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**Operations on Functions and Analyzing Graphs**

College Algebra Chapter 3 Operations on Functions and Analyzing Graphs

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**Sums and Differences of Functions**

College Algebra Chapter 3.1 The algebra and composition of functions Sums and Differences of Functions For functions f and g with domains of P and Q respectively, the sum and difference of f and g are defined by:

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**Sums and Differences of Functions**

College Algebra Chapter 3.1 The algebra and composition of functions Sums and Differences of Functions

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**Sums and Differences of Functions**

College Algebra Chapter 3.1 The algebra and composition of functions Sums and Differences of Functions

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**Products and Quotients of Functions**

College Algebra Chapter 3.1 The algebra and composition of functions Products and Quotients of Functions

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**Products and Quotients of Functions**

College Algebra Chapter 3.1 The algebra and composition of functions Products and Quotients of Functions

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**Products and Quotients of Functions**

College Algebra Chapter 3.1 The algebra and composition of functions Products and Quotients of Functions

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**Products and Quotients of Functions**

College Algebra Chapter 3.1 The algebra and composition of functions Products and Quotients of Functions

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**Composition of Functions**

College Algebra Chapter 3.1 The algebra and composition of functions Composition of Functions

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**Composition of Functions**

College Algebra Chapter 3.1 The algebra and composition of functions Composition of Functions

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**Composition of Functions**

College Algebra Chapter 3.1 The algebra and composition of functions Composition of Functions

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**Function Decomposition**

College Algebra Chapter 3.1 The algebra and composition of functions Function Decomposition

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**Function Decomposition**

College Algebra Chapter 3.1 The algebra and composition of functions Function Decomposition

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**College Algebra Chapter 3.1 The algebra and composition of functions**

Homework pg

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**Relations Functions One to One Function**

College Algebra Chapter 3.2 one to one and inverse functions Relations Functions One to One Function If a horizontal line intersects a graph at only one point, the function is one to one

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**Relations Functions 1 to 1 functions**

College Algebra Chapter 3.2 one to one and inverse functions Relations Functions 1 to 1 functions

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**College Algebra Chapter 3.2 one to one and inverse functions**

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**Inverse functions An inverse function is denoted by This does not mean**

College Algebra Chapter 3.2 one to one and inverse functions Inverse functions An inverse function is denoted by This does not mean If given coordinates (x,y) the inverse would have coordinates (y,x) (3,4) (-2,8) (-7,10)

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**College Algebra Chapter 3.2 one to one and inverse functions**

An inverse must undo operations taking place in the original equation

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**Inverse functions How to find an inverse Algebraically**

College Algebra Chapter 3.2 one to one and inverse functions Inverse functions How to find an inverse Algebraically Use y instead of f(x) Interchange x and y Solve for y The result is the inverse

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**College Algebra Chapter 3.2 one to one and inverse functions**

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**College Algebra Chapter 3.2 one to one and inverse functions**

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**College Algebra Chapter 3.2 one to one and inverse functions**

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**College Algebra Chapter 3.2 one to one and inverse functions**

Homework pg

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**Vertical and Horizontal Shifts**

College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Vertical shift or vertical translation Given any function whose graph is determined by and k>0, The graph of is the graph of shifted upward k units. The graph of is the graph of shifted downward k units. The amount of shift is equal to the constant added to the function

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**Vertical and Horizontal Shifts**

College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Vertical shift or vertical translation

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**Vertical and Horizontal Shifts**

College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Vertical shift or vertical translation

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**Vertical and Horizontal Shifts**

College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Vertical shift or vertical translation

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**Vertical and Horizontal Shifts**

College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Vertical shift or vertical translation

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**Vertical and Horizontal Shifts**

College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Horizontal shift or horizontal translation Given any function whose graph is determined by and h>0, The graph of is the graph of shifted to the left h units. The graph of is the graph of shifted to the right h units. -Happens when the input values are affected -Direction of shift is opposite the sign

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**Vertical and Horizontal Shifts**

College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Horizontal shift or horizontal translation

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**Vertical and Horizontal Shifts**

College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Horizontal shift or horizontal translation Graph

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**Vertical and Horizontal Shifts**

College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Horizontal shift or horizontal translation Graph

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**Vertical and Horizontal Shifts**

College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Horizontal shift or horizontal translation Graph

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**Vertical Reflection over x-axis**

College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical Reflection over x-axis

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**Vertical Reflection over x-axis**

College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical Reflection over x-axis

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**Horizontal Reflections over y-axis**

College Algebra Chapter 3.3 Toolbox functions and Transformation Horizontal Reflections over y-axis

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**Horizontal Reflections over y-axis**

College Algebra Chapter 3.3 Toolbox functions and Transformation Horizontal Reflections over y-axis

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**College Algebra Chapter 3.3 Toolbox functions and Transformation**

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**Ways to graph transformations**

College Algebra Chapter 3.3 Toolbox functions and Transformation Ways to graph transformations Using a table of values Applying transformations to a parent graph Apply stretch or compression Reflect result Apply horizontal or vertical shifts usually applied to a few characteristic points

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**College Algebra Chapter 3.3 Toolbox functions and Transformation**

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**College Algebra Chapter 3.3 Toolbox functions and Transformation**

Homework pg

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**Shifted Form/Vertex Form**

College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Horizontal shift is h units, vertical shift is k units To put a quadratic equation in shifted form can be done by completing the square

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**Shifted Form/Vertex Form**

College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square Group variable terms Factor our “a” Add and subtract then regroup Factor trinomial Distribute and simplify

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**Shifted Form/Vertex Form**

College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square

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**Shifted Form/Vertex Form**

College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square

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**Shifted Form/Vertex Form**

College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square Go back 3 pages to find zero’s of each function Set equation equal to zero and then solve for x

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**Shifted Form/Vertex Form**

College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square Group variable terms Factor our “a” Add and subtract then regroup Factor trinomial Distribute and simplify

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**Shifted Form/Vertex Form**

College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square

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**Shifted Form/Vertex Form**

College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square

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**Shifted Form/Vertex Form**

College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square

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**Standard form for a quadratic function has a vertex at**

College Algebra Chapter 3.4 Graphing General Quadratic Functions Standard form for a quadratic function has a vertex at

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**College Algebra Chapter 3.4 Graphing General Quadratic Functions**

Homework pg

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**Reciprocal Quadratic Functions**

College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions Reciprocal Functions Reciprocal Quadratic Functions Asymptotes are not part of the graph, but can act as guides when graphing Asymptotes appear as dashed lines guiding the branches of the graph

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**Direction/Approach Notation**

College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions Direction/Approach Notation As x becomes an infinitely large negative number, y becomes a very small negative number

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**Horizontal and Vertical asymptotes**

College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions Horizontal and Vertical asymptotes The line y=k is a horizontal asymptote if, as x increases or decreases without bound, y approaches k The line x=h is a vertical asymptote if, as x approaches h, |y| increases or decreases without bound

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**Horizontal and vertical shifts of rational functions**

College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions Horizontal and vertical shifts of rational functions First apply them to the asymptotes, then calculate the x- and y-intercepts as usual

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**To find x intercept; solve**

College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions To find x intercept; solve

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**College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions**

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**To find y-intercept; solve**

College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions To find y-intercept; solve To find x-intercept; solve

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**College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions**

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**College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions**

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**College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions**

Homework pg

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**College Algebra Chapter 3.6 Direct and inverse Variation**

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**College Algebra Chapter 3.6 Direct and inverse Variation**

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**College Algebra Chapter 3.6 Direct and inverse Variation**

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**College Algebra Chapter 3.6 Direct and inverse Variation**

Homework pg

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**Piecewise-Defined Functions**

College Algebra Chapter 3.7 Piecewise – Defined Functions Piecewise-Defined Functions The effective domain is the part of the domain that each piece is graphed over.

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**Piecewise-Defined Functions**

College Algebra Chapter 3.7 Piecewise – Defined Functions Piecewise-Defined Functions What is the piece-wise function?

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**Piecewise-Defined Functions**

College Algebra Chapter 3.7 Piecewise – Defined Functions Piecewise-Defined Functions Show what happens when the ends of each line do not meet at the same point

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**Piecewise-Defined Functions**

College Algebra Chapter 3.7 Piecewise – Defined Functions Piecewise-Defined Functions How to handle which function the end points go with

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**College Algebra Chapter 3.7 Piecewise – Defined Functions**

Homework pg

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**Composition of functions Inverse function One-to-one function **

College Algebra Chapter 3 Review Composition of functions Inverse function One-to-one function Transformation Translation Reflection Quadratic Absolute value Linear Reciprocal Reciprocal quadratic function Piecewise-defined functions Effective domain

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**Composition of functions**

College Algebra Chapter 3 Review Composition of functions Domain and Range?

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**Know them and their graphs**

College Algebra Chapter 3 Review Toolbox Functions Know them and their graphs

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**College Algebra Chapter 3 Review**

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**College Algebra Chapter 3 Review**

Variation The weight of an object on the moon varies directly with the weight of the object on Earth. A 96-kg object on Earth would weigh only 16 kg on the moon. How much would a 250-kg astronaut weigh on the moon?

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**Piece-Wise Defined Functions**

College Algebra Chapter 3 Review Piece-Wise Defined Functions

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**College Algebra Chapter 3 Review**

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