1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 2 Graphs and Functions

OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 A Library of Functions SECTION Relate linear functions to linear equations. Graph square root and cube root functions. Graph additional basic functions. Evaluate and graph piecewise functions.

3 © 2010 Pearson Education, Inc. All rights reserved LINEAR FUNCTIONS Let m and b be real numbers. The function f (x) = mx + b is called a linear function. If m = 0, the function f (x) = b is called a constant function. If m = 1 and b = 0, the resulting function f (x) = x is called the identity function.

4 © 2010 Pearson Education, Inc. All rights reserved GRAPH OF f (x) = mx + b The graph of a linear function is a nonvertical line with slope m and y-intercept b.

5 © 2010 Pearson Education, Inc. All rights reserved GRAPH OF f (x) = mx + b

6 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Writing a Linear Function Write a linear function g for which g(1) = 4 and g(–3) = –2. Solution Find the equation of the line passing through the points (1, 4) and (–3, –2). First, find the slope. Next, use the point-slope form of the line.

7 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Writing a Linear Function Solution continued Here’s the graph of y = g (x).

8 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Determining the Length of the “Megatooth” Shark The largest known “Megatooth” specimen is a tooth that has a total height of 15.6 cm. Calculate the length of the shark it came from by using the formula Shark length = (0.96)(height of tooth) – 0.22 where shark length is measured in m and tooth height is measured in cm. Shark length = (0.96)(height of tooth) – 0.22 Solution Shark length = (0.96)(15.6) – 0.22 = m (≈ 48.6 ft)

9 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Graphing the Square Root Function Solution Make a table of values. For convenience, select the values of x that are perfect squares.

10 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Graphing the Square Root Function Solution continued Then plot the ordered pairs (x, y) and draw a smooth curve through the plotted points.

11 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Graphing the Cube Root Function Solution Make a table of values. For convenience, select the values of x that are perfect cubes.

12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Solution continued Then plot the ordered pairs (x, y) and draw a smooth curve through the plotted points. Graphing the Cube Root Function

13 © 2010 Pearson Education, Inc. All rights reserved BASIC FUNCTIONS The following are some of the common functions of algebra, along with their properties, and should be included in a library of basic functions.

14 © 2010 Pearson Education, Inc. All rights reserved Constant Function f (x) = c Domain: (–∞, ∞) Range: {c} Constant on (–∞, ∞) Even function (y–axis symmetry)

15 © 2010 Pearson Education, Inc. All rights reserved Domain: (–∞, ∞) Range: (–∞, ∞) Increasing on (–∞, ∞) Odd function (origin symmetry) Identity Function f (x) = x

16 © 2010 Pearson Education, Inc. All rights reserved Domain: (–∞, ∞) Range: [0, ∞) Decreasing on (–∞, 0) Increasing on (0, ∞) Even function (y–axis symmetry) Squaring Function f (x) = x 2

17 © 2010 Pearson Education, Inc. All rights reserved Domain: (–∞, ∞) Range: (–∞, ∞) Increasing on (–∞, ∞) Odd function (origin symmetry) Cubing Function f (x) = x 3

18 © 2010 Pearson Education, Inc. All rights reserved Absolute Value Function Domain: (–∞, ∞) Range: [0, ∞) Decreasing on (–∞, 0) Increasing on (0, ∞) Even function (y–axis symmetry)

19 © 2010 Pearson Education, Inc. All rights reserved Square Root Function Domain: [0, ∞) Range: [0, ∞) Increasing on (0, ∞) Neither even nor odd (no symmetry)

20 © 2010 Pearson Education, Inc. All rights reserved Cube Root Function Domain: (–∞, ∞) Range: (–∞, ∞) Increasing on (–∞, ∞) Odd function (origin symmetry)

21 © 2010 Pearson Education, Inc. All rights reserved Reciprocal Function Domain: (–∞, 0) U (0, ∞) Range: (–∞, 0) U (0, ∞) Odd function (origin symmetry) Decreasing on (–∞, 0) U (0, ∞)

22 © 2010 Pearson Education, Inc. All rights reserved Reciprocal Function Domain: (–∞, 0) U (0, ∞) Range: (0, ∞) Increasing on (–∞, 0) Decreasing on (0, ∞) Even function (y–axis symmetry)

23 © 2010 Pearson Education, Inc. All rights reserved PIECEWISE FUNCTIONS In the definition of some functions, different rules for assigning output values are used on different parts of the domain. Such functions are called piecewise functions.

24 © 2010 Pearson Education, Inc. All rights reserved 24 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Evaluating a Piecewise Function OBJECTIVE Evaluate F(a) for piecewise function F. Step 1 Determine which line of the function applies to the number a. Step 2 Evaluate F(a) using the line chosen in Step 1. EXAMPLE Let Find F(0) and F(2). 1. Let a = 0. Because a < 1, use the first line, F(x) = x 2. Let a = 2. Because a > 1, use the second line, F(x) = 2x F(0) = 0 2 = 0 F(2) = 2(2) + 1 = 5

25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Graphing a Piecewise Function Let In the definition of F the formula changes at x =1. We call such numbers the breakpoints of the formula. For the function F the only breakpoint is 1. Solution Sketch the graph of y = F(x).

26 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Graphing a Piecewise Function Solution continued Graph the function separately over the open intervals determined by the breakpoints and then graph the function at the breakpoints themselves. For the function y = F(x), the formula for F specifies that the equation y = x 2 be used on the interval (–∞, 1)

27 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Graphing a Piecewise Function Solution continued and the equation y = 3x + 1 be used on the interval (1, ∞) and also at the breakpoint 1, so that y = F(1) = 3(1) + 1 = 4.

28 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Graphing a Step Function Solution Graph the greatest integer function Choose a typical closed interval between two consecutive integers, say [2, 3]. If 2 ≤ x < 3, then If 1 ≤ x < 2, then The values of are constant between each pair of consecutive integers and jump by one unit at each integer. Here’s the graph.

29 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Graphing a Step Function Solution continued