SECTION 2.5 Graphing Techniques: Transformations 125 91. Temperature Measurements The relationship between the Celsius C ( ) ° and Fahrenheit F ( ) ° scales for measuring temperature is given by the equation = + F C 9 5 32 The relationship between the Celsius C ( ) ° and Kelvin (K) scales is K C 273. = + Graph the equation = + F C 9 5 32 using degrees Fahrenheit on the y-axis and degrees Celsius on the x-axis. Use the techniques introduced in this section to obtain the graph showing the relationship between Kelvin and Fahrenheit temperatures. 92. Period of a Pendulum The period T (in seconds) of a simple pendulum is a function of its length l (in feet) defined by the equation T l g 2π = where g 32.2 ≈ feet per second per second is the acceleration due to gravity. (a) Use a graphing utility to graph the function T T l . ( ) = (b) Now graph the functions T T l T T l 1 , 2 , ( ) ( ) = + = + and T T l 3 . ( ) = + (c) Discuss how adding to the length l changes the period T. (d) Now graph the functions T T l T T l 2 , 3 , ( ) ( ) = = and T T l4 . ( ) = (e) Discuss how multiplying the length l by factors of 2, 3, and 4 changes the period T. 93. The equation y x c 2 ( ) = − defines a family of parabolas, one parabola for each value of c. On one set of coordinate axes, graph the members of the family for c c 0, 3, = = and c 2. = − 94. Repeat Problem 93 for the family of parabolas y x c. 2 = + 95. Challenge Problem If a function f is increasing on the intervals 3, 3 [ ] − and 11, 19 [ ] and decreasing on the interval 3, 11 [ ], determine the interval(s) on which g x f x 3 2 5 ( ) ( ) = − − is increasing. 96. Challenge Problem In statistics, the standard normal density function is given by π ( ) = ⋅ ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ f x x 1 2 exp 2 . 2 This function can be transformed to describe any general normal distribution with mean, ,μ and standard deviation, .σ A general normal density function is given by f x x 1 2 exp 2 . 2 2 π σ μ σ ( ) ( ) = ⋅ ⋅ − ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Describe the transformations needed to get from the graph of the standard normal function to the graph of a general normal function. 89. Thermostat Control Energy conservation experts estimate that homeowners can save 5% to 10% on winter heating bills by programming their thermostats 5 to 10 degrees lower while sleeping. In the graph below, the temperature T (in degrees Fahrenheit) of a home is given as a function of time t (in hours after midnight) over a 24-hour period. t T 0 4 Time (hours after midnight) Temperature (°F) 56 60 64 68 72 76 80 8 (6, 65) (21, 72) 12 16 20 24 (a) At what temperature is the thermostat set during daytime hours? At what temperature is the thermostat set overnight? (b) The homeowner reprograms the thermostat to y T t 2. ( ) = − Explain how this affects the temperature in the house. Graph this new function. (c) The homeowner reprograms the thermostat to y T t 1 . ( ) = + Explain how this affects the temperature in the house. Graph this new function. Source: Roger Albright, 547 Ways to Be Fuel Smart, 2000 90. Daylight The average number of daylight hours per month in Orlando, Florida, can be modeled by the function f x x x 0.12 1.29 9.96, 2 ( ) = − + + where x is the number of months after January (x 0 = for January, x 1 = for February, and so on). (a) Find f f 0, 6, ( ) ( ) and f 11 , ( ) and explain what each value represents. (b) The values of F x f x – 6 ( ) ( ) = estimate the average number of daylight hours per month in Alice Springs, Australia. What does x represent in the function F x ? ( ) (c) Estimate the average number of hours of daylight in January in Alice Springs. (d) What might be the physical reason for the horizontal shift between F x( ) and f x( ) when modeling the average number of daylight hours in the two cities? 97. Suppose that the graph of a function f is known. Explain how the graph of y f x 4 ( ) = differs from the graph of y f x4 . ( ) = 98. Suppose that the graph of a function f is known. Explain how the graph of y f x 2 ( ) = − differs from the graph of y f x 2 . ( ) = − 99. The area under the curve y x = bounded from below by the x-axis and on the right by x 4 = is 16 3 square units. Using the ideas presented in this section, what do you think is the area under the curve of y x = − bounded from below by the x-axis and on the left by x 4? = − Justify your answer. 100. Explain how the range of the function f x x2 ( ) = compares to the range of g x f x k. ( ) ( ) = + 1 01. Explain how the domain of g x x ( ) = compares to the domain of g x k ( ) − , where k 0. ≥ Explaining Concepts
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