5.6 Translations and Combinations of Transformations

Transcription

1 5.6 Translations and Combinations of Transformations The highest tides in the world are found in the Ba of Fund. Tides in one area of the ba cause the water level to rise to 6 m above average sea level and to drop to 6 m below average sea level. The tide completes one ccle approimatel ever 1 h. The depth of the water can be modelled b a sine function. This function will be modelled in Eample 6. INVESTIGATE & INQUIRE 1. Cop and complete the table b finding decimal values for sin and sin +. Round values to the nearest tenth, if necessar. (degrees) sin sin Sketch the graphs of the functions = sin and = sin + on the same grid, like the one shown For the graphs from question, find a) the amplitudes b) the periods 4. What transformation can been applied to = sin to give = sin +? 5. a) Make a conjecture about how the graph of = sin 1 compares with the graph of = sin. b) Use our conjecture to sketch the graphs of = sin and = sin 1 on the same grid for the domain Test our conjecture from question 5a) b graphing = sin and = sin 1, 36, on a graphing calculator. Compare the result with our graph from question 5b). 378 MHR Chapter 5

2 7. For the graphs from question 6, find a) the amplitudes b) the periods 8. What transformation can been applied to = sin to give = sin 1? 9. Write a statement about the transformational effect of c on the graph of = sin + c. 1. Write a conjecture about the transformational effect of c on the graph of = cos + c. 11. Test our conjecture from question 1 b graphing = cos, = cos +, and = cos 1, 36, on a graphing calculator. 1. For the graphs from question 11, find a) the amplitudes b) the periods INVESTIGATE & INQUIRE 1. Cop and complete the tables b finding values of sin and sin ( + 45 ). Round values to the nearest tenth, if necessar. (degrees) sin sin ( + 45) Sketch the graphs of the functions = sin and = sin ( + 45 ) on the same grid, like the one shown For the graphs from question, find a) the amplitudes b) the periods 4. What transformation can been applied to = sin to give = sin ( + 45 )? 5. a) Make a conjecture about how the graph of = sin ( 45 ) compares with the graph of = sin. b) Use our conjecture to sketch the graph of = sin and = sin ( 45 ) on the same grid for the domain Test our conjecture from question 5a) b graphing = sin and = sin ( 45 ), 36, on a graphing calculator. Compare the result with our graph from question 5b). 5.6 Translations and Combinations of Transformations MHR 379

3 7. For the graphs from question 6, find a) the amplitudes b) the periods 8. What transformation can been applied to = sin to give = sin ( 45 )? 9. Write a statement about the transformational effect of d on the graph of = sin ( d). 1. Write a conjecture about the transformational effect of d on the graph of = cos ( d). 11. Test our conjecture from question 1 b graphing = cos, = cos ( + 45 ), = cos ( 45 ), 36, on a graphing calculator. 1. For the graphs from question 11, find a) the amplitudes b) the periods The vertical translations that appl to algebraic functions also appl to trigonometric functions. If c >, the graphs of = sin + c and = cos + c are translated upward b c units. If c <, the graphs of = sin + c and = cos + c are translated downward b c units. 4 = sin = sin = sin 3 4 As with algebraic transformations, combinations of trigonometric transformations are performed in the following order. epansions and compressions reflections translations EXAMPLE 1 Sketching = asin + c Sketch one ccle of the graph of = sin + 3. State the domain and range of the ccle. 38 MHR Chapter 5

4 SOLUTION First, sketch the graph of = sin. The graph of = sin is the graph of = sin epanded verticall b a factor of. The amplitude is, so the maimum value is and the minimum value is. The period of the function = sin is π. Use the five-point method to sketch the graph. The five ke points divide the period into quarters. Therefore, the coordinates of the five ke points are (, ), π,, (π, ), 3 π,, and (π, ). Plot the 5 ke points in the ccle. Draw a smooth curve through the points. Translate the graph three units upward to obtain the graph of = sin + 3. Label the graph. The domain of the ccle is π. The range is = sin + 3 π π = sin The horizontal translations that appl to algebraic functions also appl to trigonometric functions. If d >, the graphs of = sin( d) and = cos( d) are translated d units to the right. If d <, the graphs of = sin( d) and = cos( d) are translated d units to the left. For trigonometric functions, a horizontal translation is often called the phase shift or phase angle. EXAMPLE Sketching = acos ( d ) Sketch one ccle of the graph of =.5cos + π. State the domain, range, and phase shift of the ccle. 5.6 Translations and Combinations of Transformations MHR 381

5 SOLUTION First, sketch the graph of =.5cos. The graph of =.5cos is the graph of = cos compressed verticall b a factor of.5. The amplitude is.5, so the maimum value is.5 and the minimum value is.5. The period of the function =.5cos is π. Use the five-point method to sketch the graph. The five ke points divide the period into quarters. Therefore, the coordinates of the five ke points are (,.5), π,, (π,.5), 3 π,, and (π,.5). Plot the 5 ke points in the ccle. Draw a smooth curve through the points. Translate the graph π units to the left to obtain the graph of =.5cos + π. Label the graph. 1 =.5cos + π π π π =.5cos The domain of the ccle is π 3. π The range is.5.5. The phase shift is π units to the left. EXAMPLE 3 Sketching = asin k( d ) Sketch one ccle of the graph of = 3sin π 4. State the domain, range, and phase shift of the ccle. SOLUTION First sketch the graph of = 3sin. The graph of = 3sin is the graph of = sin epanded verticall b a factor of 3 and compressed horizontall b a factor of 1. The amplitude is 3, so the maimum value is 3 and the minimum value is 3. The period of the function = 3sin is π, or π. 38 MHR Chapter 5

6 Use the five-point method to sketch the graph. The five ke points divide the period into quarters. Therefore, the coordinates of the five ke points are (, ), π 4, 3, π,, 3 4π, 3, and (π, ). Plot the 5 ke points in the ccle. Draw a smooth curve through the points. Translate the graph π 4 units to the right to obtain the graph of = 3sin π 4. Label the graph. The domain of the ccle is π π The range is 3 3. The phase shift is π 4 units to the right. = 3sin π 4 π π = 3sin If necessar, factor the coefficient of the -term to identif the characteristics of a function more easil. EXAMPLE 4 Sketching = acos k( d ) + c Sketch the graph of = 4cos 1 + π 1, 4π 4π. SOLUTION Factor the coefficient of the -term. = 4cos 1 + π 1 becomes = 4cos 1 ( +π) 1. Now, sketch the graph of = 4cos 1. The graph of = 4cos 1 is the graph of = cos epanded verticall b a factor of 4 and epanded horizontall b a factor of. The amplitude is 4, so the maimum value is 4 and the minimum value is 4. The period of the function = 4cos 1 is π, or 4π Translations and Combinations of Transformations MHR 383

7 Use the five-point method to sketch the graph. The five ke points divide the period into quarters. Therefore, the coordinates of the five ke points are (, 4), (π, ), (π, 4), (3π, ), and (4π, 4). Plot the 5 ke points in the ccle. Draw a smooth curve through the points. Translate the graph π units to the left and one unit downward to obtain the graph of = 4cos 1 ( +π) 1, π 3π. Use the pattern to sketch the graph over the domain 4π 4π. Label the graph. 4 1 = 4cos 4π 3π π π π π 3π 4π 4 = 4cos 1 + π 1 EXAMPLE 5 Sketching for a < 1 Sketch the graph of = 4sin π, 4π. SOLUTION First sketch the graph of = 4sin. The graph of = 4sin is the graph of = sin epanded verticall b a factor of 4. The amplitude is 4, so the maimum value is 4 and the minimum value is 4. The period of the function = 4sin is π. Use the five-point method to sketch the graph. The five ke points divide the period into quarters. Therefore, the coordinates of the five ke points are (, ), π, 4, (π, ), 3, 4, and (π, ). π Plot the 5 ke points in the ccle. Draw a smooth curve through the points. 384 MHR Chapter 5

8 Recall that the graph of = f() is the graph of = f() reflected in the -ais. So, reflect the graph of = 4sin in the -ais to obtain the graph of = 4sin. Translate the reflected graph π units to the right to obtain the graph of 4 = 4sin π π π = 4sin 3π 4π = 4sin π, π 5 π. Use the pattern to sketch the graph over the domain 4π. Label the graph. Note that all of the graphs required in Eamples 1 5 can be drawn directl using a graphing calculator. The graph shown is = 4sin π, 4π, from Eample 5. The calculator is in radian mode, and the window variables include Xmin =, Xma = 4π, Ymin = 5, and Yma = 5. EXAMPLE 6 Ba of Fund Tides In one area of the Ba of Fund, the tides cause the water level to rise to 6 m above average sea level and to drop to 6 m below average sea level. One ccle is completed approimatel ever 1 h. Assume that changes in the depth of the water over time can be modelled b a sine function. a) If the water is at average sea level at midnight and the tide is coming in, draw a graph to show how the depth of the water changes over the net 4 h. Assume that at low tide the depth of the water is m. b) Write an equation for the graph. c) If the water is at average sea level at :, and the tide is coming in, write an equation for the graph that shows how the depth changes over the net 4 h. 4 = 4sin SOLUTION a) The depth of water at low tide is m. At low tide, the water level is 6 m below average sea level. So, the depth of water for average sea level is 8 m. This is the depth at midnight. The maimum depth of the water is 8 + 6, or 14 m. Use the known values to sketch a 1-h ccle of depth versus time. Use the pattern to show the changes over 4 h. Water Depth (m) d : 4: 8: 1: 16: : 4: Time of Da 5.6 Translations and Combinations of Transformations MHR 385

9 b) The amplitude, a, is 6 m. The period is 1 h. 1 = kπ k = π 6 The graph has been translated 8 units upward, so c = 8. The equation that shows how the depth of the water changes over time is h = 6sin π t c) When the water is at average sea level at :, the depth is 8 m at :. The graph is translated h to the right. The equation is h = 6sin π 6 (t ) + 8. Ke Concepts Perform combinations of transformations in the following order. * epansions and compressions * reflections * translations For trigonometric functions, a horizontal translation is called the phase shift or phase angle. If necessar, factor the coefficient of the -term to identif the characteristics of a function more easil. Communicate Your Understanding 1. Describe how ou would identif the transformations on = sin that result in each of the following functions. a) = 3sin 1 3 b) = 6sin 3( π) c) = sin ( +π) +. Describe how ou would identif the transformations on = cos that result in the function = 3cos (4 π). 3. Describe how ou would sketch the graph of one ccle of the function = sin + π MHR Chapter 5

10 Practise A 1. Determine the vertical translation and the phase shift of each function with respect to = sin. a) = sin + 3 b) = sin 1 c) = sin ( 45 ) d) = sin 3 4 e) = sin ( 6 ) + 1 f) = sin + π g) = sin + 3 8π.5 h) = sin ( 15 ) 4.5 π. Determine the vertical translation and the phase shift of each function with respect to = cos. a) = cos + 6 b) = cos 3 c) = cos + π d) = cos ( + 7 ) e) = cos ( 3 ) f) = cos + π g) = cos ( + 11 ) + 5 h) = cos 5 π Sketch one ccle of the graph of each of the following. State the amplitude, period, domain, and range of the ccle. a) = 3sin + b) = cos c) = 1.5sin 1 d) = 1 cos Sketch one ccle of the graph of each of the following. State the amplitude, period, domain, range, and phase shift of the ccle. a) = sin ( π) b) = cos π c) = 1 sin + π d) = 3cos + π 4 e) = cos + π 5. Determine the amplitude, period, vertical translation, and phase shift for each function with respect to = sin. a) = sin 3 b) =.5sin () 1 c) = 6sin 3( ) d) = 5sin π Determine the amplitude, period, vertical translation, and phase shift for each function with respect to = cos. a) = cos + 3 b) = cos 3( 9 ) c) = 3cos 4 π d) =.8cos 3 π Sketch one ccle of the graph of each of the following. State the amplitude, period, domain, range, and phase shift of the ccle. a) = sin + π 4 b) = cos π c) = 3sin 1 ( π) d) = 4cos 1 3 ( + π) 4 e) = 3sin π Translations and Combinations of Transformations MHR 387

11 8. Communication Sketch one ccle of the graph of each of the following. State the amplitude, period, domain, range, and phase shift of the ccle. a) = sin π b) = cos 1 π c) = sin (3 π) + d) = 3cos ( 4π) 1 9. Write an equation for the function with the given characteristics, where T is the tpe, A is the amplitude, P is the period, V is the vertical shift, and H is the horizontal shift. a) b) c) d) T sine cosine sine cosine A P π π 4π π V 6 3 none H none none π right π left e) = sin π 3 + 1, π π f) = 5cos 1 3 π 3 +, π 3π g) = sin + π 8, π π 11. Each graph shows part of the sine function of the form = asin k( d) + c. Determine the values of a, k, d, and c for each graph. Check b graphing. a) 1 π π π π π 1. Sketch the graph of each of the following. State the range. a) = sin +, < 3π b) = cos 3, π π c) = 3cos π 6, π π d) = 4sin + π 4 1, π π b) 6 4 π π π 3π Appl, Solve, Communicate B 1. Ocean ccles The water depth in a harbour is 1 m at high tide and 11 m at low tide. One ccle is completed approimatel ever 1 h. a) Find an equation for the water depth as a function of the time, t hours, after low tide. b) Draw a graph of the function for 48 h after low tide, which occurred at 14:. 388 MHR Chapter 5

12 c) State the times at which the water depth was i) a maimum ii) a minimum iii) at its average value d) Estimate the depth of the water at i) 17: ii) 1: e) Estimate the times at which the depth of the water was i) 14 m ii) m iii) at least 18 m 13. Application An object attached to the end of a spring is oscillating up and down. The displacement of the object,, in centimetres, is a function of the time, t, in seconds, and is given b =.4cos 1t + π 6. a) Sketch two ccles of the function. b) What is the maimum distance through which the object oscillates? c) What is the period of the function? Give our answer as an eact number of seconds, in terms of π, and as an approimate number of seconds, to the nearest hundredth. 14. Ocean ccles On a certain da, the depth of water off a pier at high tide was 6 m. After 6 h, the depth of the water was 3 m. Assume a 1-h ccle. a) Find an equation for the depth of water, with respect to its average depth, in terms of the time, t hours, since high tide. b) Draw a graph of the depth of water versus time for 48 h after high tide. c) Find the depth of water at t = 8 h, 15 h, h, and 3 h. d) Predict how the equation will change if the first period begins at low tide. e) Test our prediction from part d) b drawing the graph and finding the equation. 15. Spring An object suspended from a spring is oscillating up and down. The distance from the high point to the low point is 3 cm, and the object takes 4 s to complete 5 ccles. For the first few ccles, the distance from the mean position, d(t) centimetres, with respect to the time, t seconds, is modelled b a sine function. a) Sketch a graph of this function for two ccles. b) Write an equation that describes the distance of the object from its mean position as a function of time. 5.6 Translations and Combinations of Transformations MHR 389

13 16. Ferris wheel A carnival Ferris wheel with a radius of 7 m makes one complete revolution ever 16 s. The bottom of the wheel is 1.5 m above the ground. a) Draw a graph to show how a person s height above the ground varies with time for three revolutions, starting when the person gets onto the Ferris wheel at its lowest point. b) Find an equation for the graph. c) Predict how the graph and the equation will change if the Ferris wheel turns more slowl. d) Test our predictions from part c) b drawing the graph for three revolutions and finding an equation, if the wheel completes one revolution ever s. 17. Ferris wheel A Ferris wheel has a radius of 1 m and makes one revolution ever 1 s. Draw a graph and find an equation to show a person s height above or below the centre of rotation for two counterclockwise revolutions starting at a) point A b) point B c) point C C B A 18. For the sine function epressed in the form = asin k( d) + c, does the value of c affect each of the following? Eplain. a) the amplitude? b) the period? c) the maimum and minimum values of the function? d) the phase shift? 19. Ocean ccles The depth of water, d(t) metres, in a seaport can be approimated b the sine function d(t) =.5sin.164π(t 1.5) , where t is the time in hours. a) Graph the function for t 4 using a graphing calculator. b) Find the period, to the nearest tenth of an hour. c) A cruise ship needs a depth of at least 1 m of water to dock safel. For how man hours in each period can the ship dock safel? Round our answer to the nearest tenth of an hour. 39 MHR Chapter 5

14 . Pose and solve problems Pose a problem related to each of the following. Check that ou are able to solve each problem. Then, have a classmate solve it. a) the vertical motion of a spring b) the motion of a wheel C 1. The equation of a sine function can be epressed in the form = asin k( d) + c. Describe what ou know about a, k, d, and c for each of the following statements to be true. a) The period is greater than π. b) The amplitude is less than one unit. c) The graph passes through the origin. d) The graph has no -intercepts.. a) Predict how the graphs of = sin ( + 45 ) and = sin( 315 ) are related. b) Test our prediction using a graphing calculator, and eplain our observations. 3. a) Predict how the graphs of = sin and = cos π are related. b) Test our prediction using a graphing calculator, and eplain our observations. A CHIEVEMENT Check The rodent population in a particular region varies with the number of predators that inhabit the region. At an time, ou could predict the rodent population, r(t), using the function t, where t is r(t) = sin π 4 the number of ears that have passed since a) In the first ccle of this function, what was the maimum number of rodents and in which ear did this occur? b) What was the minimum number of rodents in a ccle? c) What is the period of this function? d) How man rodents would ou predict in the ear 14? Knowledge/Understanding Thinking/Inquir/Problem Solving Communication Application Web Connection To investigate a simulation of a predator-pre relationship, visit the above web site. Go to Math Resources, then to MATHEMATICS 11, to find out where to go net. Report on our findings. e) Change the function to model a rodent population ccle that lasts 5 ears. 5.6 Translations and Combinations of Transformations MHR 391