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Algebra 2 Regents Exam Review 2
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Gravity
Terms in this set (235)
Sequence
an ordered list of numbers
Formally defined as a function that has as its domain the set the set of positive integers
f(1) or a₁
1st term
previous term
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)
d
common difference
(the difference between any two consecutive terms)
r
common ratio
(the ratio between any two consecutive terms)
Recursive formula for an Arithmetic Sequence
Explicit formula for an Arithmetic Sequence
Recursive formula for an Geometric Sequence
Explicit formula for an Geometric 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
Series
the sum of the terms of a sequence
Arithmetic Series Formula
Geometric Series Formula
Horizontal Asymptote
A horizontal line the graph approaches
Domain
set of all inputs
x's
Range
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
where b > 1
Exponential Regression
fitting data to an exponential function
Performing an Exponential Regression on the TI-Nspire
On the scatter plot page:
menu, 4, 6, 8
(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
5⁰ = 1
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
Product Log Law
Quotient Log Law
Power Log Law
Exponential Function Graph
Logarithmic Function Graph
Inverse Functions
switch the x and y
(symmetric with the line y = x)
Common Log
base 10
y = log(x)
Natural Log
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 ...
clockwise
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
sine
sec(θ) is ____________'s reciprocal
cosine
cot(θ) is _____________'s reciprocal
tangent
tan(θ) =
sine graph
cosine graph
tangent graph
The Pythagorean Identity
Putting TI-Nspire in Degree Mode
Period
minimum distance along the x-axis for the cycle to repeat
Frequency
B (how many cycles in 2π radians)
Amplitude
|A| (distance the sinusoidal model rises and falls above and below its midline)
Midline
C (average y-value of the sinusoidal model)
Amplitude Formula
Midline Formula
Sinusoidal
Any graph primarily comprised of either the sine or cosine function
Range
[min, max]
Maximum of a sinusoidal function
midline + amplitude
Minimum of a sinusoidal function
midline - amplitude
cosine is an _______ function
even
sine and tangent are ________ functions
odd
Vertical Asymptote
a vertical line a graph is approaching (formed when a function is undefined)
sin(π)
0
cos(2π)
1
sin(3π/2)
-1
cos(π/3)
½
sin(7π/6)
-½
cos(11π/6)
√3/2
sin(5π/3)
-√3/2
cos(π/4)
√2/2
sin(5π/4)
-√2/2
tan(π/3)
√3
tan(2π/3)
-√3
tan(π/6)
√3/3
tan(5π/6)
-√3/3
tan(π/2)
undefined
Rational Functions
ratio of polynomial functions
Undefined Fraction
when the denominator of a fraction is 0
Inverse Function
switch the x and y
-1
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
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.
No Solutions
∅ or { }
Cube Root
One of three equal factors
Square Root
One of two equal factors
Radicand
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
Rationalizing
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 Number
Complex Conjugates
Discriminant
Sum of the Roots
Product of the Roots
If b²-4ac < 0, then...
Conjugate, Imaginary Roots
If b²-4ac = 0, then...
Equal, Real Roots
If b²-4ac > 0 and a perfect square, then...
Unequal, Real, Rational Roots
If b²-4ac > 0 and a non-perfect square, then...
Conjugate, Real, Irrational Roots
y = f(x) + k
Vertical shift up k units
y = f(x) - k
Vertical shift down k units
y = f(x + k)
Horizontal shift left k units
y = f(x - k)
Horizontal shift right k units
y = -f(x)
Reflection over the x-axis
y = f(-x)
Reflection over the y-axis
If f(-x) = f(x) then the function is ...
even
If f(-x) = -f(x) then the function is ...
odd
Even functions are symmetric to the ...
y-axis
Odd functions are symmetric to the ...
origin
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...
cbad
The graph of the parent function y = |x|
The graph of the parent function y = x²
Parabola
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
GCF Factoring
3x²+6x = 3x(x + 2)
Conjugates
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 (Guess and Check / AC-Split the Middle)
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
Locus Definition of a Parabola
Collection of all points equidistant from a fixed point (focus) and a fixed line (directrix).
Equation of a Parabola using the Locus Definition
y - k = 1/4p (x - h)²
(h, k) is the vertex
p is ½ the distance between the focus and the directrix
Direct Variation
When two variables, x and y, have the following relationship:
y = kx or y/x = k
Average Rate of Change
Slope
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
No Solutions
∅ or { }
Cubic Polynomial Function
Degree 3
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
Degree 4
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.
x = 3
Factor and Roots: If (2x + 1) is a factor, then ______________ is a root.
x = -1/2
Identity
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.
End Behavior
The degree is even, and the leading coefficient is negative.
The degree is even, and the leading coefficient is positive.
End Behavior
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
Degree 2
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
Function
A rule that assigns exactly one output for each input.
Domain
Set of inputs
Range
Set of outputs
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.
x-intercept
when y = 0
known as roots or zeros
y-intercept
when x = 0
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
Probability Formula
Complement
all of the events in the sample space that do not fall into a particular set
Mutually Exclusive
When 2 events cannot occur at the same time
Sample Space
A collection of all of the outcomes of an experiment
Theoretical Probability
Probability can be found where you don't actually have to run the experiment
Tree Diagram
Organizational tool to represent a sample space
Empirical Probability
Probability can be found by using observations that have been made in the real-world
Venn Diagram
Organizational tool to represent a sample space
(elements are shown in their own circle within an enclosing rectangle (the universal set). Common elements between multiple sets are represented by the areas of overlap among the circles)
Sets in roster form
elements are listed numerically without repetition in braces { }
Complement Probability Formula
Union
OR
Symbol ∪
Intersection
AND
Symbol ∩
Probability Addition Rule
Conditional Probability Formula
Product Test for Independence
Conditional Test for Independence
Probability of an event is always between...
0 and 1
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