A Library of Parent Functions. Linear and Squaring Functions. Writing a Linear Function. Write the linear function f for which f 1 3 and f 4 0.


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1 0_006.qd 66 /7/0 Chapter.6 8:0 AM Page 66 Functions and Their Graphs A Librar of Parent Functions What ou should learn Identif and graph linear and squaring functions. Identif and graph cubic, square root, and reciprocal functions. Identif and graph step and other piecewisedefined functions. Recognize graphs of parent functions. Wh ou should learn it Step functions can be used to model reallife situations. For instance, in Eercise 6 on page 7, ou will use a step function to model the cost of sending an overnight package from Los Angeles to Miami. Linear and Squaring Functions One of the goals of this tet is to enable ou to recognize the basic shapes of the graphs of different tpes of functions. For instance, ou know that the graph of the linear function f a b is a line with slope m a and intercept at 0, b. The graph of the linear function has the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all real numbers. The graph has an intercept of b m, 0 and a intercept of 0, b. The graph is increasing if m > 0, decreasing if m < 0, and constant if m 0. Eample Writing a Linear Function Write the linear function f for which f and f 0. Solution To find the equation of the line that passes through,, and,, 0, first find the slope of the line. m 0 Net, use the pointslope form of the equation of a line. m Pointslope form Substitute for,, and m. Simplif. f Function notation The graph of this function is shown in Figure.6. Gett Images f() = + FIGURE.6 Now tr Eercise.
2 0_006.qd /7/0 8:0 AM Page 67 Section.6 A Librar of Parent Functions 67 Additional Eample Write the linear function f for which f 0 and f 8. Solution m m There are two special tpes of linear functions, the constant function and the identit function. A constant function has the form f c and has the domain of all real numbers with a range consisting of a single real number c. The graph of a constant function is a horizontal line, as shown in Figure.66. The identit function has the form f. Its domain and range are the set of all real numbers. The identit function has a slope of m and a intercept 0, 0. The graph of the identit function is a line for which each coordinate equals the corresponding coordinate. The graph is alwas increasing, as shown in Figure.67 f() = f() = c FIGURE.66 FIGURE.67 The graph of the squaring function f is a Ushaped curve with the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all nonnegative real numbers. The function is even. The graph has an intercept at 0, 0. The graph is decreasing on the interval, 0 and increasing on the interval 0,. The graph is smmetric with respect to the ais. The graph has a relative minimum at 0, 0. The graph of the squaring function is shown in Figure.68. f() = (0, 0) FIGURE.68
3 0_006.qd /7/0 8:0 AM Page Chapter Functions and Their Graphs Cubic, Square Root, and Reciprocal Functions The basic characteristics of the graphs of the cubic, square root, and reciprocal functions are summarized below.. The graph of the cubic function f has the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all real numbers. The function is odd. The graph has an intercept at 0, 0. The graph is increasing on the interval,. The graph is smmetric with respect to the origin. The graph of the cubic function is shown in Figure.69.. The graph of the square root function f has the following characteristics. The domain of the function is the set of all nonnegative real numbers. The range of the function is the set of all nonnegative real numbers. The graph has an intercept at 0, 0. The graph is increasing on the interval 0,. The graph of the square root function is shown in Figure.70.. The graph of the reciprocal function f has the following characteristics. The domain of the function is, 0 0,. The range of the function is, 0 0,. The function is odd. The graph does not have an intercepts. The graph is decreasing on the intervals, 0 and 0,. The graph is smmetric with respect to the origin. The graph of the reciprocal function is shown in Figure.7. f() = f() = f() = (0, 0) (0, 0) Cubic function FIGURE.69 Square root function FIGURE.70 Reciprocal function FIGURE.7
4 0_006.qd /7/0 8:0 AM Page 69 Section.6 A Librar of Parent Functions 69 Demonstrate the reallife nature of step functions b discussing Eercises If writing is a part of our course, this section provides a good opportunit for students to find other eamples of step functions and write brief essas on their applications. FIGURE.7 f () =[[ ]] Technolog When graphing a step function, ou should set our graphing utilit to dot mode. Step and PiecewiseDefined Functions Functions whose graphs resemble sets of stairsteps are known as step functions. The most famous of the step functions is the greatest integer function, which is denoted b and defined as f the greatest integer less than or equal to. Some values of the greatest integer function are as follows. greatest integer greatest integer 0 greatest integer 0 0. greatest integer. The graph of the greatest integer function f has the following characteristics, as shown in Figure.7. The domain of the function is the set of all real numbers. The range of the function is the set of all integers. The graph has a intercept at 0, 0 and intercepts in the interval 0,. The graph is constant between each pair of consecutive integers. The graph jumps verticall one unit at each integer value. Eample Evaluating a Step Function FIGURE.7 f ()=[[ ]] + Evaluate the function when,, and f Solution For, the greatest integer is, so f 0. For, the greatest integer is, so f. For, the greatest integer is, so f. You can verif our answers b eamining the graph of f shown in Figure.7. Now tr Eercise 9. Recall from Section. that a piecewisedefined function is defined b two or more equations over a specified domain. To graph a piecewisedefined function, graph each equation separatel over the specified domain, as shown in Eample..
5 0_006.qd /7/0 8:0 AM Page Chapter Functions and Their Graphs = + FIGURE = + Eample Sketch the graph of f,, Solution Graphing a PiecewiseDefined Function This piecewisedefined function is composed of two linear functions. At and to the left of the graph is the line, and to the right of the graph is the line, as shown in Figure.7. Notice that the point, is a solid dot and the point, is an open dot. This is because f. Parent Functions >. Now tr Eercise. The eight graphs shown in Figure.7 represent the most commonl used functions in algebra. Familiarit with the basic characteristics of these simple graphs will help ou analze the shapes of more complicated graphs in particular, graphs obtained from these graphs b the rigid and nonrigid transformations studied in the net section. f() = f() = f() = c f() = (a) Constant Function (b) Identit Function (c) Absolute Value Function (d) Square Root Function f() = f() = f() = f () =[[ ]] (e) Quadratic Function FIGURE.7 (f) Cubic Function (g) Reciprocal Function (h) Greatest Integer Function
6 0_006.qd /7/0 8:0 AM Page 7 Section.6 A Librar of Parent Functions 7.6 Eercises VOCABULARY CHECK: Match each function with its name.. f. f. f. f. f 6. f c f 9. f a b f (a) squaring function (b) square root function (c) cubic function (d) linear function (e) constant function (f) absolute value function (e) greatest integer function (h) reciprocal function (i) identit function PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Eercises 8, (a) write the linear function f such that it has the indicated function values and (b) sketch the graph of the function.. f, f 0 6. f 8, f. f, f 7. f 9, f. 6. f, f f 0, f 6 7. f 6, f 8. f, f In Eercises 9 8, use a graphing utilit to graph the function. Be sure to choose an appropriate viewing window. 9. f 0. f. f 6. f 6. f. f 8. h 6. g f 8. f 8 9. f 0. g. f. f. g. h. f 6. f 7. h 8. k In Eercises 9 6, evaluate the function for the indicated values. 9. f (a) f. (b) f.9 (c) f. (d) 0. g (a) g (b) g 0. (c) g 9. (d) f 7 g. h (a) h (b) h (c) h. (d) h.6. f 7 (a) f 0 (b) f. (c) f 6 (d) f. h (a) h. (b) h. (c) h 7 (d) h. k 6 (a) k (b) k 6. (c) k 0. (d) k. g (a) g.7 (b) g (c) g 0.8 (d) g. 6. g 7 6 (a) (b) g 9 (c) g (d) In Eercises 7, sketch the graph of the function. 7. g 8. g 9. g 0. g. g. g In Eercises 0, graph the function g 8 f,, g 6,, > f,, < 0 0 f,, < f,, > > g
7 0_006.qd /7/0 8:0 AM Page 7 7 Chapter Functions and Their Graphs h,, h,,,, k,, In Eercises and, (a) use a graphing utilit to graph the function, (b) state the domain and range of the function, and (c) describe the pattern of the graph. s.. In Eercises 60, (a) identif the parent function and the transformed parent function shown in the graph, (b) write an equation for the function shown in the graph, and (c) use a graphing utilit to verif our answers in parts (a) and (b) < 0 0 < < 0 0 < > g 6. Communications The cost of a telephone call between Denver and Boise is $0.60 for the first minute and $0. for each additional minute or portion of a minute. A model for the total cost C (in dollars) of the phone call is C t, t > 0 where t is the length of the phone call in minutes. (a) Sketch the graph of the model. (b) Determine the cost of a call lasting minutes and 0 seconds. 6. Communications The cost of using a telephone calling card is $.0 for the first minute and $0.8 for each additional minute or portion of a minute. (a) A customer needs a model for the cost C of using a calling card for a call lasting t minutes. Which of the following is the appropriate model? Eplain. C t t C t t (b) Graph the appropriate model. Determine the cost of a call lasting 8 minutes and seconds. 6. Deliver Charges The cost of sending an overnight package from Los Angeles to Miami is $0.7 for a package weighing up to but not including pound and $.9 for each additional pound or portion of a pound. A model for the total cost C (in dollars) of sending the package is C 0.7.9, > 0 where is the weight in pounds. (a) Sketch a graph of the model. (b) Determine the cost of sending a package that weighs 0. pounds. 6. Deliver Charges The cost of sending an overnight package from New York to Atlanta is $9.80 for a package weighing up to but not including pound and $.0 for each additional pound or portion of a pound. (a) Use the greatest integer function to create a model for the cost C of overnight deliver of a package weighing pounds, > 0. (b) Sketch the graph of the function. 6. Wages A mechanic is paid $.00 per hour for regular time and timeandahalf for overtime. The weekl wage function is given b W h h, 8 h 0 80, 0 < h 0 h > 0 where h is the number of hours worked in a week. (a) Evaluate W 0, W 0, W, and W 0. (b) The compan increased the regular work week to hours. What is the new weekl wage function?
8 0_006.qd /7/0 8:0 AM Page 7 Section.6 A Librar of Parent Functions Snowstorm During a ninehour snowstorm, it snows at a rate of inch per hour for the first hours, at a rate of inches per hour for the net 6 hours, and at a rate of 0. inch per hour for the final hour. Write and graph a piecewisedefined function that gives the depth of the snow during the snowstorm. How man inches of snow accumulated from the storm? Model It 67. Revenue The table shows the monthl revenue (in thousands of dollars) of a landscaping business for each month of the ear 00, with representing Januar. Volume (in gallons) V (60, 00) 00 (0, 7) (0, 7) 7 (, 0) 0 (, 0) (0, 0) (0, ) (0, ) (0, 0) Time (in minutes) FIGURE FOR 68 t Month, Revenue, Snthesis A mathematical model that represents these data is f (a) What is the domain of each part of the piecewisedefined function? How can ou tell? Eplain our reasoning. (b) Sketch a graph of the model. (c) Find f and f, and interpret our results in the contet of the problem. (d) How do the values obtained from the model in part (b) compare with the actual data values? 68. Fluid Flow The intake pipe of a 00gallon tank has a flow rate of 0 gallons per minute, and two drainpipes have flow rates of gallons per minute each. The figure shows the volume V of fluid in the tank as a function of time t. Determine the combination of the input pipe and drain pipes in which the fluid is flowing in specific subintervals of the hour of time shown on the graph. (There are man correct answers.) True or False? In Eercises 69 and 70, determine whether the statement is true or false. Justif our answer. 69. A piecewisedefined function will alwas have at least one intercept or at least one intercept. 70. f,, 6, can be rewritten as f, <. Eploration In Eercises 7 and 7, write equations for the piecewisedefined function shown in the graph (0, 6) (, ) 6 8 Skills Review < < < (8, 0) In Eercises 7 and 7, solve the inequalit and sketch the solution on the real number line > 6 9 L In Eercises 7 and 76, determine whether the lines and L passing through the pairs of points are parallel, perpendicular, or neither. 7. L :,,, L :, 7,, L :,,, 9 L :,,, (, ) (, ) (7, 0) (, ) 6 (0, 0)
1.6 A LIBRARY OF PARENT FUNCTIONS. Copyright Cengage Learning. All rights reserved.
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