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

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Presentation on theme: "4.3 Vertical and Horizontal Translations OBJ: Graph sine and cosine with vertical and horizontal translations."— Presentation transcript:

1 4.3 Vertical and Horizontal Translations OBJ: Graph sine and cosine with vertical and horizontal translations

2 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.

3 5 EX:  Graph y = 2 – 2 sin x 0ππ3π 2π 2

4 6 EX:  Graph y = – sin x 0ππ3π 2π 2

5 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.

6 7 EX:  Graph y = sin (x – π/3)

7 8 EX:  Graph y = 3cos (x + π/4)

8 9 EX:  Graph y = 4 – sin (x – π/3)

9 10 EX: Graph y =-3 + 3cos(x+π/4)

10 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 

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

12 11 EX: Graph y = sin 2x

13 12 EX: Graph y = -2cos 3x

14 13 EX: Graph y = 3 – 2cos 3x

15 14 EX: Graph y = –2cos(3x+π)

16 15 EX: Graph y = cos(2x/3)

17 16 EX: Graph y = –2 sin 3x

18 17 EX: Graph y = 3 cos ½ x

19 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.

20 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

21 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

22 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

23 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

24 33 8  8  4  2 Graphing a Vertical Translation Graph y = – 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 =. 22 4  2

25 33 8  8  4  2 Graphing a Vertical Translation The graph oscillates 3 units up and down from its center line y = – 2. S OLUTION 33 8  8  4  2 Therefore, the maximum value of the function is – = 1 and the minimum value of the function is – 2 – 3 = –5 y = – 2 Graph y = – sin 4 x.

26 33 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 33 8 S OLUTION Graph y = – sin 4 x.

27 33 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 = – sin 4 x.

28 Graphing a Vertical Translation Graph y = 2 cos x –.  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 2

29 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 S OLUTION Notice that the maximum occurs unit to the right of the y-axis.  4

30 Graphing a Horizontal Translation The five key points are: Graph y = 2 cos x –.  S OLUTION  On y = k : 3  +, 0 = ( , 0); 3  +, 0 =, 0 55  4 Minimum: 3  +, – 2 =, – 2 77 4  Maximum: 0 +, 2 =, 2 ; 13  4 3  +, 2 =, 2 ;  4  4  4

31 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

32 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

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

34 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 –  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?

35 S OLUTION Modeling Circular Motion The amplitude is 25 and the period is = 30. 22  15 h = 25 sin t –  15 The wheel turns = 4.5 times in 135 seconds, so the graph shows 4.5 cycles

36 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 –  15 S OLUTION

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

38 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. ||

39 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 .

40 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

41 Combining a Translation and a Reflection Asymptotes: On y = k: Halfway points: x = – – = – ; x = – =   4 33 4  4  4  (h, k) = –, 0  4 – –, 2 = –, 2 ; –, – 2 = (0, – 2)   4   4  Graph y = – 2 tan x +.  4


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