1.5. Analyzing Graphs of Functions. The Graph of a Function. What you should learn. Why you should learn it. 54 Chapter 1 Functions and Their Graphs

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1 0_005.qd /7/05 8: AM Page 5 5 Chapter Functions and Their Graphs.5 Analzing Graphs of Functions What ou should learn Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals on which functions are increasing or decreasing and determine relative maimum and relative minimum values of functions. Determine the average rate of change of a function. Identif even and odd functions. Wh ou should learn it Graphs of functions can help ou visualize relationships between variables in real life. For instance, in Eercise 8 on page, ou will use the graph of a function to represent visuall the temperature for a cit over a -hour period. The Graph of a Function In Section., ou studied functions from an algebraic point of view. In this section, ou will stud functions from a graphical perspective. The graph of a function f is the collection of ordered pairs, f such that is in the domain of f. As ou stud this section, remember that the directed distance from the -ais f the directed distance from the -ais as shown in Figure.5. f() = f() FIGURE.5 Eample Finding the Domain and Range of a Function 5 = f( ) (0, ) (, ) (5, ) Range (, ) Domain 5 FIGURE.5 Use the graph of the function f, shown in Figure.5, to find (a) the domain of f, (b) the function values f and f, and (c) the range of f. a. The closed dot at, indicates that is in the domain of f, whereas the open dot at 5, indicates that 5 is not in the domain. So, the domain of f is all in the interval, 5. b. Because, is a point on the graph of f, it follows that f. Similarl, because, is a point on the graph of f, it follows that f. c. Because the graph does not etend below f or above f 0, the range of f is the interval,. Now tr Eercise. The use of dots (open or closed) at the etreme left and right points of a graph indicates that the graph does not etend beond these points. If no such dots are shown, assume that the graph etends beond these points.

2 0_005.qd /7/05 8: AM Page 55 Section.5 Analzing Graphs of Functions 55 B the definition of a function, at most one -value corresponds to a given -value. This means that the graph of a function cannot have two or more different points with the same -coordinate, and no two points on the graph of a function can be verticall above or below each other. It follows, then, that a vertical line can intersect the graph of a function at most once. This observation provides a convenient visual test called the Vertical Line Test for functions. Vertical Line Test for Functions A set of points in a coordinate plane is the graph of as a function of if and onl if no vertical line intersects the graph at more than one point. Eample Vertical Line Test for Functions Use the Vertical Line Test to decide whether the graphs in Figure.5 represent as a function of. (a) FIGURE.5 (b) (c) a. This is not a graph of as a function of, because ou can find a vertical line that intersects the graph twice. That is, for a particular input, there is more than one output. b. This is a graph of as a function of, because ever vertical line intersects the graph at most once. That is, for a particular input, there is at most one output. c. This is a graph of as a function of. (Note that if a vertical line does not intersect the graph, it simpl means that the function is undefined for that particular value of. ) That is, for a particular input, there is at most one output. Now tr Eercise 9.

3 0_005.qd /7/05 8: AM Page 5 5 Chapter Functions and Their Graphs Zeros of a Function If the graph of a function of has an -intercept at a, 0, then a is a zero of the function. f () = + 0 (, 0) 5 (, 0) 8 Zeros of f:, 5 FIGURE.55 8 g () = 0 ( 0, 0) 0, 0 Zeros of g: ±0 FIGURE.5 (, 0) ( ) ht () = t t Zero of h: t FIGURE.57 t Zeros of a Function The zeros of a function f of are the -values for which f 0. Eample Find the zeros of each function. Finding the Zeros of a Function a. f 0 b. g 0 c. ht To find the zeros of a function, set the function equal to zero and solve for the independent variable. a. 0 0 Set f equal to Factor. Set st factor equal to 0. Set nd factor equal to 0. The zeros of f are 5 and In Figure.55, note that the graph of has 5, 0. f and, 0 as its -intercepts. b ±0 Set g equal to 0. Square each side. Add to each side. Etract square roots. The zeros of g are 0 and 0. In Figure.5, note that the graph of g has 0, 0 and 0, 0 as its -intercepts. c. t t 5 0 Set ht equal to 0. t 0 t t Multipl each side b t 5. Add to each side. Divide each side b. The zero of h is t. In Figure.57, note that the graph of h has as its t-intercept. Now tr Eercise 5. t t 5, 0

4 0_005.qd /7/05 8: AM Page 57 Decreasing FIGURE.58 Constant Increasing Section.5 Analzing Graphs of Functions 57 Increasing and Decreasing Functions The more ou know about the graph of a function, the more ou know about the function itself. Consider the graph shown in Figure.58. As ou move from left to right, this graph falls from to 0, is constant from 0 to, and rises from to. Increasing, Decreasing, and Constant Functions A function f is increasing on an interval if, for an and in the interval, < implies f < f. A function f is decreasing on an interval if, for an and in the interval, < implies f > f. A function f is constant on an interval if, for an and in the interval, f f. Eample Increasing and Decreasing Functions Use the graphs in Figure.59 to describe the increasing or decreasing behavior of each function. a. This function is increasing over the entire real line. b. This function is increasing on the interval,, decreasing on the interval,, and increasing on the interval,. c. This function is increasing on the interval, 0, constant on the interval 0,, and decreasing on the interval,. f() = f() = (, ) (0, ) (, ) t (, ) f(t) = t +, t < 0, 0 t t +, t > (a) FIGURE.59 (b) (c) Now tr Eercise. To help ou decide whether a function is increasing, decreasing, or constant on an interval, ou can evaluate the function for several values of. However, calculus is needed to determine, for certain, all intervals on which a function is increasing, decreasing, or constant.

5 0_005.qd /7/05 8: AM Page Chapter Functions and Their Graphs A relative minimum or relative maimum is also referred to as a local minimum or local maimum. Relative maima The points at which a function changes its increasing, decreasing, or constant behavior are helpful in determining the relative minimum or relative maimum values of the function. Definitions of Relative Minimum and Relative Maimum A function value fa is called a relative minimum of f if there eists an interval, that contains a such that < < implies fa f. A function value fa is called a relative maimum of f if there eists an interval, that contains a such that < < implies f a f. Relative minima Figure.0 shows several different eamples of relative minima and relative maima. In Section., ou will stud a technique for finding the eact point at which a second-degree polnomial function has a relative minimum or relative maimum. For the time being, however, ou can use a graphing utilit to find reasonable approimations of these points. FIGURE.0 Eample 5 Approimating a Relative Minimum 5 FIGURE. f () = Use a graphing utilit to approimate the relative minimum of the function given b f. The graph of f is shown in Figure.. B using the zoom and trace features or the minimum feature of a graphing utilit, ou can estimate that the function has a relative minimum at the point 0.7,.. Relative minimum Later, in Section., ou will be able to determine that the eact point at which the relative minimum occurs is, 0. Now tr Eercise 9. You can also use the table feature of a graphing utilit to approimate numericall the relative minimum of the function in Eample 5. Using a table that begins at 0. and increments the value of b 0.0, ou can approimate that the minimum of f occurs at the point 0.7,.. Technolog If ou use a graphing utilit to estimate the - and -values of a relative minimum or relative maimum, the zoom feature will often produce graphs that are nearl flat. To overcome this problem, ou can manuall change the vertical setting of the viewing window. The graph will stretch verticall if the values of Ymin and Yma are closer together.

6 0_005.qd /7/05 8: AM Page 59 (, f( )) (, f( )) Secant line f f( ) f( ) Average Rate of Change Section.5 Analzing Graphs of Functions 59 In Section., ou learned that the slope of a line can be interpreted as a rate of change. For a nonlinear graph whose slope changes at each point, the average rate of change between an two points, f and, f is the slope of the line through the two points (see Figure.). The line through the two points is called the secant line, and the slope of this line is denoted as m sec. Average rate of change of f from to f f change in change in m sec FIGURE. Eample Average Rate of Change of a Function f() = (0, 0) (, ) (, ) FIGURE. Find the average rates of change of f (a) from to 0 and (b) from 0 to (see Figure.). a. The average rate of change of f from to 0 is f f b. The average rate of change of f from 0 to is f f f0 f 0 f f0 0 Now tr Eercise Secant line has positive slope. Secant line has negative slope. Eample 7 Finding Average Speed Eploration Use the information in Eample 7 to find the average speed of the car from t 0 to t 9 seconds. Eplain wh the result is less than the value obtained in part (b). The distance s (in feet) a moving car is from a stoplight is given b the function st 0t, where t is the time (in seconds). Find the average speed of the car (a) from t 0 to t seconds and (b) from t to t 9 seconds. a. The average speed of the car from t 0 to t seconds is st st t t 0 feet per second. b. The average speed of the car from t to t 9 seconds is st st t t s s0 0 s9 s 9 Now tr Eercise feet per second.

7 0_005.qd /7/05 8: AM Page 0 0 Chapter Functions and Their Graphs Even and Odd Functions In Section., ou studied different tpes of smmetr of a graph. In the terminolog of functions, a function is said to be even if its graph is smmetric with respect to the -ais and to be odd if its graph is smmetric with respect to the origin. The smmetr tests in Section. ield the following tests for even and odd functions. Eploration Graph each of the functions with a graphing utilit. Determine whether the function is even, odd, or neither. f g h 5 j 8 k 5 p 9 5 What do ou notice about the equations of functions that are odd? What do ou notice about the equations of functions that are even? Can ou describe a wa to identif a function as odd or even b inspecting the equation? Can ou describe a wa to identif a function as neither odd nor even b inspecting the equation? Tests for Even and Odd Functions A function f is even if, for each in the domain of f, f f. A function f is odd if, for each in the domain of f, f f. Eample 8 Even and Odd Functions a. The function g is odd because g g, as follows. g Substitute for. Simplif. Distributive Propert g Test for odd function b. The function h is even because h h, as follows. h h Substitute for. Simplif. Test for even function The graphs and smmetr of these two functions are shown in Figure.. Additional Eample Is the function given b f even, odd, or neither? Substituting for, f. Because f and f, f f and f f. The function is neither even nor odd. g() = (, ) (, ) (a) Smmetric to origin: Odd Function FIGURE. Now tr Eercise 7. 5 (, ) (, ) h() = + (b) Smmetric to -ais: Even Function

8 0_005.qd /7/05 8: AM Page Section.5 Analzing Graphs of Functions.5 Eercises VOCABULARY CHECK: Fill in the blanks.. The graph of a function f is the collection of or, f such that is in the domain of f.. The is used to determine whether the graph of an equation is a function of in terms of.. The of a function f are the values of for which f 0.. A function f is on an interval if, for an and in the interval, < implies f > f. 5. A function value fa is a relative of f if there eists an interval, containing a such that < < implies fa f.. The between an two points, f and, f is the slope of the line through the two points, and this line is called the line. 7. A function f is if for the each in the domain of f, f f. 8. A function f is if its graph is smmetric with respect to the -ais. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Eercises, use the graph of the function to find the domain and range of f..... In Eercises 5 8, use the graph of the function to find the indicated function values. 5. (a) f (b) f. (a) f (b) f (c) (d) f (c) f 0 (d) f = f() f = f() = f() = f() = f() = f() 7. (a) f (b) f 8. (a) f (b) f (c) f 0 (d) f (c) f (d) f In Eercises 9, use the Vertical Line Test to determine whether is a function of.to print an enlarged cop of the graph, go to the website = f().. 5 = f()

9 0_005.qd /7/05 8: AM Page Chapter Functions and Their Graphs... f. f In Eercises 5, find the zeros of the function algebraicall. 5. f 7 0. f f 9 f f f 9 f f 9 5 f f 8 5.., f,, (0, ) (, ) f,, 0 0 < > > (, 0) (, 0) In Eercises 5 0, (a) use a graphing utilit to graph the function and find the zeros of the function and (b) verif our results from part (a) algebraicall f 5. f 7 7. f 8. f 8 f f 9 In Eercises 8, determine the intervals over which the function is increasing, decreasing, or constant.. f. f f (, ) (, ) f (0, ) (, ) (, )

10 0_005.qd /7/05 8: AM Page Section.5 Analzing Graphs of Functions In Eercises 9 8, (a) use a graphing utilit to graph the function and visuall determine the intervals over which the function is increasing, decreasing, or constant, and (b) make a table of values to verif whether the function is increasing, decreasing, or constant over the intervals ou identified in part (a). 9. f 0. g. gs s. h. f t t. f 5. f. f 7. f 8. f In Eercises 9 5, use a graphing utilit to graph the function and approimate (to two decimal places) an relative minimum or relative maimum values. 9. f 50. f 5 5. f 5. f f f In Eercises 55, graph the function and determine the interval(s) for which f f 5. f 57. f 58. f 59. f 0. f.. f In Eercises 70, find the average rate of change of the function from to. Function. f 5. f( 8 5. f. f 8 7. f 8. f 9. f f f -Values 0, 0,, 5, 5,,,, 8 In Eercises 7 7, determine whether the function is even, odd, or neither. Then describe the smmetr. 7. f 7. h 5 7. g 5 7. f 75. f t t t 7. gs s In Eercises 77 80, write the height h of the rectangle as a function of = = 80. (, ) h In Eercises 8 8, write the length L of the rectangle as a function of (, ) (, ) L = = = L (8, ) 8 (, ) 85. Electronics The number of lumens (time rate of flow of light) L from a fluorescent lamp can be approimated b the model L , h = (, ) (, ) = (, ) L 0 90 where is the wattage of the lamp. (a) Use a graphing utilit to graph the function. (b) Use the graph from part (a) to estimate the wattage necessar to obtain 000 lumens. h (, ) L h = (8, ) 8 =

11 0_005.qd /7/05 8: AM Page Chapter Functions and Their Graphs 8. Data Analsis: Temperature The table shows the temperature (in degrees Fahrenheit) of a certain cit over a -hour period. Let represent the time of da, where 0 corresponds to A.M. Time, Model It Temperature, A model that represents these data is given b , 0. (a) Use a graphing utilit to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? (c) Use the graph to approimate the times when the temperature was increasing and decreasing. (d) Use the graph to approimate the maimum and minimum temperatures during this -hour period. (e) Could this model be used to predict the temperature for the cit during the net -hour period? Wh or wh not? 87. Coordinate Ais Scale Each function models the specified data for the ears 995 through 005, with t 5 corresponding to 995. Estimate a reasonable scale for the vertical ais (e.g., hundreds, thousands, millions, etc.) of the graph and justif our answer. (There are man correct answers.) (a) f t represents the average salar of college professors. (b) f t represents the U.S. population. (c) f t represents the percent of the civilian work force that is unemploed. 88. Geometr Corners of equal size are cut from a square with sides of length 8 meters (see figure). (a) Write the area A of the resulting figure as a function of. Determine the domain of the function. (b) Use a graphing utilit to graph the area function over its domain. Use the graph to find the range of the function. (c) Identif the figure that would result if were chosen to be the maimum value in the domain of the function. What would be the length of each side of the figure? 89. Digital Music Sales The estimated revenues r (in billions of dollars) from sales of digital music from 00 to 007 can be approimated b the model r 5.9t 0.75t 0.5t 0, where t represents the ear, with t corresponding to 00. (Source: Fortune) (a) Use a graphing utilit to graph the model. (b) Find the average rate of change of the model from 00 to 007. Interpret our answer in the contet of the problem. 90. Foreign College Students The numbers of foreign students F (in thousands) enrolled in colleges in the United States from 99 to 00 can be approimated b the model. F 0.00t 0.t., t t 7 where t represents the ear, with t corresponding to 99. (Source: Institute of International Education) (a) Use a graphing utilit to graph the model. (b) Find the average rate of change of the model from 99 to 00. Interpret our answer in the contet of the problem. (c) Find the five-ear time periods when the rate of change was the greatest and the least. 8 8

12 0_005.qd /7/05 8: AM Page 5 Section.5 Analzing Graphs of Functions 5 Phsics In Eercises 9 9, (a) use the position equation s t v 0 t s 0 to write a function that represents the situation, (b) use a graphing utilit to graph the function, (c) find the average rate of change of the function from t to t, (d) interpret our answer to part (c) in the contet of the problem, (e) find the equation of the secant line through t and t, and (f) graph the secant line in the same viewing window as our position function. 9. An object is thrown upward from a height of feet at a velocit of feet per second. t 0, t 9. An object is thrown upward from a height of.5 feet at a velocit of 7 feet per second. t 0, t 9. An object is thrown upward from ground level at a velocit of 0 feet per second. t, t 5 9. An object is thrown upward from ground level at a velocit of 9 feet per second. t, t An object is dropped from a height of 0 feet. t 0, t 9. An object is dropped from a height of 80 feet. t, t Snthesis True or False? In Eercises 97 and 98, determine whether the statement is true or false. Justif our answer. 97. A function with a square root cannot have a domain that is the set of real numbers. 98. It is possible for an odd function to have the interval 0, as its domain. 99. If f is an even function, determine whether g is even, odd, or neither. Eplain. (a) g f (b) g f (c) g f (d) g f 00. Think About It Does the graph in Eercise represent as a function of? Eplain. Think About It In Eercises 0 0, find the coordinates of a second point on the graph of a function f if the given point is on the graph and the function is (a) even and (b) odd , ,, 5, Writing Use a graphing utilit to graph each function. Write a paragraph describing an similarities and differences ou observe among the graphs. (a) (b) (c) (d) (e) 5 (f) 0. Conjecture Use the results of Eercise 05 to make a conjecture about the graphs of 7 and 8. Use a graphing utilit to graph the functions and compare the results with our conjecture. Skills Review In Eercises 07 0, solve the equation In Eercises, evaluate the function at each specified value of the independent variable and simplif.. f 5 8 (a) f9 (b) f (c) f 7. f 0 (a) f (b) f 8 (c) f. f 9 (a) f (b) f 0 (c) f. f 5 (a) f (b) f (c) f In Eercises 5 and, find the difference quotient and simplif our answer. f h f 5. f 9,, h 0 h f h f. f 5,, h 0 h