3.3 Horizontal and Vertical Translations of Functions

Transcription

1 . Horizontal and Vertical Translations of Functions When an object is dropped from the top of a bridge over a bod of water, the approimate height of the falling object above the water is given b the function h(t) = 5t + d where h(t) metres is the height of the object t seconds after it is dropped, and d metres is the height of the bridge. INVESTIGATE & INQUIRE. The table includes the approimate heights, in metres, of three famous Canadian bridges. Write the function that describes the height of a falling object above the water t seconds after it is dropped from the top of each bridge. Bridge Ambassador Bridge Confederation Bridge Capilano Canon Suspension Bridge Height (m) 5 7. Graph h(t) versus t for the three functions on the same set of aes or in the same viewing window of a graphing calculator.. In what quadrant do the three graphs appear? Eplain wh.. For a given t-coordinate, how does the h-coordinate of a point on the graph for the Confederation Bridge compare with the h-coordinate of a point on the graph for the Ambassador Bridge? Eplain wh. 5. For a given t-coordinate, how does the h-coordinate of a point on the graph for the Capilano Bridge compare with the h-coordinate of a point on the graph for the Ambassador Bridge? Eplain wh.. Graph the three functions = 5 + c for c = 5, c =, and c = 7 on the same set of aes or in the same viewing window of a graphing calculator. If the domain of the three functions is the set of real numbers, how do the three graphs compare with the three graphs of h(t) versus t from question? Eplain wh. 7. How do the three graphs from question compare with the graph of = over the same domain? Eplain wh. 8 MHR Chapter

2 EXAMPLE Positive Vertical Translation a) Graph the functions = and = + on the same set of aes. b) How does the graph of = compare to the graph of = +? SOLUTION a) = = = + b) The graphs of = and = + are congruent. The graph of the function = + is obtained when the graph of the function = undergoes a vertical translation of units in the positive direction, that is, upward. The graph of the function = + can also be obtained from the graph of = b adding to each -value on the graph of =. The point (, ) on the graph of = is transformed to become the point (, + ) on the graph of = +. The graphs show that the functions have the same domain but different ranges. The domain of each function is the set of real numbers. The range of = is. The range of = + is. 8 = The window variables include Xmin =, Xma =, Ymin =, and Yma =. Note that the transformations shown in this chapter can be performed using a graphing software program, such as Zap-a-Graph. For details of how to do this, refer to the Zap-a-Graph section of Appendi C. EXAMPLE Vertical Translations How do the graphs of = + and = compare with the graph of =, where.. Horizontal and Vertical Translations of Functions MHR 85

3 SOLUTION = = + = 9 = The window variables include Xmin =, Xma = 8, Ymin =, and Yma = 8. = 8 = The graph of the function = + is the graph of the function = with a vertical translation of units upward. The point (, ) on the graph of = is transformed to become the point (, + ) on the graph of = +. Similarl, the graph of the function = is the graph of the function = with a vertical translation of units downward. The point (, ) on the graph of = is transformed to become the point (, ) on the graph of =. All three graphs are congruent and have domain. The range of = is, of = + is, and of = is. The results from Eamples and can be generalized for all functions as follows. The graph of = f() + k is congruent to the graph of = f(). If k >, the graph of = f() + k is the graph of = f() translated upward b k units. If k <, the graph of = f() + k is the graph of = f() translated downward b k units. 8 MHR Chapter = f() + = f() 8 = f()

4 EXAMPLE Horizontal Translations How do the graphs of = + and = compare to the graph of =. SOLUTION Complete tables of values using convenient values for, or use a graphing calculator. = = + = 9 7 = The window variables include Xmin =, Xma = 8, Ymin =, and Yma = 5. = 8 = The graphs of =, =, + and = are congruent. The graph of = + is obtained when the graph of = is translated horizontall units to the left. The graph of = + is also obtained from the graph of = b subtracting from each -value on the graph of =. The point (, ) on the graph of = is transformed to become the point (, ) on the graph of =. + Similarl, the graph of = is obtained when the graph of = is translated horizontall units to the right. The graph of = is also obtained from the graph of = b adding to each -value on the graph of =. The point (, ) on the graph of = is transformed to become the point ( +, ) on the graph of =. All three graphs have the same range,. The domain of = is, of = + is, and of = is.. Horizontal and Vertical Translations of Functions MHR 87

5 The results from Eample can be generalized for all functions as follows. The graph of = f( h) is congruent to the graph of = f(). If h >, the graph of = f( h) is the graph of = f() translated to the right b h units. If h <, the graph of = f( h) is the graph of = f() translated to the left b h units. =f( + ) 8 =f() =f( ) 8 EXAMPLE Horizontal and Vertical Translations Sketch the graph of = ( ) +. SOLUTION Sketch the graph of =. Translate the graph of = three units to the right to obtain the graph of = ( ). Translate the graph of = ( ) four units upward to obtain the graph of = ( ) +. The point (, ) on the function = is transformed to become the point ( +, + ). For eample, (, ) becomes (, ). 8 = = ( ) + = ( ) 8 Note that, in Eample, ou could graph the three functions using a graphing calculator. However, it is not necessar to graph = or = ( ) before graphing = ( ) +. The window variables include Xmin =, Xma = 7, Ymin =, and Yma =. 88 MHR Chapter

6 Ke Concepts A function and its translation image are congruent. The table summarizes translations of the function = f(). Translation Vertical Horizontal Mathematical Form = f () + k = f ( h) Effect If k >, then k units upward If k <, then k units downward If h >, then h units to the right If h <, then h units to the left Communicate Your Understanding. Starting with the graph of =, describe how ou would sketch each of the following graphs. a) = + b) = + c) =. Starting with the graph of =, describe how ou would sketch each of the following graphs. a) = 5 b) = ( 5) c) = ( + 5) +. Describe how ou would find the domain and range of each of the following functions. a) = + b) = c) = 5 Practise A. The function = f() is given. Describe how the graphs of the following functions can be obtained from the graph of = f(). a) = f() + 5 b) = f() c) = f( ) d) = f( + 8) e) = f() f) + 7 = f() g) = f( + ) 5 h) = f( ) + i) = f( 5) 7 j) = f( + ) + 9. The function = f() has been transformed to = f( h) + k. Determine the values of h and k for each of the following transformations. a) units upward b) 8 units downward c) units to the right d) 5 units to the left e) units to the left and units downward f) 7 units to the right and 7 units upward. State the domain and range of each function. a) = + b) = c) = d) = ( ) e) = ( + 5) f) = + g) = 5 h) = +. Horizontal and Vertical Translations of Functions MHR 89

7 . The graph of a function = f() is shown. Sketch the graph of each of the following. a) = f() b) = f() + c) = f( ) d) = f( + ) e) = f( ) f) = f( + ) + i) a) b) = ii) c) = iii) iv) 5. The graph of the function drawn in blue is a translation image of the function drawn in red. Write an equation for each function drawn in blue. Check each equation using a graphing calculator. =. Use transformations to sketch the graph of each of the following functions, starting with the graph of =. a) = + b) = 5 c) = ( ) d) = ( + ) e) = ( + 5) f) = ( ) + 7. Use transformations to sketch the graph of each of the following functions, starting with the graph of =. a) = + 7 b) + = c) = + d) = e) = + f) = Use transformations to sketch the graph of each of the following functions, starting with the graph of =. a) = + b) + = c) = ( 7) d) = ( + ) e) = ( + ) f) = ( 5) MHR Chapter

8 Appl, Solve, Communicate 9. Falling objects The approimate height above the ground of a falling object dropped from the top of a building is given b the function h(t) = 5t + d where h(t) metres is the height of the object t seconds after it is dropped, and d metres is the height from which it is dropped. The table shows the heights of three tall buildings in Canada. a) Write the three functions, f(p-c), f(tbw), and f(cg), that describe the height of a falling object above the ground t seconds after it is dropped from the top of each building. b) Graph h(t) versus t for the three functions on the same set of aes or in the same viewing window of a graphing calculator. c) How could ou transform the graph of f(p-c) onto the graph of f(tbw)? d) How could ou transform the graph of f(cg) onto the graph of f(tbw)? e) How could ou transform the graph of f(p-c) onto the graph of f(cg)? B Building Petro-Canada, Calgar Two Bloor West, Toronto Complee G, Québec Cit. Service calls Elena and Mario both repair kitchen appliances. Elena charges $5 for a service call, plus $5/h for labour. Mario charges $ for a service call, plus $5/h for labour. Write an equation for the cost, C dollars, of a service call in terms of the number of hours worked, t a) for Elena b) for Mario c) How are the graphs of the two equations related? Eplain.. Communication When the graph of = + is translated unit to the right and units upward, how is the resulting graph related to the graph of = +? Eplain.. Application Man companies pa their emploees using a salar scale that depends on the number of ears worked. One salar scale is modelled b the function S( ) = 5 + 5, where S( ) dollars is the salar and is the number of ears worked for the compan. The emploees union negotiates an increase of $ for each emploee. a) How is the graph of S( ) transformed b the increase? b) Write the function that models the salar scale after the increase. c) State a reasonable domain and range for the function in part b). Justif our reasoning. Height (m) 8. Horizontal and Vertical Translations of Functions MHR 9

9 . Greatest integer function The greatest integer function is defined b [] = the greatest integer that is less than or equal to. For eample, [] =, [.8] =, and [ 5.] =. The graph of = [] is shown. a) Eplain the meanings of the open and closed dots on the graph of = []. b) State the domain and range of = []. c) Use transformations to sketch the graph of = [] + ; = [ ]; = [ + ].. Parking costs EZ-Park determines its parking charges based on the greatest integer function = [ + ] +, where is the parking charge, in dollars, and is the number of hours that a vehicle is in the parking garage. a) Sketch the graph of the function. b) How much would a driver pa to park for min? for h? for h 5 min? for h min? 5. What transformation relates the graph of a) = f( ) to its image, = f( + )? b) = f() + 5 to its image, = f() 7?. Describe a vertical translation that could be applied to the graph of = so that the translation image passes through the point (, ). 7. Inquir/Problem Solving The function = + could be a vertical translation of = three units upward or a horizontal translation of = three units to the left. Eplain wh. C 8. Chemistr a) One wa to describe the concentration of an acid is as a percent b volume. For eample, in ml of a % acid solution, the volume of pure acid is or ml, and the volume of water is or 8 ml. If 5 ml of pure acid is mied with ml of water to give 5 ml of acid solution, the concentration of the solution is given b 5 % = %. If water is mied with 5 ml of % acid solution, 5 write an equation that describes the acid concentration, C(), as a function of the volume of water added,. b) Graph C() versus. c) What is the acid concentration after ml of water have been added? 9 MHR Chapter

10 d) Write an equation that describes the acid concentration as a function of the volume of water added to ml of 5% acid solution. e) Graph C() versus for the function from part d). f) How could ou transform the graph from part e) onto the graph from part b)? CAREER CONNECTION Veterinar Medicine There are man more domestic animals in Canada than there are people. For eample, in addition to the millions of dogs and cats in Canadian homes, there are over cattle and pigs on Canadian farms. Medical services for these and other animals are provided b workers in the field of veterinar medicine.. Ages of cats and dogs As with humans, the medical needs of domestic animals change as the age. However, humans and domestic animals age differentl. For a small dog, aged ears or more and with a mass up to about kg, the number of human ears equivalent to the age of the dog is given b the formula h(a) = a + where h is the equivalent number of human ears, and a is the age of the dog. For a domestic cat aged ears or more, the number of human ears equivalent to the age of the cat is given b the formula h(a) = a + 5 where h is the equivalent number of human ears, and a is the age of the cat. a) Graph h versus a for cats and for small dogs over the domain ears to 5 ears on the same aes or in the same viewing window of a graphing calculator. b) Describe how the graphs are related b a transformation. c) A cat and a small dog were born on the same da and are over ears old. How do the numbers of human ears equivalent to their ages compare? Eplain. d) If their ages are epressed as equivalent human ears, do cats and small dogs age at the same rate after the age of? Eplain. e) If their ages are epressed as equivalent human ears, do cats and small dogs age at the same rate from birth? How do ou know?. Research Use our research skills to investigate a) the training needed to become a veterinarian, also know as a doctor of veterinar medicine b) the organizations that emplo veterinarians c) other careers that involve animal care. Horizontal and Vertical Translations of Functions MHR 9