be represented as -A/B
positive slope
line with this kind of slope rises from left to right ; m > 0 ; for 2 lines with this kind of slopes, the line with the greater slope is steeper ; goes up and right
negative slope
line with this kind of slope falls from left to right ; m < 0 ; for 2 lines with this kind of slopes, the line with the slope of greater absolute value is steeper ; goes down and left
zero slope
line with this kind of slope is horizontal ; m = 0 ; 0/#
undefined slope
line with this kind of slope is vertical ; m is undefined ; #/0
parallel lines
2 lines in a plan that don't intersect ; have the same slope ; m1 = m2 ; plug in the point given and slope into point-slope form
perpendicular lines
2 lines in a plane that intersect to form a right angle ; slopes are negative reciprocals of each other ; m1 = -1/m2 or m1m2 = -1 ; slopes = -1 ; plug in the point given and slope into point-slope form
slope-intercept form
y = mx + b for a linear equation where m is the slope and b is the y-intercept ; steps for graphing equations in this form: 1) write the equation in this form by solving for y 2) find the y-intercept and use it to plot the point where the line crosses the y-axis 3) find the slope and use it to plot a 2nd point on the line 4) draw a line through the 2 points
y-intercept
b in y = mx + b ; starting point when graphing equations in slope-intercept form ; for slope-intercept form, (0, b) when written as an ordered pair ; point where the line crosses the y-axis ; for standard form, (0, y) when written as an ordered pair or represented as C/A
standard form
Ax + By = C for a linear equation where A and B are not both zero ; quick way to graph an equation in this form is to plot its intercepts (when they exist) ; can convert to slope-intercept form if you want ; find the x-intercept and the y-intercept by using an x and y tee chart or using the shortcut of dividing the C value by A for the x-intercept or the C value by B for the y-intercept ; steps for graphing equations in this form: 1) write the equation in this form 2) find the x-intercept by letting y = 0 and solving for x and use it to plot the point where the line crosses the x-axis 3) find the y-intercept by letting x = 0 and solving for y and use it to plot the point where the line crosses the y-axis 4) draw a line through the 2 points ; every linear equation can be written in this form, including the equation of a vertical line ; y = ax^2 + bx + c for a quadratic function ; a doesn't equal 0 ; when given the solutions to a quadratic equation, you're trying to get it in this form ; real + imaginary for equations with i in them
x-intercept
the x-coordinate of the point where the line intersects the x-axis ; for standard form, (x, 0) when written as an ordered pair or represented as C/B
vertical line
equation of it can't be written in slope-intercept form because the slope of a it isn't defined ; the graph of x = c is this line through (c, 0) where c is a constant number
horizontal line
the graph of y = c is this line through (0, c) where c is a constant number
point-slope form
given the slope m and a point (x1, y1) use y - y1 = m(x - x1) to write an equation of a line ; need x and y in the final equation
two points
given 2 points (x1, y1) and (x2, y2) use m = y2-y1/x2-x1 to find the slope m ; then us point slope form with the slope and either of the given points to write an equation of a line ; look out for if the points given are intercepts
1) simplify
2) evaluate
used with expressions ; 2 of the most common directions in algebra
solve
used with equations ; 1 of the most common directions in algebra ; when doing this to an equation, we're looking for a variable to satisfy the variable given in the equation ; this value for the variable is the solution to the equation ; discovering the solution set
solution
a number or ordered pair (x, y) that makes a mathematical equation or inequality a true statement ; exponent value tells us how many there are for an equation ; any point on a function is considered one to the function ; one of a system of linear equations in 2 variables is an ordered pair (x, y) that satisfies each equation
linear inequalitiy
graph of one in 2 variables is a half-plane ; steps for graphing one:
1) graph line
2) dashed or solid line?
3) test point (most popular one is (0,0) ; do one that isn't on the line)
4) shading
piecewise function
"real-life" functions that are represented by a combination of equations with each corresponding to a part of the domain ; defined by multiple equations ; includes conditions with a boundary ; have to graph them in pieces or they won't pass the vertical line test ; graph both lines completely then erase the unneeded pieces
step function
function that's composed of multiple line segments ; pay attention to its condition's signs ; graph resembles a set of stair steps
greatest integer function
parent function that's also a type of step function ; denoted by g(x) = [[x]] ; for every real number x, g(x) is the greatest integer less than or equal to x
absolute value functions
the graph of a piecewise function consisting of 2 rays, is V-shaped, and opens up if a > 0 and down if a < 0 ; basic equation for it is a|x - h| + k where (h, k) is the vertex and symmetric to the line x = h, h goes left and right, k goes up and down, and a goes steeper and wider and equals the slope of the right side ray of its graph ; wider than the graph of y = |x| if |a| < 1 and narrower than the graph of y = |x| if |a| > 1
vertex/minimum
the corner point of an absolute value function's graph
system of 2 linear equations
in 2 variables x and y, consists of 2 equations in standard form or slope intercept form ; have a point of intersection where the 2 lines meet on a graph ; graph by 1) changing the equation from standard form to slope-intercept form or 2) plugging the intercepts (x, 0) and (y 0) into the equations ; graph whole lines across whole graph
consistent system
system of equations that has at least 1 solution ; one with exactly 1 solution is independent and has lines that intersect and have different slopes ; one with infinitely many solutions in dependent and has lines with the same slope and the same y-intercept
inconsistent system
system of equations with no solutions ; lines have the same slope and different y-intercepts ; consists of parallel lines
graphing
visual way to solve a system of linear equations
substituation
algebraic way to solve a system of linear equations ; steps of it are:
1) solve 1 of the equations for 1 of its variables 2) substitute the expression from step 1 into the other equation and solve for the other variable 3) substitute the value from step 2 into the revised equation from step 1 and solve ; look for a coefficient of 1 or -1 as an indicator to solve using this method ; order of it matters
linear combination (elimination)
algebraic way to solve a system of linear equations ; steps of it are: 1) multiply 1 or both of the equations by a constant to obtain coefficients that differ only in sign for 1 of the variables 2) add the revised equations form step 1 ; combining like terms will eliminate 1 of the variables ; solve for the remaining variable 3) substitute the value obtained in step 2 into either of the original equations and solve for the other variables ; goal of it is to add the equations to obtain an equation in 1 variable ; uses the "stacked method" when adding the 2 equations together
matrix
a rectangular arrangement of numbers in rows and columns ; some have special names because of their dimensions or entries ; two of them are equal if their dimensions are the same and the entries in corresponding positions are equal ; must have the same dimensions for you to add or subtract them ; can't divide them ; order is very important when dealing with them ; the product of 2 of them A and B is defined provided the number of columns in A is equal to the number of rows in B (cancel out the middle number) ; commutative property doesn't work with them
dimensions
rows x columns in a matrix
entries
the numbers in a matrix
row matrix
a matrix with only 1 row
column matrix
a matrix with only 1 column
square matrix
a matrix with the same number of rows and columns
zero matrix
a matrix whose entries are all zeroes
rows
go side to side
columns
go up and down
associative process of matrix multiplication
A(BC) = (AB)C ; A, B, and C are matrices
left distributive property
A(B + C) = AB + AC ; A, B, and C are matrices
right distributive property
(A + B)C = AC + BC ; A, B, and C are matrices
associative property of scalar multiplcation
c(AB) = (cA)B = A(cB)
system of linear inequalities
2 inequalities ; solution of one is an ordered pair that's a solution of each inequality in the system ; graph of one is the graph of all of the solutions of the system, which is the region common to all half-planes ; steps to graph it are 1) graph the line that corresponds to the inequality ; use a dashed line for an inequality with < or > and a solid line for an inequality with ≥ or ≤ 2) lightly shade the half-plane that's the graph of the inequality ; colored pencils may help you distinguish the different half-planes
point symmetry
where 2 distinct points P and P' are symmetric with respect to point M if and only if M is the midpoint of PP' ; point M is symmetric with respect to itself ; when the definition of it is extended to a set of points, such as the graph of a function, then each point P in the set of numbers must have an image point P' that's also in the set ; a figure that's symmetric with respect to a given point can be rotated 180 degrees about that point and appear unchanged ; f(-x) = -f(x) for graphs with this
origin
the lowest point on the graph of y = x^2 and the highest point on the graph of y = -x^2 ; a common point of symmetry
symmetry with respect to the origin
function has a graph that's this if and only if f(-x) = -f(x) for all x in the domain of f
line symmetry
2 distinct points P and P' have this if and only if e is the perpendicular bisector of PP' ; a point P is symmetric to itself with respect to line e if and only if P is on e ; graphs that have this can be folded along the line of symmetry so that the 2 halves match exactly ; some common lines of symmetry are the x-axis, y-axis, y = x, and y = -x
x-axis
line symmetry along this where (x, y) --> (x, -y) ; formula for it is y = 0
y-axis
line symmetry along this where (x, y) --> (-x, y) ; formula for it is x = 0
y = x
line symmetry along this where (x, y) --> (y, x)
y = -x
line symmetry along this where (x, y) --> (-y, -x)
even functions
functions whose graphs are symmetric with respect to the y-axis ; where f(-x) = (f, x) ; have even exponents ; stop working if it's this
odd functions
functions whose graphs are symmetric with respect to the origin ; where f(-x) = -f(x) ; have odd exponents ; even if it's this, keep working
family of graphs
a group of graphs that display 1 or more similar characteristics
parent graph
a basic graph that's transformed to create other members in a family of graphs
constant function
parent function that's a horizontal line ; f(x) = a
identity function
parent function that's a linear function ; f(x) = x
cubic function
parent function ; f(x) = x^3 ; graph shows a curvy shape
square root function
parent function ; f(x) = √x ; graph shows 1/2 of a sideways quadratic function
rational/reciprocal function
parent function ; f(x) = 1/x ; graphs don't touch the barriers
absolute value function
parent function ; f(x) = |x| ; graph shows a V shape
exponential function
parent function that's the inverse of a logarithmic function ; f(x) = a^x
logarithmic function
parent function that's the inverse of an exponential function ; f(x) = log(small a)x
horizontal translations
y = f(x + h) translates the graph h units left ; y = f(x - h) translates the graph h units right
reflections
y = -f(x) reflects over the x-axis
vertical translations
y = f(x) + k translates the graph k units up ; y = f(x) - k translates the graph k units down
rigid transformation
transformation that doesn't change the shape of a graph translating horizontally or vertically are this
nonrigid transformation
transformation that does change the shape of a graph ; stretching and shrinking are this
continuous functions
functions where you can trace the graph of it without lifting your pencil ; graphs are smooth, continuous curves ; linear and quadratic functions are this at all points
discontinuous functions
functions where you can't trace the graph of it without lifting your pencil ; graphs aren't smooth, continuous curves
infinite discontinuity
type of discontinuity where |f(x)| becomes greater and greater s the graph approaches a given x value ; creates an asymptote of a curve, which is a line such that the distance between the curve and the line approaches 0 as they tend to infinity
jump discontinuity
type of discontinuity that indicates that the graph stops at a given value of the domain and then begins again at a different range value for the same value of the domain ; some piecewise functions have this
point discontinuity
type of discontinuity where there's a value in the domain for which the function is undefined, but the pieces of graph match up
continuity test
a function is continuous at x = c if it satisfies these conditions:
1) the function is defined at c ; in other words, f(c) exists
2) the function approaches the same y-value on the left and right sides of x = c
3) the y-value that the function approaches from each side is f(c)
end behavior
describes what the y-values of a function do as |x| becomes greater and greater ; a tool for analyzing functions
down and up
graph of the end behavior of functions where the leading coefficient is positive and the degree is odd ; example of it is f(x) = x ; when x --> -infinity, f(x) --> -infinity ; when x --> infinity, f(x) --> +infinity
up and down
graph of the end behavior of functions where the leading coefficient is negative and the degree is odd ; example of it is f(x) = -x ; when x -- > -infinity, f(x) --> +infinity ; when x --> infinity, f(x) --> -infinity
up and up
graph of the end behavior of functions where the leading coefficient is positive and the degree is even ; example of it is f(x) = x^2 ; when x --> -infinity, f(x) --> +infinity ; when x --> infinity, f(x) --> +infinity
down and down
graph of the end behavior of functions where the leading coefficient is negative and the degree is even ; example of it is f(x) = -x^2 ; when x --> -infinity, f(x) --> -infinity ; when x --> infinity, f(x) --> -infinity
increasing
a function f is doing this on an interval I if and only if for ever a and b contained in I, f(a) < f(b) whenever a < b
decreasing
a function f is doing this on an interval I if and only if for every a and b contained in I, f(a) > b whenever a < b
constant
a function f remains this on an interval I if and only if for every a and b contained in I, f(a) = f(b) whenever a < b
critical points
special points in the domain of a function where the function changes from increasing to decreasing or vice versa ; function is considered constant at these ; not ordered pairs ; points on a graph in which a line drawn tangent to the curve is horizontal or vertical ; 3 different types of them
maximum
type of critical point ; when the graph of a function is increasing to the left of x = c and decreasing to the right of x = c, then there's this at x = c ; graph is an upside down U and increases then decreases ; highest point on the graph of a quadratic function ; finding it when the graph goes down and a is -
minimum
type of critical point ; when the graph of a function is decreasing to the left of x = c and increasing to the right of x = c, then there's this at x = c ; graph is a U and decreases then increases ; lowest point on the graph of a quadratic function ; finding it when the graph goes up and a is +
point of inflection
type of critical point ; point where the graph changes its curvature ; graph is an S
absolute maximum
the greatest value that function assumes over its domain ; graph is down and down
absolute minimum
the least value that a function assumes over its domain ; graph is up and up
extremum
the general term for maximum or minimum
relative extrema
the general term for relative maximum or relative minimum
relative maximum
value of a function that may not be the greatest value of f on the domain, but is the greatest y-value on some interval of the domain
relative minimum
value of a function that may not be the least value of f on the domain, but is the least y-value on some interval of the domain
polynomial function
a function where the exponents are all whole numbers and the coefficients are all real numbers ; in standard form if its terms are written in descending order of exponents from left to right ; we can multiply them ; most common one that we multiple is linear polynomials (binomial expressions)
leading coefficient
the number in a function that's in front of the variable
constant term
the number in a function that isn't by a variable and stands alone
degree
the number in a function that's the exponent of a variable
FOIL
1 method for multiplying expressions containing 2 terms ; uses the distributive property twice ; you add the products of the first terms, outer terms, inner terms, and last terms
monomial
an expression that has only 1 term ; when factoring out a common one, you're looking for the GCF
binomial
an expression that has 2 terms ; most common expressions that we multiply
trinomail
an expression that has 3 terms ; can use factoring to write one as a product of binomials, but only use this method when the leading coefficient is 1 ; can be factored by grouping, but only use this method when the leading coefficient is greater than 1
difference of 2 squares
binomial expression ; prevalent pattern on the ACT
perfect squares trinomial
trinomial expression ; prevalent pattern on the ACT
grouping
method of factoring in which you factor out the common factors of 2 terms ; use when there's no factor common to all 4 terms ; when doing this for trinomials, find the key number and 2 factors of it and use them as the coefficients of 2 terms to be placed in the middle
square root
the inverse of squaring ; negative numbers don't have them (in the real number system) because the square of any real number is nonnegative ; principal one of 0 is 0 and 1 is 1 and a positive number is positive ; if c is greater than or equal to 0, then the square root of c squared is c
radical sign
the symbol that's used to denote the principle or nonnegative square root because there's an understood 2, which is the root in the radical sign
radicals
should always be written in simplest form, which means writing them so that 1) no perfect square factor other than 1 is under the radical sign 2) no fraction is under the radical sign 3) no fraction has a radical in its denominator ; when reducing them to simplest form, they're sometimes in the answer ; in order to add or subtract them, they must be like terms
rationalizing the denominator
the process in which a square root occurs in the denominator of a quotient and has to be rewritten so that the denominator contains no square roots ; idea is to multiply the denominator by an expression equal to 1 so that the new denominator contains no square roots ; be sure to multiply both the numerator and the denominator by the same expression
fundamental counting principle
for 2 events, if 1 event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur is m x n ; for 3 events, if 3 events can occur in m, n and p ways, then the number of ways that all 3 events can occur is m x n x p
permutation
an ordering of n objects is one of the objects ; the number of them of n items of a set arranged r items at a time is nPr = n!/(n-r)! for 0 is less than or equal to r and r is less than or equal to n ; the number of distinguishable permutations of n objects where 1 object is repeated q1 times, another is repeated q2 times, and so on is n!/q1!q2!... ; order is important
factorial
the expression n! is read as "n one" and represents the product of all integers from 1 to n
combination
a selection of r objects from a group of n objects where the order isn't important ; the number of them of n items of a set chosen r items at a time is nCr = n!/(n-r)!r! for 0 is less than or equal to r and r is less than or equal to n
probability
measures how likely it is for an event to occur ; one of an impossible event is 0 or 0% ; one of a certain event is 1 or 100% ; one of an event is a number between 0 and 1 or a percent between 0% and 100% ; can be written as a percentage
experimental probability
a calculation of the probability of an event based on performing an experiment, conducting a survey, or looking at the history of an event ; when you gather data from observations, you can calculate this ; each observation is an experiment or a trial ; P(event) = number of times the event occurs/number of trials ; sometimes estimated using a simulation
simulation
a model of the event ; in situations where actual trials are difficult or unreasonable to conduct, you can estimate the experimental probability of en event using this
sample space
the set of all possible outcomes to an experiment or activity
equally likely outcomes
outcomes in a sample space that have the same chance of occuring
theoretical probability
often simply called the probability of an event ; you can calculate it as a ratio of outcomes ; if a sample space has n equally likely outcomes and an event A occurs in m of these outcomes, then it of event A is P(A) = m/n
combinatorics
includes the fundamental counting principle and ways to count permutations and combinations ; it can be easier to use these to find theoretical probability rather than listing and counting all the equally likely outcomes
geometric probability
a type of probability found by calculating a ratio of 2 lengths, areas, or volumes
quadratic function
graph of it is U-shaped and called a parabola ; one of the parent functions (f(x) = x^2) ; can be written in standard form, vertex form, or intercept form ; when solving an equation of one, you're finding the x-values when f(x) = 0 ; this makes up an ordered pair that represents the x-intercepts, which we call the zeros of the equation or roots of the function ; steps to solving it are 1) determine how many solutions you'll have by looking at the highest degree 2) set the equation equal to 0 3) factor it 4) use the zero product property 5) solve for the x-values ; when given the solutions for it, you're trying to get it in standard form ; some can be solved with square roots ; not all have real-number solutions (for example, x^2 = -1 has no real-number solutions because the square root of any real number x is never negative) ; not all can be factored, so you use completing the square in these cases and in any other ones
vertex
lowest (minimum) or highest (maximum) point on the graph of a quadratic function ; (h, k) in vertex form
x-coordinate
formula for it of the vertex is -b/2a ; in intercept form, the axis of symmetry
axis of symmetry
reflection flips a figure over this line ; also called a line of symmetry ; the vertical line x = -b/2a ; x-coordinate of the vertex ; when graphing a parabola, pick x's on the same side of it to plug in to the quadratic function ; x = h in vertex form ; 1/2 way between (p, 0) and (q, 0) in intercept form
x and y intercepts
the points where the parabola hits the x and y axes ; usually in ordered pairs
vertex form
y = a(x-h)^2 + k for a quadratic function ; vertex is (h, k) ; axis of symmetry is x = h
intercept form
y = a(x-p)(x-q) for a quadratic function ; x-intercepts are p and q ; axis of symmetry is 1/2 way between (p, 0) and (q, 0)
x-intercepts
p and q in intercept form ; change signs
zero product property
if ab = 0, then a = 0 or b = o ; a and b are real numbers or algebraic expresisons
root
another word for solution
square roots
some types of quadratic equations can be solved with them ; for example, if s > 0, then the quadratic equation x^2 = s has 2 real-number solutions, which are x = √s and x = -√s (these solutions are often written in the condensed form of x = ±√s, which is read as "plus or minus the square root of s") ; for one of a negative number, two properties are true, which are 1) if r is a positive real number, then √r = i√r 2) by property 1, it follows that (i√r)^2 = -r
i
to overcome the issue of not all quadratic equations having real-number solutions, mathematicians created an expanded system of numbers using this "imaginary unit" ; defined as it = √-1 ; it^2 = -1 ; can be used to write the square root of any negative number ; can't leave it in the denominator
complex number
a number a + bi where a and b are real numbers when written in standard form (the number a is the real part of the complex number ; the number bi is the imaginary part of the complex number) ; 2 of them a + bi and c + di are equal if and only if a = c and b = d ; to add or subtract 2 of them, add or subtract their real parts and their imaginary parts separately ; to find the quotient of 2 of them, multiply the numerator and the denominator by the complex conjugate of the denominator
complex conjugates
2 factors in the form a + bi and a - bi ; product of them is always a real number ; you can use them to write the quotient of 2 complex numbers in standard form (to find the quotient of 2 complex numbers, multiply the numerator and the denominator by this of the denominator)
imaginary number patterns
i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1
negative exponent
when given this, flip it and make it positive
completing the square
a process that allows you to write an expression of the form x^2 + bx as the square of a binomial ; to do it for x^2 + bx, you need to add (b/2)^2 (making x^2 + bx + (b/2)^2 = (x + b/2)^2 ; a method that lets you solve any quadratic equation ; you can use it to write a function in standard form in vertex form
magic number
(b/2)^2 for completing the square ; must be the same on both sides of the equation
quadratic formula
x = -b ± √b^2 -4ac / 2a ; a, b, and c are real numbers such that a doesn't equal 0
discriminant
in the quadratic formula, the expression b^2 - 4ac under the radical sign ; you can use it to determine the equation's number and type of solutions (1) if it's > 0, there's 2 real solutions 2) if it = 0, there's 1 real solution 3) if it's < 0, there's 2 imaginary solutions)
distance
it between the points (x1, y1) and (x2, y2) is d = √(x2-x1)^2 + (y2-y1 )^2 ; can be used to obtain an equation of the circle whose center is the origin and whose radius is "r" ; because it between any point (x, y) on the circle and the center (0, 0) is r, you can find it using the formula for it
midpoint
it of the line segment joining A(x1, y1) and B(x2, y2) is m = (x1 + x2/2, y1 + y2/2) ; each coordinate of it is the mean of the corresponding coordinates of A and B
circle
the set of all points (x, y) that're equidistant from a fixed point, called the center of the circle ; standard form of an equation of it with center (h, k) and radius "r" is (x-h)^2 + (y-k)^2 = r^2
center of a circle
a fixed point that all points in a circle are equidistant from
radius
the distance "r" between the center and any point (x, y) on the circle
ellipse
the set of all points P such that the sum of the distances between P and 2 distinct fixed points, called the foci, is constant ; can have a horizontal major axis or a vertical major axis with the center at the origin ; standard form of the equation of it with center at (h, k) and a > b is (x-h)^2/a^2 + (y-k)^2/a^2 = 1 for a horizontal major axis and (x-h)^2/b^2 + (y-k)^2/a^2 = 1 for a vertical major axis
focus/foci
2 distinct fixed points in an ellipse that're located on the major axis ; equation to find them is a^2 - b^2 = c^2 ; for a horizontal ellipse, (+- c, 0) on the x-axis ; for a vertical ellipse, (0, +-c) on the y-axis ; 2 fixed points in a hyperbola ; equation to find them is a^+ b^2 = c^2 ; lie on the traverse axis of a hyperbola, 2 units from the center ; 1 fixed point of a parabola ; line from it to a point on the parabola is the same as the line from the point to the directrix ; for a vertical parabola, (h, k+c) ; for a horizontal parabola, (h+c, k)
vertices
2 points that the line through the foci intersects the ellipse at ; for a horizontal ellipse, (+-a, 0) ; for a vertical ellipse, (0, +-a) ; 2 points that the line through the foci intersects the hyperbola at ; for a horizontal hyperbola, (+-a, 0) ; for a vertical hyperbola, (0, +-a)
major axis
the line segment joining the vertices of an ellipse ; length of it is 2a
center
the midpoint of the major axis of an ellipse ; the midpoint of the transverse axis of a hyperbola
co-vertices
2 points that the line perpendicular to the major axis at the center intersects the ellipse at ; for a vertical ellipse, (+- b, 0) ; for a horizontal ellipse, (0, +-b)
minor axis
the line segment that joins the co-vertices of an ellipse ; length of it is 2b
hyperbola
the set of all points P such that the difference of the distances from P to 2 fixed points, called the foci, is constant ; has 2 branches and 2 asymptotes ; standard form of the equation of it with center at (0, 0) is x^2/a^2 - y^2/b^2 = 1 for a horizontal one ; standard form of the equation of it with center at (0, 0) is y^2/a^2 - x^2/b^2 = 1 for a vertical one; when graphing it, can't touch the box or asymptotes
transverse axis
the line segment joining the vertices of a hyperbola
asymptotes
a hyperbola has 2 branches and 2 of them ; contain the diagonals of a rectangle centered at the hyperbola's center ; for a horizontal hyperbola, y = +-b/a x ; for a vertical hyperbola, y = +-a/b x
conjugate axis
the line segment with endpoints (+-b, 0) and (0, +-b) of a hyperbola
parabola
the set of all points in a plane that're the same distance from a fixed line and a fixed point not on the line ; the graph of a quadratic function that's U-shaped ; line from the focus to a point on it is the same as the line from the point to the directrix ; vertical ones (a > 0) open up and horizontal ones (a < 0) open down ; equation of a vertical one with the vertex at (h, k) is y = 1/4c(x-h)^2 + k ; equation of a horizontal one with the vertex at (h, k) is x = 1/4c(y-k)^2 + h ; wider than the graph of y = x^2 if |a| < 1 and narrower than the graph of y = x^2 if |a| > 1
directrix
the fixed line of a parabola ; line from the focus to a point on the parabola is the same as the line from the point to it ; for a vertical parabola, y = k-c ; for a horizontal parabola, x = h-c
focal length
the distance between the vertex and the focus of a parabola ; for a horizontal parabola, x = h-c
conic section
a curve you get by intersecting a plane and a double cone ; by changing the inclination of this plane, you get a circle, parabola, ellipse, or hyperbola ; Ax^2 + By^2 + Cx + Dy + E = 0 is the general form conic equation and can apply to all of them ; you can determine which one it is based on the following questions: 1) are both variables squared? (no --> it's a parabola ; yes --> next) 2) do the squared terms have opposite signs? (yes --> it's a hyperbola ; no --> next) 3) are the squared terms multiplied by the same number? (yes --> it's a circle ; no --> it's an ellipse)
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