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 Properties of Functions SECTION Determine whether a function is increasing or decreasing on an interval. Use a graph to locate relative maximum and minimum values. Identify even and odd functions. Find the average rate of change of a function.

3 © 2010 Pearson Education, Inc. All rights reserved INCREASING, DECREASING, AND CONSTANT FUNCTIONS Let f be a function, and let x 1 and x 2 be any two numbers in an open interval (a, b) contained in the domain of f. The symbols a and b may represent real numbers, –∞, or ∞. Then

4 © 2010 Pearson Education, Inc. All rights reserved INCREASING, DECREASING, AND CONSTANT FUNCTIONS (i) f is an increasing function on (a, b) if x 1 < x 2 implies f (x 1 ) < f (x 2 ).

5 © 2010 Pearson Education, Inc. All rights reserved INCREASING, DECREASING, AND CONSTANT FUNCTIONS (ii)f is a decreasing function on (a, b) if x 1 f (x 2 ).

6 © 2010 Pearson Education, Inc. All rights reserved INCREASING, DECREASING, AND CONSTANT FUNCTIONS (iii) f is a constant on (a, b) if x 1 < x 2 implies f (x 1 ) = f (x 2 ).

7 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Tracking the Behavior of a Function Solution From the graph of g, find the intervals over which g is increasing, decreasing, or is constant. a.increasing on the interval (–∞, –2) b.constant on the interval (–2, 3) c.decreasing on the interval (3, ∞)

8 © 2010 Pearson Education, Inc. All rights reserved DEFINITION OF RELATIVE MAXIMUM AND RELATIVE MINIMUM If a is in the domain of a function f, we say that the value f (a) is a relative minimum of f if there is an interval (x 1, x 2 ) containing a such that f (a) ≤ f (x) for every x in the interval (x 1, x 2 ). We say that the value f (a) is a relative maximum of f if there is an interval (x 1, x 2 ) containing a such that f (a) ≥ f (x) for every x in the interval (x 1, x 2 ).

9 © 2010 Pearson Education, Inc. All rights reserved RELATIVE MAXIMUM AND RELATIVE MINIMUM

10 © 2010 Pearson Education, Inc. All rights reserved Definitions The value f(a) is called an extreme value of f if it is either a relative maximum value or a relative minimum value. At a turning point, a graph changes direction from increasing to decreasing or from decreasing to increasing.

11 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Approximating Relative Extrema Use a graphing utility to approximate the relative maximum and the relative minimum point on the graph of the function f(x) = x 3 – x 2. Solution Use the TRACE and ZOOM features to see that the function has: Relative minimum pt ≈ (0.67, –0.15) Relative maximum pt ≈ (0, 0)

12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Algebra in Coughing The average flow velocity, v, of outgoing air through the windpipe is modeled by where r 0 is the rest radius of the windpipe, r is its contracted radius, and c is a positive constant. For Mr. Osborn, assume that c = 1 and r 0 = 13 mm. Use a graphing utility to estimate the value of r that will maximize the airflow v when Mr.Osborn coughs.

13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Algebra in Coughing By using the TRACE and ZOOM features, we see that the maximum point on the graph is estimated at (8.67, 325). So Mr. Osborn’s windpipe contracts to a radius of 8.67 mm to maximize the airflow velocity. Solution

14 © 2010 Pearson Education, Inc. All rights reserved EVEN FUNCTION A function f, is called an even function if, for each x in the domain of f, –x is also in the domain of f and f (–x) = f (x). The graph of an even function is symmetric about the y-axis.

15 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Graphing the Squaring Function Show that the squaring function f (x) = x 2 is an even function, and sketch its graph. Solution The function f (x) = x 2 is even because To graph f (x) = x 2, make a table of values. x0123 f (x)= x

16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Graphing the Squaring Function Solution continued symmetry to plot additional points. Plot the points, then use y-axis

17 © 2010 Pearson Education, Inc. All rights reserved ODD FUNCTION A function f, is an odd function if, for each x in the domain of f, –x is also in the domain of f and f (–x) = – f (x). The graph of an odd function is symmetric about the origin.

18 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Graphing the Cubing Function Show that the cubing function defined by g(x) = x 3 is an odd function, and sketch its graph. Solution The function g(x) = x 3 is odd because We sketch the graph of g(x) = x 3 by plotting points in the first quadrant and then use symmetry in the origin to extend the graph to the third quadrant.

19 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Graphing the Cubing Function Solution continued

20 © 2010 Pearson Education, Inc. All rights reserved THE AVERAGE RATE OF CHANGE OF A FUNCTION Let (a, f (a)) and (b, f (b)) be points on the graph of a function f. Then the average rate of change of f (x) as x changes from a to b is defined by

21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 OBJECTIVE Find the average rate of change of a function f as x changes from a to b. Step 1 Find f (a) and f (b). Step 2 Use the values from Step 1 in the definition of average rate of change. Finding the Average Rate of Change EXAMPLE Find the average rate of change of f (x) = 2  3x 2 as x changes from x = 1 to x = 3. =  12

22 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Finding the Average Rate of Change Find the average rate of change of f (x) = 2t 2  3 as t changes from t = 5 to t = x, x  5. Solution Average rate of change

23 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Finding the Average Rate of Change Solution continued Average rate of change

24 © 2010 Pearson Education, Inc. All rights reserved DIFFERENCE QUOTIENT For a function f, the difference quotient is

25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Let f (x) = 2x 2 – 3x + 5. Find and simplify Evaluating and Simplifying a Difference Quotient Solution Find Now substitute into the difference quotient.

26 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Evaluating and Simplifying a Difference Quotient Solution continued