Algebra 2 Regents Exam Review

Terms in this set (206)

an ordered list of numbers

Formally defined as a function that has as its domain the set the set of positive integers

recursive formulas

terms of a sequence are found by performing operations on previous terms

explicit formulas

terms of a sequence are found by using the term's index (position)

common difference
(the difference between any two consecutive terms)

common ratio
(the ratio between any two consecutive terms)

Explicit formula for an Arithmetic Sequence

Geometric Sequence

based on constant multiplying to get the next term
(×/÷ pattern)

Arithmetic Sequence

based on constant
addition to get the next term
(+/- pattern)

Summation (Sigma) Notation

the sum of the terms of a sequence

Arithmetic Series Formula

Geometric Series Formula

Horizontal Asymptote

A horizontal line the graph approaches

set of all inputs
x's

set of all outputs
y's

Basic Form of an Exponential Function

where a is the y-intercept and b is the base (multiplier)

Decreasing Exponential Function

where 0 < b < 1

Increasing Exponential Function

(x²)(x³) = x⁵

Example of Exponent Law 1

x⁸÷x⁵ = (x⁸)/(x⁵) = x³

Example of Exponent Law 2

4⁻² = 1/4² = 1/16

Example of Exponent Law 3

Example Of Exponent Law 4

(x³)⁵ = x¹⁵

Example Of Exponent Law 5

(2xy²)³ = 8x³y⁶

Example Of Exponent Law 6

(3/x⁴)³ = 27/x¹²

Example Of Exponent Law 7

25 ^ ½ = √25 = 5

Example Of Exponent Law 8

4^ 3/2 = (√4)³ = 2³ = 8

Example Of Exponent Law 9

vertical asymptote

A vertical line the graph approaches (formed when a function is undefined)

Exponential Growth and Decay Model

Solving an Exponential Equation Using Logs

Exponential Function Graph

Logarithmic Function Graph

Inverse Functions

switch the x and y
(symmetric with the line y = x)

base 10
y = log(x)

base e
y = ln(x)

Compound Interest Formula

Continuous Compound Interest Formula

Half-Life Formula

Method of Common Bases

(1) Find a common base
(2) Rewrite each expression using the common base
(3) Use the power exponent law to simplify any expression
(4) Set exponents equal and solve

Trig Facts (Unit Circle)

Equation of Unit Cirlce

x² + y² =1
center (0, 0)
radius = 1

sin(θ) in the unit circle is always...

the y-coordinate

cos(θ) in the unit circle is always...

the x-coordinate

Reference Angle

The positive acute angle formed by the terminal ray and the x-axis.

Coterminal Angles

Any two angles drawn in standard position that share a terminal ray.

positive angles are drawn...

counter-clockwise

negative angles are drawn ...

Angle drawn in Standard Position

vertex is at the origin and its initial ray points along the positive x-axis.

Converting from Radians to Degrees and Degrees to Radians

The radian angle, θ, created by rotating an arc length of s in a circle with radius r is

sin(θ) is positive when θ lies in

Quadrant I and II

cos(θ) is positive when θ lies in

Quadrant I and IV

tan(θ) is positive when θ lies in

Quadrant I and III

csc(θ) is ____________'s reciprocal

sec(θ) is ____________'s reciprocal

cot(θ) is _____________'s reciprocal

The Pythagorean Identity

Putting TI-Nspire in Degree Mode

minimum distance along the x-axis for the cycle to repeat

B (how many cycles in 2π radians)

|A| (distance the sinusoidal model rises and falls above and below its midline)

C (average y-value of the sinusoidal model)

Amplitude Formula

Midline Formula

Any graph primarily comprised of either the sine or cosine function

Maximum of a sinusoidal function

midline + amplitude

Minimum of a sinusoidal function

midline - amplitude

cosine is an _______ function

sine and tangent are ________ functions

Vertical Asymptote

a vertical line a graph is approaching (formed when a function is undefined)

Rational Functions

ratio of polynomial functions

Undefined Fraction

when the denominator of a fraction is 0

Inverse Function

switch the x and y

Simplifying a Fraction (simplest rational form)

finding all common factors of the numerator and denominator and dividing them out

Division of Rational Expressions

multiplying the first fraction by the reciprocal of the second fraction (keep-change-flip)

Multiplication of Rational Expressions

factor first (if needed), reduce and/or cross reduce, then multiply their respective numerators and denominators

Adding/Subtracting Rational Expressions

find a common denominator - make new fractions - add/subtract the numerators - keep the denominator - reduce if needed

Complex Fraction

fractions (major) that have other fractions (minor) within their numerators and/or denominators

Solving a Proportion

when 2 fractions are equal to each other; cross multiply

"Clearing" Denominators in an Equation

find the LCD (least common denominator), and multiply both sides of the equation by the LCD

least common denominator (the lowest common multiple, LCM, of all the denominators)

Solving a Rational Inequality

Step 1: Find the values of x that satisfy the equation.

Step 2: Find the values of x that make any denominators equal to zero

Step 3: Plot these critical values on a one-dimensional
number line and test the different regions in the original inequality to determine the solution set.

Graph of a Square Root Function

Solving a Square Root Equation

Extraneous Solutions

Solutions that are introduced by various algebraic techniques that for one reason or another are not valid solutions of the original equations.

One of three equal factors

One of two equal factors

Expression under the radical

Simplest Radical Form

When the radicand has no factor that can be raised to a power greater than or equal to the index and there is no radical in the denominator. In other words, the number in the radical cannot have "perfect root" factors.

Radical Conjugates

Multiplying radical conjugates eliminates the radicals and results in a rational number

Solving a Quadratic Equation by the Square Root Method

Solving a Quadratic Equation by Completing the Square

Quadratic Formula

Imaginary Number

Complex Conjugates

Sum of the Roots

Product of the Roots

If b²-4ac > 0 and a perfect square, then...

Unequal, Real, Rational Roots

If b²-4ac = 0, then...

Equal, Real Roots

If b²-4ac > 0 and a non-perfect square, then...

Conjugate, Real, Irrational Roots

Vertical shift up k units

Vertical shift down k units

Horizontal shift left k units

Horizontal shift right k units

Reflection over the x-axis

Reflection over the y-axis

If f(-x) = f(x) then the function is ...

If f(-x) = -f(x) then the function is ...

Even functions are symmetric to the ...

Odd functions are symmetric to the ...

y = k f(x), where k > 1

Vertical stretch by k

y = k f(x) where 0 < k < 1

Vertical compression by k

y = f(kx) where k > 1

Horizontal compression by k

y = f(1/k x) where k > 1

Horizontal stretch by k

When y = a f (b(x + c)) + d, transformations are applied in this order...

The graph of the parent function y = |x|

The graph of the parent function y = x²

Graph of a Quadratic Function

Leading Coefficient

In f(x)=ax²+bx+c, the a is referred to as the ...

Axis of Symmetry

Maximum Vertex (turning point)

In f(x)=ax²+bx+c , if a > 0, the f(x) has a ...

Minimum Vertex (turning point)

In f(x)=ax²+bx+c , if a < 0, the f(x) has a ...

Root / x-intercept / zero

Soluiton to the equation f(x) = 0.

Factor (the verb)

To rewrite a quantity as an equivalent product

3x²+6x = 3x(x + 2)

Opposite Binomials
Example: (x + 5)(x - 5)

Conjugate Pair Factoring

4x²-25 = (2x + 5)(2x - 5)

Complete Factoring

Factoring an expression until it cannot be factored anymore.
Example: 40-250x² = 10(4 - 25x²) =
10(2 + 5x)(2 - 5x)

Trinomial Factoring and Guess & Check

x² + 2x - 35 = (x + 7)(x - 5)
2x² + 5x - 6 = (4x - 3)(x + 2)

Zero Product Law

If the product of multiple factors is equal to zero then at least one of the factors must be zero.

Solving a System of Equations Graphically

Find the points of intersection /
You are solving for both x and y

Completed Square Form / Vertex Form

y = a(x - h)² + k
(h, k) is the vertex

Locus Definition of a Circle

The collection of all points equidistant from a given point.

Equation of a Circle in Center-Radius Form

(x - h)² + (y - k)² = r²
(h, k) is the center
r is the radius

Direct Variation

When two variables, x and y, have the following relationship:
y = kx or y/x = k

Average Rate of Change

Slope-Intercept Form of a Line

Point-Slope Form of a Line

Inverses of Linear Functions

Switch the x and y
Reflection in the line y = x
Notation: y = f⁻¹(x)

Piece-wise Linear Function

A function defined by several linear functions over different domain intervals.

How to solve: |-2x+7|=25

Absolute Value Function

Elimination and Substitution Method

Ways to solve a system of linear equations

x = 7
y = 3
z = -2

Solution to:
2x + y + z = 15
6x - 3y - z = 35
-4x + 4y - z = -14

Cubic Polynomial Function

Degree of a polynomial

Highest power of a polynomial

Even Power Functions

Symmetric to the y-axis and have an even exponent

Odd Power Functions

Symmetric to the origin and have an odd exponent

Quartic Polynomial Function

The degree of a polynomial function gives...

The maximum number of roots a polynomial function can have.

Factor and Roots: If (x - 3) is a factor, then ______________ is a root.

Factor and Roots: If (2x + 1) is a factor, then ______________ is a root.

An equation that is true for all values of the replacement variable or variables.

Let p(x) be a polynomial function
and p(8)=0, then ...

(x - 8) is a factor and x = 8 is a root.

The degree is odd, and the leading coefficient is negative.

The degree is odd, and the leading coefficient is positive.

The degree is even, and the leading coefficient is negative.

The degree is even, and the leading coefficient is positive.

Remainder Theorem

When the polynomial p(x) is divided by the linear factor (x - a) or (bx - a), then the remainder will always be p(a) or p(b/a).

Let p(x) be a polynomial function
and p(-2)=12, then ...

When p(x) is divided by (x + 2), the remainder will be 12.

Let p(x) be a polynomial function
and p(2)=12, then ...

When p(x) is divided by (x - 2), the remainder will be 12.

Quotient Remainder Form

q(x) is the quotient, r is the remainder, and (x - a) is the divisor

Quadratic Polynomial Function

Let p(x) be a polynomial function
and p(2/3)=0, then ...

(3x - 2) is a factor and x = 2/3 is a root

A rule that assigns exactly one output for each input.

Composition of Functions

The output of one function becomes the input to another function. Notation: f(g(x))

One-to-One Functions

A function where each input has only one output AND each output has only one input.

Inverse Functions

Functions that literally "undo" one another. Created by switching the x and y.

A function is increasing when...

the y-values are going up.

A function is decreasing when ...

the y-values are going down.

A function is positive when...

the y-values are positive (above the x-axis)

A function is negative when...

the y-values are negative (below the x-axis)

Vertical Line Test

A line test used to determine if a relation is a function.

Horizontal Line Test

A line test used to determine if a function is one-to-one.

when y = 0
known as roots or zeros

Relative Max and Mins

high and low points on a graph and/or endpoints

Absolute Max and Mins

the highest and lowest point on a graph

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