College Algebra Chapter 2 Functions and Graphs Section 2.7 Analyzing Graphs of Functions and Piecewise-Defined Functions Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Concepts Test for Symmetry Identify Even and Odd Functions Graph Piecewise-Defined Functions Investigate Increasing, Decreasing, and Constant Behavior of a Function Determine Relative Minima and Maxima of a Function

Concept 1 Test for Symmetry

Test for Symmetry Consider an equation in the variables x and y. Symmetric with respect to the y-axis: Substituting –x for x results in equivalent equation. Symmetric with respect to the x-axis: Substituting –y for y results in equivalent equation. Symmetric with respect to the origin: Substituting –x for x and –y for y results in equivalent equation.

Example 1 Determine whether the graph of the equation is symmetric to the x-axis, y-axis, origin, or none of these.

Example 2 Determine whether the graph of the equation is symmetric to the x-axis, y-axis, origin, or none of these.

Example 3 Determine whether the graph of the equation is symmetric to the x-axis, y-axis, origin, or none of these.

Example 4 Determine whether the graph of the equation is symmetric to the x-axis, y-axis, origin, or none of these. Y=|x|-2 Solution: Y-axis : y =|-x|-2=|x|-2 yes X-axis : -y = |x|-2→ y=-|x|+2 No Origin :-y=|-x|-2 Y=-|x|+2 No

Skill Practice 1 (1 of 2) Determine whether the graph is symmetric to the y-axis, x-axis, or origin.

Skill Practice 1 (2 of 2) Determine whether the graph is symmetric to the y-axis, x-axis, or origin.

Concept 2 Identify Even and Odd Functions

Identify Even and Odd Functions Even function: f(–x) = f(x) for all x in the domain of f. (Symmetric with respect to the y-axis) Odd function: f(–x) = –f(x) for all x in the domain of f. (Symmetric with respect to the origin)

Example 5 Determine if the function is even, odd, or neither.

Example 6 Determine if the function is even, odd, or neither.

Example 7 Determine if the function is even, odd, or neither.

Example 8 Determine if the function is even, odd, or neither.

Example 9 Determine if the function is even, odd, or neither.

Skill Practice 3 (1 of 2) Determine if the function is even, odd , or neither. a.

Skill Practice 3 (2 of 2) b. c.

Skill Practice 4 Determine if the function is even, odd, or neither.

Concept 3 Graph Piecewise-Defined Functions

Example 10 Evaluate the function for the given values of x.

Example 11 Evaluate the function for the given values of x.

Skill Practice 5 Evaluate the function for the given values of x.

Example 12 (1 of 2) Graph the function. Y = - x - 1 x ≥ 2 x < 2 x y -2 1 2 x y 2 -3 3 -4 4 -5 x < 2 x ≥ 2

Example 12 (2 of 2)

Example 13 Y = - 1 Y =x - 1 Graph the function. Y =-2x + 6 x y -1 -2 x -1 -2 x y -1 1 2 x y 2 3 4 -2

Skill Practice 6 Graph the function.

Skill Practice 7 Graph the function.

Graph Piecewise-Defined Functions Greatest integer function: is the greatest integer less than or equal to x.

Example 14 Evaluate.

Example 15 Graph.

Skill Practice 8 Evaluate f(x)=[x] for the given values of x. f(1,7)

Example 16 A new job offer in sales promises a base salary of $3000 a month. Once the sales person reaches $50,000 in total sales, he earns his base salary plus a 4.3% commission on all sales of $50,000 or more. Write a piecewise-defined function (in dollars) to model the expected total monthly salary as a function of the amount of sales, x. Solution: When under 50000 m sales, only salary $50,000+in sales means salary plus comm.

Skill Practice 9 A retail story buys T-shirts from the manufacturer. The cost is $7.99 per shirt for 1 to 100 shirts, inclusive. Then the price is decreased to $6.99 per shirt thereafter. Write a piecewise-defined function that expresses the cost C(x) (in $)to buy x shirts.

Concept 4 Investigate Increasing, Decreasing, and Constant Behavior of a Function

Investigate Increasing, Decreasing, and Constant Behavior of a Function (1 of 3)

Investigate Increasing, Decreasing, and Constant Behavior of a Function (2 of 3)

Investigate Increasing, Decreasing, and Constant Behavior of a Function (3 of 3)

Example 17 Use interval notation to write the interval(s) over which f(x) is increasing, decreasing, and constant.

Example 18 Use interval notation to write the interval(s) over which f(x) is increasing, decreasing, and constant. Increasing: never Decreasing: always (-∞,∞) Constant: never

Skill Practice 10 Use interval notation to write the interval(s) over which f is Increasing Decreasing Constant

Concept 5 Determine Relative Minima and Maxima of a Function

Determine Relative Minima and Maxima of a Function

Example 19 (1 of 2) Identify the location and value of any relative maxima or minima of the function. The point _(-2,-1)_ is the lowest point in a small interval surrounding x =_-2 . At x =_-2 the function has a relative minimum of _-1_. The point (0,1) is the highest point in a small interval surrounding x = _0_. At x =_0_ the function has a relative maximum of _1_.

Example 19 (2 of 2)

Example 20 (1 of 2) Identify the location and value of any relative maxima or minima of the function. At x =_-1_ the function has a relative minimum of _-2__. At x = _4_the function has a relative minimum of _0_. At x =_2_ the function has a relative maximum of _3_.

Example 20 (2 of 2)

Example 21 Identify the location and value of any relative maxima or minima of the function. no max no min (inflection point)

Skill Practice 11 For the graph shown, Determine the location and value of any relative maxima. Determine the location and value of any relative minima.