2.4 Library of Functions; Piecewise-defined Functions. 1 Graph the Functions Listed in the Library of Functions

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1 80 CHAPTER Functions and Their Graphs Problems 8 88 require the following discussion of a secant line. The slope of the secant line containing the two points, f and + h, f + h on the graph of a function = f ma be given as m sec = f + h - f + h - = f + h - f h h Z 0 In calculus, this epression is called the difference quotient of f. (a) Epress the slope of the secant line of each function in terms of and h. Be sure to simplif our answer. (b) Find m sec for h = 0.5, 0., and 0.0 at =. What value does m sec approach as h approaches 0? (c) Find the equation for the secant line at = with h = 0.0. (d) Use a graphing utilit to graph f and the secant line found in part (c) on the same viewing window. 8. f = f = f = + 8. f = f = f = f = 88. f = Eplaining Concepts: Discussion and Writing 89. Draw the graph of a function that has the following properties: domain: all real numbers; range: all real numbers; intercepts: 0, - and, 0; a local maimum value of - is at - ; a local minimum value of - 6 is at. Compare our graph with those of others. Comment on an differences. 90. Redo Problem 89 with the following additional information: increasing on - q, -,, q; decreasing on -,. Again compare our graph with others and comment on an differences. 9. How man -intercepts can a function defined on an interval have if it is increasing on that interval? Eplain. 9. Suppose that a friend of ours does not understand the idea of increasing and decreasing functions. Provide an eplanation, complete with graphs, that clarifies the idea. 9. Can a function be both even and odd? Eplain. 9. Using a graphing utilit, graph = 5 on the interval -,. Use MAXIMUM to find the local maimum values on -,. Comment on the result provided b the calculator. 95. A function f has a positive average rate of change on the interval, 5. Is f increasing on, 5? Eplain. 96. Show that a constant function f() = b has an average rate of change of 0. Compute the average rate of change of = - on the interval -,. Eplain how this can happen. Are You Prepared? Answers smmetric with respect to the -ais. + = , 0,, 0, 0, - 9. Librar of Functions; Piecewise-defined Functions PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section., pp. ) Graphs of Ke Equations (Section.: Eample, p. 0; Eample 0, p. 5; Eample, p. 5; Eample, p. 6) Now Work the Are You Prepared? problems on page 87. OBJECTIVES Graph the Functions Listed in the Librar of Functions (p. 80) Graph Piecewise-defined Functions (p. 85) Graph the Functions Listed in the Librar of Functions First we introduce a few more functions, beginning with the square root function. On page 5, we graphed the equation =. Figure 8 shows a graph of the function f =. Based on the graph, we have the following properties:

2 SECTION. Librar of Functions; Piecewise-defined Functions 8 Figure 8 6 (, ) (, ) 5 (9, ) 0 Properties of f(). The domain and the range are the set of nonnegative real numbers.. The -intercept of the graph of f = is 0. The -intercept of the graph of f = is also 0.. The function is neither even nor odd.. The function is increasing on the interval 0, q. 5. The function has an absolute minimum of 0 at = 0. EXAMPLE Solution Graphing the Cube Root Function (a) Determine whether f = is even, odd, or neither. State whether the graph of f is smmetric with respect to the -ais or smmetric with respect to the origin. (b) Determine the intercepts, if an, of the graph of f =. (c) Graph f =. (a) Because f- = - = - = - f the function is odd. The graph of f is smmetric with respect to the origin. (b) The -intercept is f0 = 0 = 0. The -intercept is found b solving the equation f = 0. f = 0 = 0 f() = = 0 Cube both sides of the equation. The -intercept is also 0. (c) Use the function to form Table and obtain some points on the graph. Because of the smmetr with respect to the origin, we find onl points, for which Ú 0. Figure 9 shows the graph of f =. Table f() L.6 (, ) a 8, b (, ) A, B 8 (8, ) Figure 9 (, ) 8 (, ) From the results of Eample and Figure 9, we have the following properties of the cube root function. (, ) (, ) (, ) (, ) 8 Properties of f(). The domain and the range are the set of all real numbers.. The -intercept of the graph of f = is 0.The -intercept of the graph of f = is also 0.. The graph is smmetric with respect to the origin. The function is odd.. The function is increasing on the interval - q, q. 5. The function does not have an local minima or an local maima.

3 8 CHAPTER Functions and Their Graphs EXAMPLE Solution Table 5 Graphing the Absolute Value Function (a) Determine whether f = ƒƒ is even, odd, or neither. State whether the graph of f is smmetric with respect to the -ais or smmetric with respect to the origin. (b) Determine the intercepts, if an, of the graph of f = ƒƒ. (c) Graph f = ƒƒ. (a) Because f- = ƒ - ƒ = ƒƒ = f the function is even. The graph of f is smmetric with respect to the -ais. (b) The -intercept is f0 = ƒ0ƒ = 0. The -intercept is found b solving the equation f = 0 or ƒƒ = 0. So the -intercept is 0. (c) Use the function to form Table 5 and obtain some points on the graph. Because of the smmetr with respect to the -ais, we need to find onl points, for which Ú 0. Figure 0 shows the graph of f = ƒƒ. f (), 0 0 (, ) (, ) (, ) Figure 0 (, ) (, ) (, ) From the results of Eample and Figure 0, we have the following properties of the absolute value function. (, ) (, ) (, ) Properties of f() ƒ ƒ. The domain is the set of all real numbers. The range of f is 5 Ú 06.. The -intercept of the graph of f = ƒƒ is 0. The -intercept of the graph of f = ƒƒ is also 0.. The graph is smmetric with respect to the -ais. The function is even.. The function is decreasing on the interval - q, 0. It is increasing on the interval 0, q. 5. The function has an absolute minimum of 0 at = 0. Seeing the Concept Graph = ƒƒ on a square screen and compare what ou see with Figure 0. Note that some graphing calculators use abs for absolute value. Below is a list of the ke functions that we have discussed. In going through this list, pa special attention to the properties of each function, particularl to the shape of each graph. Knowing these graphs along with ke points on each graph will la the foundation for further graphing techniques. Figure Constant Function f() = b (0,b) Constant Function See Figure. f = b b is a real number

4 SECTION. Librar of Functions; Piecewise-defined Functions 8 The domain of a constant function is the set of all real numbers; its range is the set consisting of a single number b.its graph is a horizontal line whose -intercept is b. The constant function is an even function. Identit Function Figure Identit Function f = (, ) f() = (, ) See Figure. The domain and the range of the identit function are the set of all real numbers. Its graph is a line whose slope is and whose -intercept is 0. The line consists of all points for which the -coordinate equals the -coordinate. The identit function is an odd function that is increasing over its domain. Note that the graph bisects quadrants I and III. Figure Square Function Square Function (, ) f() = (, ) f = (, ) (, ) See Figure. The domain of the square function f is the set of all real numbers; its range is the set of nonnegative real numbers. The graph of this function is a parabola whose intercept is at 0, 0. The square function is an even function that is decreasing on the interval - q, 0 and increasing on the interval 0, q. Figure Cube Function f() = Cube Function f = (, ) (, ) See Figure. The domain and the range of the cube function are the set of all real numbers. The intercept of the graph is at 0, 0. The cube function is odd and is increasing on the interval - q, q. Figure 5 Square Root Function Square Root Function (, ) f() = (, ) f = (, ) Figure 6 Cube Root Function (, ) 8 5 (, ) (, ) (, ) (, ) 8 See Figure 5. The domain and the range of the square root function are the set of nonnegative real numbers. The intercept of the graph is at 0, 0. The square root function is neither even nor odd and is increasing on the interval 0, q. Cube Root Function f = See Figure 6. The domain and the range of the cube root function are the set of all real numbers. The intercept of the graph is at 0, 0. The cube root function is an odd function that is increasing on the interval - q, q.

5 8 CHAPTER Functions and Their Graphs Reciprocal Function f = Figure 7 Reciprocal Function (, ) (, ) f() = (, ) Refer to Eample, page 6, for a discussion of the equation = See. Figure 7. The domain and the range of the reciprocal function are the set of all nonzero real numbers. The graph has no intercepts. The reciprocal function is decreasing on the intervals - q, 0 and 0, q and is an odd function. (, ) Absolute Value Function f = ƒƒ Figure 8 Absolute Value Function f() = (, ) (, ) (, ) (, ) DEFINITION See Figure 8. The domain of the absolute value function is the set of all real numbers; its range is the set of nonnegative real numbers. The intercept of the graph is at 0, 0. If Ú 0, then f =, and the graph of f is part of the line = ; if 6 0, then f = -, and the graph of f is part of the line = -. The absolute value function is an even function; it is decreasing on the interval - q, 0 and increasing on the interval 0, q. The notation int stands for the largest integer less than or equal to. For eample, int =, int.5 =, inta b = 0, inta - b = -, intp = This tpe of correspondence occurs frequentl enough in mathematics that we give it a name. Greatest Integer Function f = int* = greatest integer less than or equal to Table f () int() (, ) - - (-, - ) a -, - b We obtain the graph of f = int b plotting several points. See Table 6. For values of, - 6 0, the value of f = int is - ; for values of, 0 6, the value of f is 0. See Figure 9 for the graph. Figure 9 Greatest Integer Function a - a, 0b a, 0b 0 a, 0b, - b The domain of the greatest integer function is the set of all real numbers; its range is the set of integers. The -intercept of the graph is 0. The -intercepts lie in the interval 0,. The greatest integer function is neither even nor odd. It is constant on ever interval of the form k, k +, for k an integer. In Figure 9, we use a solid dot to indicate, for eample, that at = the value of f is f = ; we use an open circle to illustrate that the function does not assume the value of 0 at =. * Some books use the notation f = instead of int.

6 SECTION. Librar of Functions; Piecewise-defined Functions 85 Figure 0 f = int 6 6 (a) Connected mode 6 6 (b) Dot mode Although a precise definition requires the idea of a limit, discussed in calculus, in a rough sense, a function is said to be continuous if its graph has no gaps or holes and can be drawn without lifting a pencil from the paper on which the graph is drawn. We contrast this with a discontinuous function. A function is discontinuous if its graph has gaps or holes so that its graph cannot be drawn without lifting a pencil from the paper. From the graph of the greatest integer function, we can see wh it is also called a step function. At = 0, = ;, = ;, and so on, this function is discontinuous because, at integer values, the graph suddenl steps from one value to another without taking on an of the intermediate values. For eample, to the immediate left of =, the -coordinates of the points on the graph are, and at = and to the immediate right of =, the -coordinates of the points on the graph are. So, the graph has gaps in it. COMMENT When graphing a function using a graphing utilit, ou can choose either the connected mode, in which points plotted on the screen are connected, making the graph appear without an breaks, or the dot mode, in which onl the points plotted appear. When graphing the greatest integer function with a graphing utilit, it ma be necessar to be in the dot mode. This is to prevent the utilit from connecting the dots when f changes from one integer value to the net. See Figure 0. The functions discussed so far are basic. Whenever ou encounter one of them, ou should see a mental picture of its graph. For eample, if ou encounter the function f =, ou should see in our mind s ee a picture like Figure. Now Work PROBLEMS 9 THROUGH 6 Graph Piecewise-defined Functions Sometimes a function is defined using different equations on different parts of its domain. For eample, the absolute value function f = ƒƒ is actuall defined b two equations: f = if Ú 0 and f = - if 6 0. For convenience, these equations are generall combined into one epression as f = ƒƒ = e if Ú 0 - if 6 0 When a function is defined b different equations on different parts of its domain, it is called a piecewise-defined function. EXAMPLE Analzing a Piecewise-defined Function The function f is defined as f = c - + if - 6 if = if 7 (a) Find f-, f, and f. (b) Determine the domain of f. (c) Locate an intercepts. (d) Graph f. (e) Use the graph to find the range of f. (f) Is f continuous on its domain? Solution (a) To find f-, observe that when = - the equation for f is given b f = - +. So f- = - (- ) + = 5 When =, the equation for f is f =. That is, f = When =, the equation for f is f =. So f = =

7 86 CHAPTER Functions and Their Graphs (b) To find the domain of f, look at its definition. Since f is defined for all greater than or equal to -, the domain of f is { Ú - }, or the interval -, q. (c) The -intercept of the graph of the function is f0. Because the equation for f when = 0 is f = - +, the -intercept is f0 = =. The -intercepts of the graph of a function f are the real solutions to the equation f = 0. To find the -intercepts of f, solve f = 0 for each piece of the function and then determine if the values of, if an, satisf the condition that defines the piece. f = = 0 - = - = - 6 f = 0 = 0 No solution = f = 0 = 0 = 0 7 Figure (, 0) (0,) 8 (,) (,) (, ) The first potential -intercept, =, satisfies the condition - 6, so = is an -intercept. The second potential -intercept, = 0, does not satisf the condition 7, so = 0 is not an -intercept. The onl -intercept is. The intercepts are (0, ) and a., 0b (d) To graph f, graph each piece. First graph the line = - + and keep onl the part for which - 6. Then plot the point, because, when =, f =. Finall, graph the parabola = and keep onl the part for which 7. See Figure. (e) From the graph, we conclude that the range of f is 5 ƒ 7-6, or the interval -, q. (f) The function f is not continuous because there is a jump in the graph at =. Now Work PROBLEM 9 EXAMPLE Cost of Electricit In the summer of 009, Duke Energ supplied electricit to residences of Ohio for a monthl customer charge of $.50 plus.5 per kilowatt-hour (kwhr) for the first 000 kwhr supplied in the month and 5.6 per kwhr for all usage over 000 kwhr in the month. (a) What is the charge for using 00 kwhr in a month? (b) What is the charge for using 500 kwhr in a month? (c) If C is the monthl charge for kwhr, develop a model relating the monthl charge and kilowatt-hours used. That is, epress C as a function of. Source: Duke Energ, 009. Solution (a) For 00 kwhr, the charge is $.50 plus.5 = $0.05 per kwhr. That is, Charge = $.50 + $ = $7.0 (b) For 500 kwhr, the charge is $.50 plus.5 per kwhr for the first 000 kwhr plus 5.6 per kwhr for the 500 kwhr in ecess of 000. That is, Charge = $.50 + $ $ = $7.66 (c) Let represent the number of kilowatt-hours used. If 0 000, the monthl charge C (in dollars) can be found b multipling times $0.05 and adding the monthl customer charge of $.50. So, if 0 000, then C =

8 SECTION. Librar of Functions; Piecewise-defined Functions 87 Figure Charge (dollars) C (00, 7.0) (500, 7.66) Usage (kwhr) (000, 6.85) 500. Assess Your Understanding Are You Prepared? Answers are given at the end of these eercises. If ou get a wrong answer, read the pages listed in red.. Sketch the graph of =. (p. 5). List the intercepts of the equation = - 8. (pp. ). Sketch the graph of = (p. 6). Concepts and Vocabular For 7 000, the charge is , since equals the usage in ecess of 000 kwhr, which costs $0.056 per kwhr. That is, if 7 000, then C = The rule for computing C follows two equations: if C = e if See Figure for the graph. = = The Model. The function f = is decreasing on the interval 7. True or False The cube root function is odd and is. decreasing on the interval - q, q. 5. When functions are defined b more than one equation, the are called functions. 6. True or False The cube function is odd and is increasing on the interval - q, q. Skill Building In Problems 9 6, match each graph to its function. 8. True or False The domain and the range of the reciprocal function are the set of all real numbers. A. Constant function B. Identit function C. Square function D. Cube function E. Square root function F. Reciprocal function G. Absolute value function H. Cube root function In Problems 7, sketch the graph of each function. Be sure to label three points on the graph. 7. f = 8. f = 9. f = 0. f =. f =. f = ƒƒ. f =. f = 5. If 7. If if 6 0 f = c if = 0 + if 7 0 find: (a) f- (b) f0 (c) f f = e - if - - if 6 find: (a) f0 (b) f (c) f (d) f - if 6-6. If f = c 0 if = - + if 7-8. If find: (a) f- (b) f- (c) f0 f = e if if find: (a) f- (b) f0 (c) f (d) f()

9 88 CHAPTER Functions and Their Graphs In Problems 9 0: (a) Find the domain of each function. (b) Locate an intercepts. (c) Graph each function. (d) Based on the graph, find the range. (e) Is f continuous on its domain? if Z 0 9. f = b if = 0. f = b + if if Ú - 5. f = b + if 6 0 if Ú 0 8. f = b - if - 6 if 7 In Problems, the graph of a piecewise-defined function is given. Write a definition for each function..... (, ) (, ) (, ) (, ) (, ) if Z 0 0. f = b if = 0 + if - 6. f = c 5 if = - + if 7 6. f = c if 6 0 if Ú 0 (, ) (, 0) - + if 6. f = b - if Ú + 5 if f = c - if = 0-5 if 7 0 ƒ ƒ if f = b if f = int 0. f = int (, 0) (0, ) (, ) (, ) 5. If f = int, find (a) f. (b) f.6 (c) f-.8 Applications and Etensions 7. Cell Phone Service Sprint PCS offers a monthl cellular phone plan for $9.99. It includes 50 antime minutes and charges $0.5 per minute for additional minutes. The following function is used to compute the monthl cost for a subscriber: 9.99 if 0 50 C = b if 7 50 where is the number of antime minutes used. Compute the monthl cost of the cellular phone for use of the following number of antime minutes: (a) 00 (b) 65 (c) 5 Source: Sprint PCS 8. Parking at O Hare International Airport The short-term (no more than hours) parking fee F (in dollars) for parking hours at O Hare International Airport s main parking garage can be modeled b the function if 0 6 F = c 5 int + + if if 9 Determine the fee for parking in the short-term parking garage for (a) hours (b) 7 hours (c) 5 hours (d) 8 hours and minutes Source: O Hare International Airport 6. If f = inta find b, (a) f. (b) f.6 (c) f Cost of Natural Gas In April 009, Peoples Energ had the following rate schedule for natural gas usage in singlefamil residences: Monthl service charge $5.95 Per therm service charge st 50 therms $0.606/therm Over 50 therms $0.0580/therm Gas charge $0.90/therm (a) What is the charge for using 50 therms in a month? (b) What is the charge for using 500 therms in a month? (c) Develop a model that relates the monthl charge C for therms of gas. (d) Graph the function found in part (c). Source: Peoples Energ, Chicago, Illinois, Cost of Natural Gas In April 009, Nicor Gas had the following rate schedule for natural gas usage in singlefamil residences: Monthl customer charge $8.0 Distribution charge st 0 therms $0.7/therm Net 0 therms $0.0579/therm Over 50 therms $0.059/therm Gas suppl charge $0./therm (a) What is the charge for using 0 therms in a month? (b) What is the charge for using 50 therms in a month? (c) Develop a model that gives the monthl charge C for therms of gas. (d) Graph the function found in part (c). Source: Nicor Gas, Aurora, Illinois, 009

10 SECTION. Librar of Functions; Piecewise-defined Functions Federal Income Ta Two 009 Ta Rate Schedules are given in the accompaning table. If equals taable income and equals the ta due, construct a function = f for Schedule X. Schedule X Single If Taable Income Is Over But Not Over The Ta Is This Amount REVISED 009 TAX RATE SCHEDULES Plus This % Of the Ecess Over Schedule Y- Married Filing jointl or qualifing Widow(er) If Taable Income Is Over But Not Over The Ta Is This Amount Plus This % Of The Ecess Over $0 $8,50 0% $0 $0 $6,700 0% $0 8,50,950 $ % 8,50 6,700 67,900 $, % 6,700,950 8,50, %,950 67,900 7,050 9, % 67,900 8,50 7,550 6, % 8,50 7,050 08,850 6, % 7,050 7,550 7,950,75.00 % 7,550 08,850 7,950 6,7.50 % 08,850 7,950 08,6.00 5% 7,950 7,950 00, % 7,950 Source: Internal Revenue Service 5. Federal Income Ta Refer to the revised 009 ta rate schedules. If equals taable income and equals the ta due, construct a function = f for Schedule Y-. 5. Cost of Transporting Goods A trucking compan transports goods between Chicago and New York, a distance of 960 miles. The compan s polic is to charge, for each pound, $0.50 per mile for the first 00 miles, $0.0 per mile for the net 00 miles, $0.5 per mile for the net 00 miles, and no charge for the remaining 60 miles. (a) Graph the relationship between the cost of transportation in dollars and mileage over the entire 960-mile route. (b) Find the cost as a function of mileage for hauls between 00 and 00 miles from Chicago. (c) Find the cost as a function of mileage for hauls between 00 and 800 miles from Chicago. 5. Car Rental Costs An econom car rented in Florida from National Car Rental on a weekl basis costs $95 per week. Etra das cost $ per da until the da rate eceeds the weekl rate, in which case the weekl rate applies. Also, an part of a da used counts as a full da. Find the cost C of renting an econom car as a function of the number of das used, where 7. Graph this function. 55. Minimum Paments for Credit Cards Holders of credit cards issued b banks, department stores, oil companies, and so on, receive bills each month that state minimum amounts that must be paid b a certain due date. The minimum due depends on the total amount owed. One such credit card compan uses the following rules: For a bill of less than $0, the entire amount is due. For a bill of at least $0 but less than $500, the minimum due is $0. A minimum of $0 is due on a bill of at least $500 but less than $000, a minimum of $50 is due on a bill of at least $000 but less than $500, and a minimum of $70 is due on bills of $500 or more. Find the function f that describes the minimum pament due on a bill of dollars. Graph f. 56. Interest Paments for Credit Cards Refer to Problem 55. The card holder ma pa an amount between the minimum due and the total owed. The organization issuing the card charges the card holder interest of.5% per month for the first $000 owed and % per month on an unpaid balance over $000. Find the function g that gives the amount of interest charged per month on a balance of dollars. Graph g. 57. Wind Chill The wind chill factor represents the equivalent air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is t 0 v 6.79 W = d v - v - t.0.79 v t v 7 0 where v represents the wind speed (in meters per second) and t represents the air temperature ( C). Compute the wind chill for the following: (a) An air temperature of 0 C and a wind speed of meter per second m/sec (b) An air temperature of 0 C and a wind speed of 5 m/sec (c) An air temperature of 0 C and a wind speed of 5 m/sec (d) An air temperature of 0 C and a wind speed of 5 m/sec (e) Eplain the phsical meaning of the equation corresponding to 0 v (f) Eplain the phsical meaning of the equation corresponding to v Wind Chill Redo Problem 57(a) (d) for an air temperature of - 0 C. 59. First-class Mail In 009 the U.S. Postal Service charged $.7 postage for first-class mail retail flats (such as an 8.5 b envelope) weighing up to ounce, plus $0.7 for each additional ounce up to ounces. First-class rates do not appl to flats weighing more than ounces. Develop a model that relates C, the first-class postage charged, for a flat weighing ounces. Graph the function. Source: United States Postal Service

11 90 CHAPTER Functions and Their Graphs Eplaining Concepts: Discussion and Writing In Problems 60 67, use a graphing utilit. 60. Eploration Graph =. Then on the same screen 65. Eploration Graph =. Then on the same screen graph graph = +, followed b = +, followed b = - +. Could ou have predicted the result? = -. What pattern do ou observe? Can ou predict 66. Eploration Graph =, =, and = 6 on the the graph of = -? Of = + 5? same screen. What do ou notice is the same about each 6. Eploration Graph =. Then on the same screen graph? What do ou notice that is different? graph = -, followed b = -, followed b 67. Eploration Graph =, = 5, and = 7 on the = +. What pattern do ou observe? Can ou same screen. What do ou notice is the same about each predict the graph of = +? Of = - 5? graph? What do ou notice that is different? 6. Eploration Graph = ƒƒ. Then on the same screen 68. Consider the equation graph = ƒƒ, followed b = ƒƒ, followed b = ƒƒ. What pattern do ou observe? Can ou predict the graph of = Of = 5ƒƒ? ƒƒ? 6. Eploration Graph =. Then on the same screen graph = -. What pattern do ou observe? Now tr = ƒƒ and = - ƒƒ. What do ou conclude? 6. Eploration Graph =. Then on the same screen graph = -. What pattern do ou observe? Now tr = + and = - +. What do ou conclude? Are You Prepared? Answers... 0, - 8,, 0 (, ) (, ) (, ) (, ) = b if is rational 0 if is irrational Is this a function? What is its domain? What is its range? What is its -intercept, if an? What are its -intercepts, if an? Is it even, odd, or neither? How would ou describe its graph? 69. Define some functions that pass through 0, 0 and, and are increasing for Ú 0. Begin our list with =, =, and =. Can ou propose a general result about such functions?.5 Graphing Techniques: Transformations OBJECTIVES Graph Functions Using Vertical and Horizontal Shifts (p. 90) Graph Functions Using Compressions and Stretches (p. 9) Graph Functions Using Reflections about the -Ais and the -Ais (p. 96) At this stage, if ou were asked to graph an of the functions defined b =, = =, =, =, =, or = ƒƒ, our response should, be, Yes, I recognize these functions and know the general shapes of their graphs. (If this is not our answer, review the previous section, Figures through 8.) Sometimes we are asked to graph a function that is almost like one that we alread know how to graph. In this section, we develop techniques for graphing such functions. Collectivel, these techniques are referred to as transformations. Graph Functions Using Vertical and Horizontal Shifts EXAMPLE Vertical Shift Up Use the graph of f = to obtain the graph of g = +. Solution Begin b obtaining some points on the graphs of f and g. For eample, when = 0, then = f0 = 0 and = g0 =. When =, then = f = and