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2. Functions and Their Graphs Chapter 12

3. 12 Functions and Their Graphs 12.1 Relations and Functions 12.2 Graphs of Functions and Transformations 12.3 Quadratic Functions and Their Graphs 12.4 Applications of Quadratic Functions and Graphing Other Parabolas. 12.5 The Algebra of Functions 12.6 Variation

4. 12.2 Graphs of Functions and Transformations • Some functions and their graphs appear often when studying algebra. We will look at the basic graphs of • 1. the absolute value function, . • 2. the quadratic function, . • 3. the square root function, . • It is possible to obtain the graph of any function by plotting points. But we will also see how we can graph other, similar functions by transforming the graphs of the functions above. • First, we will graph two absolute value functions. We will begin by plotting • points so that.

5. Illustrate Vertical Shifts with Absolute Value Functions Example 1 y Solution g(x)=ǀxǀ + 2 up 2 up 2 0 0 0 2 1 1 1 3 2 2 x 2 4 -1 1 -1 3 -2 2 -2 4 The domain of g(x) is (-∞,∞) The range is [2,∞)

6. Illustrate Horizontal Shifts with Quadratic Functions Example 2 y Solution g(x)=(x ‒ 2)2 right 2 right 2 0 0 2 0 1 1 3 1 2 4 x 4 4 -1 1 Vertex (0,0) 1 1 Vertex (2,0) -2 4 0 4 The domain of g(x) is (-∞,∞) The range is [0,∞)

7. Illustrate Reflecting a Graph About the x-Axis with Square Root Function Example 3 y Solution g(x)= ‒√x 0 0 0 0 1 1 1 - 1 x 4 2 4 - 2 9 3 9 - 3 The domain of g(x) is [0,∞) The range is (-∞,0]

8. Graph a Function Using a Combination of the Transformations Example 4 y Solution left 2 down 3 x Shift f(x) left 2 Shift f(x) down 3

9. Graph a Piecewise Function Example 5 Graph the piecewise function Solution Graph f(x) by making two separate tables of values, One for each rule. y f(x)=2x- 4 This will be an open circle 3 2 3 -1 4 4 2 0 Notice that 3 is not included in the domain 5 6 1 1 x 6 8 0 2

10. Define the Greatest Integer Function, f(x)= [| x |] Example 6 Let f(x)= [| x |]. Find the following functions values. Solution a) We need to find the largest integer that is less than or equal to b) since the largest integer less than or equal to 6 is 6. c) To help us understand how to find this function value, we will locate 2.3 on a number line. • The largest integer less than or equal to • ‒ 2.3 is ‒3, f (‒2.3) = [| ‒2.3 |] = ‒ 3.

11. Example 7 Graph f (x)= [|x|] Solution To understand what produces the pattern in the graph of this function, we begin by closely examining what occurs between x = 0 and x =1 (when 0 ≤ x ≤ 1). For all values of x greater than or equal to 0 and less than 1, the function value, f(x), equals zero. When x = 1, the function value changes to 1.

12. Represent an Applied Problem with the Graph of a Greatest Integer Function Example 8 To mail a large envelope within the United States in 2010, the U.S. Postal Service charged $0.88 for the first ounce and $0.17 for each additional ounce or fraction of an ounce. Let C(x) represent the cost of mailing a large envelope within the United States and let x represent the weight of the envelope, in ounces. Graph C(x) for any large envelope weighing up to (and including) 5 ounces. (www.usps.com) Solution If a large envelope weighs between 0 and 1 ounce the cost, C(x), is $0.88. If a large envelope weighs more than 1 oz but less than or equal to 2 oz , the cost, C(x), is $0.88 + $0.17= $1.05 The pattern will continue, and we get the graph at the right. $0.88 $0.17 $1.05