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Algebra 2 EOI Vocabulary Epic Charter
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Terms in this set (168)
rational exponent
represents both an integer exponent and an nth root. The root is found in the denominator (like a tree, the root is at the bottom), and the integer exponent is found in the numerator.
fractional exponent
represents both an integer exponent and an nth root. The root is found in the denominator (like a tree, the root is at the bottom), and the integer exponent is found in the numerator.
product rule (exponent)
Multiplying numbers with the same base. Simply add exponent powers. Also applies to when variable bases are the same. Such as x . x
Quotient rule (exponent)
To divide two exponents with the same base, you keep the base and subtract the powers . (This is similar to reducing fractions ; when you subtract the powers put the answer in the numerator or denominator depending on where the higher power was located. If the higher power is in the denominator, put the difference in the denominator and vice versa, )
Negative exponent rule
negative exponents are moved to the other side of the fraction bar to make them positive
Power Rule
multiply the exponents when raising a power to a power
Expanded power Rule
If an exponent is outside of parentheses, the exponent is applied
to everything inside the parentheses. You must make sure to apply any other power rules in
combination with the expanded power rule.
Zero Exponent Rule
if anything is to a power of 0, the answer will be 1
One Exponent Rule
Any base raised to the 1 power is equal to itself.
square root
a number that when multiplied by itself equals a given number
radical symbol
the symbol used to denote a root
√
radicand
The number or expression inside a radical symbol
Product Property of Radicals
The square root of ab is equal to the square root of a times the square root of b.
Quotient Property of Radicals
The square root of a quotient equals the quotient of the square roots of the numerator and denominator
Rationalize the Denominator
a method used to rewrite an expression so there are no radicals in the denominator and no fractions within radicals
Sum Property of Radicals
When adding radicals, you may only combine radicals with the same index and exactly
the same quantity under the radical
Difference Property of Radicals
When subtracting radicals, you may only combine radicals with the same index and exactly
the same quantity under the radical
Radical
any expression containing a radical (√) symbol.
cube root
a number that when multiplied three times equals a given number
polynomial
A monomial or the sum of monomials
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
FOIL Method
to multiply two binomials, find the sum of the products of the First terms, the Outer terms, the Inner terms, and the Last terms
Polynomial Long Division
a method used to divide polynomials similar to the way you divide numbers
imaginary number
i = square root of (-1)
powers of i
i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
this pattern repeats for all following powers
complex numbers
numbers with real and imaginary components.
complex numbers addition
add the two real parts and add the two imaginary parts of the complex numbers
Note that adding two complex numbers is similar to combining like terms.
complex numbers subtraction
subtract the two real parts and subtract the two imaginary parts of the complex numbers.
Note that subtracting two complex numbers is similar to combining like terms.
complex number multiplication
multiply as if multiplying two binomials.
complex conjugate
(a+bi) and (a-bi)
complex number division
multiply the dividend and the divisor by the complex conjugate of the divisor.
parent function
The simplest function in a family; all functions in the family are transformations of it
parent graphs
The graphs of parent functions. Examples: absolute value, cubic, quadratic, square root, linear
quadratic parent function
The simplest quadratic function f(x)=x^2 or y=x^2.
cubic parent function
f(x) = x^3 , the graph is a curve that rises three times faster than is moves sideways
exponential parent function
Y=b^x
logarithmic parent function
y = log x
transformations
movements of geometric figures
polynomial translations
1) f(x - k) shifts the graph of f(x) to the right k units
2) f(x + k) shifts the graph of f(x) to the left k units
3) f(x) - k shifts the graph of f(x) down k units
4) f(x) + k shifts the graph of f(x) up k units
function
A relation where each input has exactly one output
function notation
an equation in the form of 'f(x)=' to show the output value of a function, f, for an input value x
use the symbol f(x) in place of y
domain
The set of input values of a function, usually represented by x
range
The set of output values of a function, usually represented by y
evaluate
To find the value of a mathematical expression.
composition of functions
A function is performed, and then a second function is performed on the result of the first function. The composition of f and g is denoted by f ◦ g, and [f ◦ g](x) = f[g(x)] .
interval notation
The set of numbers between, possibly including, two numbers. Numbers are enclosed between parenthesis ( ) and/or brackets [ ] -
Including [ ] not including ( )
inverse functions
functions created when the domain and range of a function are interchanged.
f(f-1(x)) = x, AND f-1(f(x)) = x
Steps to find inverse of a function
1) Change f(x) to y.
2) Change the x's to y's and the y to an x.
3) Solve for y.
4) Change y to f^ -1(x).
infinity
limitless
system of equations
A set of two or more equations with the same variables.
system of inequalities
A set of two or more inequalities with the same variables
salary
A fixed amount of money paid to an employee for each pay period.
commission
An amount paid to an employee based on a percentage of the employee's sales
Substitution Method of Solving Systems of Equations
A method of solving a system of equations by replacing one variable with an equivalent expression containing the other variable.
Elimination Method of Solving Systems of Equations
solving systems by adding or subtracting equations to eliminate a variable
Graphing Method of Solving Systems of Equations
to graph both equations in a problem and the solution is the intersect of the lines
boundary line
The line that divides a plane into two half-places, ex: when graphing the inequality y <= 2x + 4 the boundary line is the graph y = 2x + 4. The boundary line may or may not be part of the solution to an inequality.
coordinate system
Also known as a coordinate grid. A 2-dimensional system in which the coordinates of a point are its distances from two intersection usually perpendicular, straight lines called axes.
coordinate pair
A pair of numbers that gives the coordinates of a point on a grid in this order (horizontal coordinate, vertical coordinate). Also know as an ordered pair.
input
A value of the independent variable.
laws of exponents
The theorem stating the elementary properties of exponents.
line of best fit
a straight line that best represents the data on a scatter plot. This line may pass through some of the points, none of the points, or all of the points. Sometimes called a trend line.
linear function
Functions that are a first-degree polynomial of one variable. The graph of the function is a line.
output
A value of the dependent variable.
rate of change
The ration of the change in the output value and change in the input value of the a function.
slope formula
The forumula used to find the slope of a line. Slope is often represented with the variable m.
slope
describes the steepness, incline, or grade of a line. A higher slope value indicates a steeper incline. The slope of a line is the ratio of the change in y over the change in x.
x-axis
In a Cartesian grid, the horizontal axis.
x-coordinate
In an ordered pair, the value that is always written first.
x-intercept
The point at which a function crosses the x-axis.
y-axis
In a Cartesion grid, the vertical axis.
y-coordinate
In an ordered pair, the value that is always written second.
y-intercept
The point at which a function crosses the y-axis.
quadratic equation
an equation that can be written in the form ax² + bx + c = 0, where a ≠ 0
roots of an equation
the solution set of an equation
The x-intercepts of an equation
factoring
a process used to write a polynomial as a product of other polynomials having equal or lesser degree
quadratic forumla
Can be used to solve a quadratic equation
completing the square
the process of converting a quadratic equation into a perfect square trinomial by adding or subtracting terms on both sides
Standard Form of a Quadratic Equation
y=ax²+ bx + c, where a ≠ 0.
If a < 0, the graph of the parabola opens downwards.
If a > 0, the graph of the parabola opens upwards.
vertex of a parabola
the lowest point (minimum) of the parabola if it opens upwards and the highest point (maximum) of the parabola if it opens downwards.
The lowest or highest point on a parabola. It lies on the axis of symmetry.
maximum of a parabola
what the highest point or vertex is called when a parabola opens downwards
y-coordinate of the vertex
minimum of a parabola
what the lowest point or vertex is called when a parabola opens upwards
y-coordinate of the vertex
leading coefficient
The coefficient of the term with the highest degree
vertex form of a quadratic equation
y=a(x-h)²+k with vertex at (h, k)
If a < 0, the parabola opens downwards.
If a > 0, the parabola opens downwards.
axis of symmetry
the line which passes through the vertex of a parabola and divides the parabola into two congruent halves.
formula for x-coordinate for the vertex of the parabola
For a quadratic equation in the form y = ax2 + bx + c, with a ≠ 0, the following formula gives the x-coordinate of the vertex of the parabola.
To find the y-value of the vertex, substitute the x-coordinate back into the original equation and solve for y.
quadratic model
quadratic function that represents a real data set
Conic Sections
four shapes, parabolas, circles, ellipses, and hyperbolas that are formed by the intersection of a plane and a double cone
Ellipse
A elongated circle, or oval shape, where the set of all points (x, y) in a plane such that the sum of the distance from (x, y) to one focus and the distance from (x, y) to the other focus is constant.
focus of an ellipse
One of the two points that can be used to define an ellipse. For every point on an ellipse, the distance from the point to one focus, plus the distance from the point to the other focus, is equal to some constant value. Another name for a focus is a focal point. The plural of focus is foci
vertex of an ellipse
the points where an ellipse intersects its major axes
major axis of an ellipse
The long axis of an ellipse connecting the vertices.
minor axis of an ellipse
The short axis of an ellipse. It is perpendicular to the major axis.
standard equation of an ellipse
shown in the picture with 0 < b < a and center (h, k).
Hyperbola
an open curve formed by a plane that cuts the base of a right circular cone, where set of all points (x, y) in a plane such that the difference of the distance from (x, y) to one focus and the distance from (x, y) to the other focus is constant.
Focus of a Hyperbola
One of the two points that can be used to define a hyperbola. For every point on a hyperbola, the distance from the point to one focus, minus the distance from the point to the other focus, is equal to some constant value. The plural of focus is foci. Another name for a focus is a focal point.
Vertex of a Hyperbola
The point on each branch of a hyperbola that is closest to the other branch of the hyperbola. The plural of vertex is vertices. The vertices are the endpoints of the transverse axis, which is a line of symmetry for a hyperbola.
Transverse Axis of a Hyperbola
line segment with the hyperbola's vertices as its endpoints.
Center of a Hyperbola
The midpoint of the transverse axis
Asymptotes of a Hyperbola
Lines that a hyperbola approaches but does not intersect. They intersect at the center of the hyperbola.
horizontal: y = ± b/a vertical: y = ± a/b
Standard Equation of a Hyperbola
The Equation with center (h, k) is given in the picture
Parabola
set of all points in a plane that are an equal distance from both a fixed point, the focus, and a fixed line, the directrix.
Focus of a Parabola
A point that can be used to define a parabola. The distance from any point on the parabola to the focus is the same as the distance from that point to a line called the directrix. Another name for the focus is the focal point.
horizontal: (h+p,k) vertical: (h,k+p)
Directrix of a Parabola
a fixed line that is equidistant from the vertex as the focus is to the vertex
horizontal: x = h-p vertical: y = k-p
Direction a Parabola Opens
If a < 0, the parabola opens downwards.
If a > 0, the parabola opens downwards.
Circle
The set of all points in a plane that are the same distance from a given point called the center
Circle Equation
(x-h)²+(y-k)²=r² , where (h, k) is the center of the circle and r is the radius of the circle.
Exponential Function
a function of the form f(x) = ab^x, where b is a positive real number not equal to one. The function intersects the y-axis at (0, a) and b is referred to as the function's base. The function has a horizontal asymptote at y = 0. The inverse of a logarithmic function.
Exponential Function's Base
b in f(x) = ab^x
Exponential Function Parent Graph
horizontal asymptote at y = 0 and intersects the y-axis at (0, a)
Properties of an Exponential Function's Graph
1)If c > 0, the graph of g(x) is the graph of f(x) = ab^x shifted c units to the right. If c < 0, the graph of g(x) is the graph of f(x) = ab^x shifted c units to the left.
2) If k > 0, the graph of g(x) is the graph of f(x) = ab^x shifted k units up. If k < 0, the graph of g(x) is the graph of f(x) = ab^x shifted k units down.
3) The graph has a horizontal asymptote at y = k.
Exponential Equation
See picture
Logarithmic Function
f(x) = LOGb(x - q) + r, where q and r are real numbers and b is a positive real number not equal to 1, has the properties listed below.
The inverse of an exponential function.
Logarithmic Graph Properties
• The graph has a vertical asymptote at x = q.
• As x continues to increase, y continues to increase; therefore, there is no horizontal asymptote.
• If q is positive, the graph of f(x) = logb(x) is shifted q units to the right. If q is negative, the graph of f(x) = logb(x) is shifted q units to the left.
• If r is positive, the graph of f(x) = logb(x) is shifted r units up. If r is negative, the graph of f(x) = logb(x) is shifted r units down.
Logarithmic Parent Graph
Graph of f(x) = log(x)
Data best represented by a Linear Models
The difference of the dependent variable is constant
Data best represented by a Quadratic Models
The second difference of the dependent variable is constant
Data best represented by a Exponential Models
Data that has a common ratio in the values of the dependent variable.
Data best represented by a Logarithmic Models
Data that as x increases, the amount by which y is increasing decreases.
Independent Variable
The experimental factor that is manipulated; the variable whose effect is being studied.
Typically goes on the Horizontal Axis. Also called the input or x variable
Dependent Variable
The outcome factor; the variable that may change in response to manipulations of the independent variable.
Typically goes on the Vertical Axis. Also called the output or y variable.
Logarithmic Properties
helpful in solving logarithmic equations.
Factoring a Quadratic
A quadratic equation in the form ax^2 + bx + c can be rewritten as a product of two factors called the "factored form". This form resembles (x + ?)(x + ?)
Difference of Two Squares
a² - b² = (a + b)(a - b)
Perfect Square Trinomials
a trinomial in the form a²+2ab+b² = (a+b)² or a²- 2ab+b² = (a-b)²
Factor by Grouping
A method of factoring that uses the distributive property to remove a common binomial factor of two pairs of terms.
1) Arrange the terms from highest order to lowest order.
2) Group the first two terms together and then the last two terms together.
3) Factor out a GCF from each separate binomial.
4) Factor out the common binomial.
FOIL method
to multiply two binomials, find the sum of the products of the First terms, the Outer terms, the Inner terms, and the Last terms
zero of a function
An x-intercept of the graph of a function. Also called a solution or root.
zero product property
If the product of two factors is zero, then at least one of the factors must be zero. Ex: If ab = 0, then a = 0 or b = 0.
Polynomial Division
follows same pattern as regular long division; used to divide polynomials by their roots
Graphing Polynomials
1) determining end behavior
2) determining x- and y-intercepts
3) determining relative maximums and relative minimums
Determine End Behavior of a Polynomial
use the following chart
Calculate x-intercept
set y equal to zero and solve for x.
Calculate y-intercept
set x equal to zero and solve for y
relative minimum
The lowest point in a particular section of a graph. Also called local minimum
relative maximum
the highest point in a particular section of a graph. Also called local maximum
turning point of a graph
a point at which the graph changes from increasing to decreasing or from decreasing to increasing. It corresponds to a relative maximum or a relative minimum. A function of degree n can have at most n - 1 turning points.
rational function
a function that can be written as a fraction, where the numerator and denominator are polynomials and whose denominator is not zero
Rational Function Parent Graph
graph touches neither the x-axis nor the y-axis.
Vertical Asymptote
a vertical line that a graph cannot cross.
Calculate Vertical Asymptote
To calculate set the denominator equal to 0 and solve for x.
Horizontal Asymptote
a horizontal line that a graph cannot touch
Calculate Horizontal Asymptote
If the degree of C(x) is greater than the degree of D(x), then there is no horizontal asymptote.
If the degree of C(x) is less than the degree of D(x), then the horizontal asymptote is y = 0.
If the degree of C(x) is equal to the degree of D(x), then the horizontal asymptote is the line defined by the ratio of leading coefficients
Solve Rational Equations
1) Cross multiply when one ratio is set equal to another.
2) Another method is to multiply both sides of the rational equation by the LCD of the denominators in the equation.
3) Check your solutions to make sure that they are not extraneous solutions.
extraneous solutions
a solution of an equation derived from an original equation but it is not a solution of the original equation.
Scatter Plots
A graph with points plotted to show a possible relationship between two sets of data.
Positive correlation
A correlation where as one variable increases, the other also increases, or as one decreases so does the other. Both variables move in the same direction.
trend line
a line on a scatter plot drawn near the points. it shows a correlation
strong correlation
that the two variables' pattern is very closely related
weak correlation
the plotted points are spread out and don't lie close together and
suggests that one variable changing does not necessarily cause the other variable's change
Curve of Best Fit
A curve that best approximates the trend on a scatter plot. It can be used to make inferences or predictions.
arithmetic sequence
a sequence in which each term is found by adding the same number to the previous term
Arithmetic Sequence Formula
a(n)= first term + (number of terms - 1)difference
arithmetic series
the indicated sum of the terms of an arithmetic sequence. where a1 is the first term and an is the nth term in the sequence.
geometric sequence
a sequence in which each term is found by multiplying the previous term by the same number
geometric sequence Formula
The nth term of a geometric sequence, an, is given with the following equation, where a1 is the first term of the sequence, r is the common ratio, and r doesn't equal 0 or 1.
Geometric series
The sum of n terms of a geometric sequence. Where a1 is the first term and r is the common ratio.
Natural Logarithm
A logarithm with base e. It is written ln x. The natural logarithm is the power of e necessary to equal
delta
Greek Letter that means the difference or change in a certain quantity
discriminant
The expression under the radical in the quadratic formula,
b²-4ac, used to find the number and type of solutions of a quadratic equation.
matrix
rectangular array of numbers or other mathematical objects
absolute minimum
A function where if there is a point on its graph that has the y-corrdinate that is lower than every other point in the graph. Also called global minimum.
absolute maximum
The y-value of a point on a graph that is higher than any of the other points on the entire graph. Also called global maximum.
sigma
Σ upper case sigma is used for the summation notation and the lower case sigma stands for standard deviation
standard deviation
A measure of variability that describes an average distance of every score from the mean.
synthetic division
A shortcut for polynomial long division
It is a method used to divide any polynomial by a divisor of the form (x-k)
variance
A measure of spread within a distribution (the square of the standard deviation).
weighted averages
Is an average of values of a set of items to each of which is accorded a weight in dilative of it's frequency or relatively importance
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