Operations on Functions and Analyzing Graphs

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Operations on Functions and Analyzing Graphs College Algebra Chapter 3 Operations on Functions and Analyzing Graphs

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:

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

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

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

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

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

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

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

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

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

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

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

College Algebra Chapter 3.1 The algebra and composition of functions Homework pg 256 1-77

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

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

College Algebra Chapter 3.2 one to one and inverse functions

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)

College Algebra Chapter 3.2 one to one and inverse functions An inverse must undo operations taking place in the original equation

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

College Algebra Chapter 3.2 one to one and inverse functions

College Algebra Chapter 3.2 one to one and inverse functions

College Algebra Chapter 3.2 one to one and inverse functions

College Algebra Chapter 3.2 one to one and inverse functions Homework pg 268 1-96

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

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

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

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

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

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

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

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

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

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

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

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

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

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

College Algebra Chapter 3.3 Toolbox functions and Transformation

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

College Algebra Chapter 3.3 Toolbox functions and Transformation

College Algebra Chapter 3.3 Toolbox functions and Transformation Homework pg 283 1-86

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

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

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

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

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

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

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

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

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

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

College Algebra Chapter 3.4 Graphing General Quadratic Functions Homework pg 295 1-60

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

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

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

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

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

College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions

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

College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions

College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions

College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions Homework pg 307 1-56

College Algebra Chapter 3.6 Direct and inverse Variation

College Algebra Chapter 3.6 Direct and inverse Variation

College Algebra Chapter 3.6 Direct and inverse Variation

College Algebra Chapter 3.6 Direct and inverse Variation Homework pg 321 1-58

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.

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

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

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

College Algebra Chapter 3.7 Piecewise – Defined Functions Homework pg 335 1-42

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

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

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

College Algebra Chapter 3 Review

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?

Piece-Wise Defined Functions College Algebra Chapter 3 Review Piece-Wise Defined Functions

College Algebra Chapter 3 Review