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4.3 Vertical and Horizontal Translations OBJ: Graph sine and cosine with vertical and horizontal translations

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DEF: Vertical Translation A function of the form y =c + a sin b x or of the form y = c + a cos b x is shifted vertically when compared with y = a sin b x or y =a cos b x.

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5 EX: Graph y = 2 – 2 sin x 0ππ3π 2π 2

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6 EX: Graph y = – 3 + 2 sin x 0ππ3π 2π 2

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DEF: Phase Shift The function y=sin (x+d) has the shape of the basic sine graph y = sin x, but with a translation d units: to the right if d < 0 and to the left if d > 0. The number d is the phase shift of the graph. The cosine graph has the same function traits.

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7 EX: Graph y = sin (x – π/3)

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8 EX: Graph y = 3cos (x + π/4)

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9 EX: Graph y = 4 – sin (x – π/3)

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10 EX: Graph y =-3 + 3cos(x+π/4)

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DEF: Period of Sine and Cosine The graph of y = sin b x will look like that of sin x, but with a period of 2 . b Also the graph of y = cos b x looks like that of y = cos x, but with a period of 2 b

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y = c + a(trig b (x + d) a (amplitude) multiply a times (0 |1 0 -1 0 1) b (period) 2π b c (vertical shift) d (starting point)

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11 EX: Graph y = sin 2x

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12 EX: Graph y = -2cos 3x

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13 EX: Graph y = 3 – 2cos 3x

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14 EX: Graph y = –2cos(3x+π)

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15 EX: Graph y = cos(2x/3)

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16 EX: Graph y = –2 sin 3x

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17 EX: Graph y = 3 cos ½ x

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G RAPHING S INE AND C OSINE F UNCTIONS In previous chapters you learned that the graph of y = a f (x – h) + k is related to the graph of y = | a | f (x) by horizontal and vertical translations and by a reflection when a is negative. This also applies to sine, cosine, and tangent functions.

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G RAPHING S INE AND C OSINE F UNCTIONS TRANSFORMATIONS OF SINE AND COSINE GRAPHS To obtain the graph of Transform the graph of y = | a | sin bx or y = | a | cos bx as follows. y = a sin b (x – h) + k or y = a cos b (x – h) + k

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G RAPHING S INE AND C OSINE F UNCTIONS TRANSFORMATIONS OF SINE AND COSINE GRAPHS VERTICAL SHIFT Shift the graph k units vertically. y = a sin bx y = a sin bx + k k

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TRANSFORMATIONS OF SINE AND COSINE GRAPHS G RAPHING S INE AND C OSINE F UNCTIONS HORIZONTAL SHIFT Shift the graph h units Vertically. y = a sin b ( x – h) h y = a sin bx

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TRANSFORMATIONS OF SINE AND COSINE GRAPHS G RAPHING S INE AND C OSINE F UNCTIONS REFLECTION If a < 0, reflect the graph in the line y = k after any vertical and horizontal shifts have been performed. y = a sin bx + k y = – a sin bx + k

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33 8 8 4 2 Graphing a Vertical Translation Graph y = – 2 + 3 sin 4 x. S OLUTION By comparing the given equation to the general equation y = a sin b(x – h) + k, you can see that h = 0, k = – 2, and a > 0. Therefore translate the graph of y = 3 sin 4x down two units. Because the graph is a transformation of the graph of y = 3 sin 4x, the amplitude is 3 and the period is =. 22 4 2

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33 8 8 4 2 Graphing a Vertical Translation The graph oscillates 3 units up and down from its center line y = – 2. S OLUTION 33 8 8 4 2 Therefore, the maximum value of the function is – 2 + 3 = 1 and the minimum value of the function is – 2 – 3 = –5 y = – 2 Graph y = – 2 + 3 sin 4 x.

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33 8 8 4 2 Graphing a Vertical Translation The five key points are: On y = k : (0, 2);, – 2 ;, – 2 4 2 Maximum:, 1 8 Minimum:, – 5 33 8 S OLUTION Graph y = – 2 + 3 sin 4 x.

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33 8 8 4 2 Graphing a Vertical Translation C HECK You can check your graph with a graphing calculator. Use the Maximum, Minimum and Intersect features to check the key points. Graph y = – 2 + 3 sin 4 x.

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Graphing a Vertical Translation Graph y = 2 cos x –. 4 2 3 S OLUTION 3 Because the graph is a transformation of the graph of y = 2 cos x, the amplitude is 2 and the period is = 3 . 2 3 22 2

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Graphing a Vertical Translation By comparing the given equation to the general equation y = a cos b (x – h) + k, you can see that h =, k = 0, and a > 0. 4 Therefore, translate the graph of y = 2 cos x right unit. 2 3 4 Graph y = 2 cos x –. π 4 2 3 S OLUTION Notice that the maximum occurs unit to the right of the y-axis. 4

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Graphing a Horizontal Translation The five key points are: Graph y = 2 cos x –. 4 2 3 S OLUTION 4 1 4 On y = k : 3 +, 0 = ( , 0); 3 +, 0 =, 0 55 2 3 4 4 Minimum: 3 +, – 2 =, – 2 77 4 4 1 2 Maximum: 0 +, 2 =, 2 ; 13 4 3 +, 2 =, 2 ; 4 4 4

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Graphing a Reflection Graph y = – 3 sin x. Because the graph is a reflection of the graph of y = 3 sin x, the amplitude is 3 and the period is 2 . When you plot the five points on the graph, note that the intercepts are the same as they are for the graph of y = 3 sin x. S OLUTION

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Graphing a Reflection However, when the graph is reflected in the x-axis, the maximum becomes a minimum and the minimum becomes a maximum. Graph y = – 3 sin x. S OLUTION

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Graphing a Reflection On y = k : (0, 0); (2 , 0); 1 2 2 , 0 = ( , 0) Minimum: 2 , – 3 =, – 3 1 4 2 Maximum: 2 , 3 =, 3 3 4 33 2 Graph y = – 3 sin x. S OLUTION The five key points are:

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Modeling Circular Motion F ERRIS W HEEL You are riding a Ferris wheel. Your height h (in feet) above the ground at any time t (in seconds) can be modeled by the following equation: h = 25 sin t – 7.5 + 30 15 The Ferris wheel turns for 135 seconds before it stops to let the first passengers off. Graph your height above the ground as a function of time. What are your minimum and maximum heights above the ground?

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S OLUTION Modeling Circular Motion The amplitude is 25 and the period is = 30. 22 15 h = 25 sin t – 7.5 + 30 15 The wheel turns = 4.5 times in 135 seconds, so the graph shows 4.5 cycles. 130 30

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Modeling Circular Motion The key five points are (7.5, 30), (15, 55), (22.5, 30), (30, 5) and (37.5, 30). h = 25 sin t – 7.5 + 30 15 S OLUTION

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Modeling Circular Motion Since the amplitude is 25 and the graph is shifted up 30 units, the maximum height is 30 + 25 = 55 feet. The minimum height is 30 – 25 = 5 feet. h = 25 sin t – 7.5 + 30 15 S OLUTION

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G RAPHING T ANGENT F UNCTIONS TRANSFORMATIONS OF TANGENT GRAPHS Shift the graph k units vertically and h units horizontally. Then, if a < 0, reflect the graph in the line y = k. To obtain the graph of y = a tan b (x – h) + k transform the graph of y = a tan bx as follows. ||

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Combining a Translation and a Reflection Graph y = – 2 tan x +. 4 S OLUTION The graph is a transformation of the graph of y = 2 tan x, so the period is .

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Combining a Translation and a Reflection Therefore translate the graph of y = 2 tan x left unit and then reflect it in the x-axis. 4 Graph y = – 2 tan x +. 4 S OLUTION By comparing the given equation to y = a tan b (x – h) + k, you can see that h = –, k = 0, and a < 0. 4

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Combining a Translation and a Reflection Asymptotes: On y = k: Halfway points: x = – – = – ; x = – = 2 12 1 4 33 4 4 4 2 12 1 (h, k) = –, 0 4 – –, 2 = –, 2 ; –, – 2 = (0, – 2) 4 14 1 4 24 14 1 4 Graph y = – 2 tan x +. 4

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