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Transforming functions: horizontal shifts 

Apply fundamental counting principle to solve problems

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Fundamental counting principle

Fundamental Counting Principle Involving Restrictions

Determine number of permutations of n elements taken r at a time to solve problems

Factorial notation

Path counting problems

Permutations

Determine number of combinations of n different elements taken r at a time to solve problems

Combinations

Problems involving both permutations and combinations

Expand powers of a binomial using binomial theorem

Pascal's triangle

Pascal's Triangle

Pascal"s Triangle

Pascal"s triangle

Binomial theorem

Demonstrate understanding of operations on, and compositions of, functions

Function notation (advanced)

Composite functions

Adding functions

Subtracting functions

Multiplying functions

Dividing functions

Operations of Functions – In a Nutshell

Operations with functions

Inequalities of combined functions

Inequalities of Combined Functions

Demonstrate understanding of effects of horizontal and vertical translations on function graphs and equations

Transformations of functions: Horizontal translations

Transformations of functions: Vertical translations

Demonstrate understanding of effects of horizontal and vertical stretches on function graphs and equations

Transformations of functions: Horizontal stretches

Transformations of functions: Vertical stretches

Apply translations and stretches to graphs and equations of functions

Combining transformations of functions

Demonstrate understanding of effects of reflections on function graphs and equations

Reflection across the y-axis: y = f(-x)

Reflection across the x-axis: y = -f(x)

Even and odd functions

Demonstrate understanding of inverses of relations

One to one functions

Inverse functions

Demonstrate understanding of logarithms

What is a logarithm?

Logarithms and Exponential Growth

Understanding Logarithms through Exponential Growth

Understanding Logarithms

Converting from logarithmic form to exponential form

Evaluating logarithms without a calculator

Common logarithms

Natural log: ln

Evaluating logarithms using change-of-base formula

Converting from exponential form to logarithmic form

Demonstrate understanding of product, quotient, and power laws of logarithms

Product rule of logarithms

Quotient rule of logarithms

Combining product rule and quotient rule in logarithms

Evaluating logarithms using logarithm rules

Graph and analyze exponential and logarithmic functions

Graphing exponential functions

Graphing transformations of exponential functions

Finding an exponential function given its graph

Graphing logarithmic functions

Finding a logarithmic function given its graph

Solve problems involving exponential and logarithmic equations

Solving logarithmic equations

Solving exponential equations with logarithms

Solving exponential equations using exponent rules

Demonstrate understanding of factoring polynomials of degree greater than 2

Factoring difference of cubes

Factoring sum of cubes

Polynomial long division

Polynomial synthetic division

Polynomial functions

Recognizing a Polynomial Function

Remainder theorem

Finding the Remainder Using Synthetic Division and the Remainder Theorem

Factor theorem

Using the Factor Theorem to Test For Factors of a Polynomial

Rational zero theorem

Rational Zero Theorem

Characteristics of polynomial graphs

Factor by taking out the greatest common factor

Factor by taking out GCF

Factor by grouping

Factoring difference of squares: \(x^2 - y^2\)

Factoring difference of squares: <katex>x^2 - y^2</katex>

Factoring difference of squares: x^2 - y^2

Factoring trinomials

Factoring Trinomials

Multiplicities of polynomials

Imaginary zeros of polynomials

Graph and analyze polynomial functions

Characteristics of quadratic functions

Transformations of quadratic functions

Quadratic function in general form: y = ax^2 + bx + c

Converting from general to vertex form by completing the square

Shortcut: Vertex formula

Converting general form into vertex form by applying the vertex formula

Vertex Formula Application

Graphing quadratic functions: General form VS. Vertex form

Finding the quadratic functions for given parabolas

Applications of quadratic functions

Graph and analyze radical functions

Radical functions and transformations

Square root of a function

Solving radical equations

Solving Radical Equations Involving One Square Root Term

Solving Radical Equations Involving More than One Square Root Term

Basic radical functions

Basic Radical Function

Basic Radical Functions

Graph and analyze rational functions

Slant asymptote

Determining the Equation of a Slant Asymptote Using Long Division

Determining the Equation of a Slant Asymptote Using Synthetic Division

Graphing reciprocals of linear functions

Vertical asymptote

Horizontal asymptote

Horizontal Asymptote

Graphs of rational functions

Graphing reciprocals of quadratic functions

Demonstrate understanding of angles in standard position, expressed in degrees and radians

Angle in standard position

Coterminal angles

Coterminal Angles

Reference angle

Converting between degrees and radians

Trigonometric ratios of angles in radians

Develop and apply the equation of the unit circle

Unit circle

Solve problems using six trigonometric ratios for angles expressed in radians and degrees

Find the exact value of trigonometric ratios

ASTC rule in trigonometry (All Students Take Calculus)

ASTC rule in trigonometry

Radian measure and arc length

Graph and analyze trigonometric functions sine, cosine, and tangent to solve problems

Sine graph: y = sin x

Cosine graph: y = cos x

Tangent graph: y = tan x

Graphing transformations of trigonometric functions

Determining trigonometric functions given their graphs

Solve algebraically and graphically first and second-degree trigonometric equations

Solving first degree trigonometric equations

Solving second degree trigonometric equations

Solving trigonometric equations involving multiple angles

Determining non-permissible values for trig expressions

Solving trigonometric equations using pythagorean identities

Solving trigonometric equations using Pythagorean identities

Solving trigonometric equations using sum and difference identities

Solving trigonometric equations using double-angle identities

Prove trigonometric identities using reciprocal, quotient, Pythagorean, sum or difference, double-angle identities

Quotient identities and reciprocal identities

Pythagorean identities

Sum and difference identities

Sum and Difference Identities

Double-angle identities

Cofunction identities

Pre-Calculus 12

Permutations, Combinations, and Binomial Theorem

Relations and Functions

math

Ex. 1:

Ex. 2:

Horizontal translations refer to movements of a graph of a function horizontally along the x-axis by changing the x values. So, if y = f(x), then y = (x &ndash;h) results in a horizontal shift. If h > 0, then the graph shifts h units to the right; while If h < 0, then the graph shifts h units to the left.

Completing the square