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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 0 π π 3π 2π 2 2

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

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y = c + a(trig b (x + d) a (amplitude) multiply a times (0 | ) 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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**y = | a | • f (x) by horizontal and vertical translations **

GRAPHING SINE AND COSINE FUNCTIONS 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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**GRAPHING SINE AND COSINE FUNCTIONS**

TRANSFORMATIONS OF SINE AND COSINE GRAPHS To obtain the graph of y = a sin b (x – h) + k or y = a cos b (x – h) + k Transform the graph of y = | a | sin bx or y = | a | cos bx as follows.

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**GRAPHING SINE AND COSINE FUNCTIONS**

TRANSFORMATIONS OF SINE AND COSINE GRAPHS VERTICAL SHIFT Shift the graph k units vertically. k y = a • sin bx + k y = a • sin bx

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**GRAPHING SINE AND COSINE FUNCTIONS**

TRANSFORMATIONS OF SINE AND COSINE GRAPHS HORIZONTAL SHIFT Shift the graph h units Vertically. h y = a • sin b(x – h) y = a • sin bx

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**GRAPHING SINE AND COSINE FUNCTIONS**

TRANSFORMATIONS OF SINE AND COSINE GRAPHS y = a • sin bx + k 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

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** Graph y = – 2 + 3 sin 4 x. SOLUTION 2 4 2**

Graphing a Vertical Translation Graph y = – sin 4 x. Because the graph is a transformation of the graph of y = 3 sin 4x, the amplitude is 3 and the period is = 2 4 2 SOLUTION 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. 3 8 4 2 Therefore translate the graph of y = 3 sin 4x down two units.

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**Graph y = – 2 + 3 sin 4 x. SOLUTION**

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

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** Graph y = – 2 + 3 sin 4 x. SOLUTION 4 2 8 3 8**

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

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**Graphing a Vertical Translation**

Graph y = – sin 4 x. CHECK 3 8 4 2 You can check your graph with a graphing calculator. Use the Maximum, Minimum and Intersect features to check the key points.

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**Graphing a Vertical Translation**

Graph y = 2 cos x – 4 2 3 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 SOLUTION

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**Graphing a Vertical Translation**

Graph y = 2 cos x – π 4 2 3 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 SOLUTION Therefore, translate the graph of y = 2 cos x right unit. 2 3 4 Notice that the maximum occurs unit to the right of the y-axis. 4

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** Graph y = 2 cos x – . 4 2 3 SOLUTION 4 1 • 3 + , 0 = , 0 5 2**

Graphing a Horizontal Translation Graph y = 2 cos x – 4 2 3 SOLUTION The five key points are: 4 1 On y = k : • 3 , 0 = (, 0); • 3 , 0 = , 0 5 2 3 Maximum: , 2 = , 2 ; 13 4 3 , 2 = , 2 ; Minimum: • 3 , – 2 = , – 2 7 4 1 2

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**When you plot the five points on the graph, note that the **

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. SOLUTION 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.

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**However, when the graph is reflected in the x-axis, the **

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

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** Graph y = – 3 sin x. SOLUTION 1 • 2, 0 = (, 0) 2 1 2 4 3 3 2**

Graphing a Reflection Graph y = – 3 sin x. SOLUTION The five key points are: 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 3 2

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**Modeling Circular Motion**

FERRIS WHEEL 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?

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**Modeling Circular Motion**

h = 25 sin t – 15 SOLUTION The amplitude is 25 and the period is = 30. 2 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**

h = 25 sin t – 15 SOLUTION The key five points are (7.5, 30), (15, 55), (22.5, 30), (30, 5) and (37.5, 30).

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**Modeling Circular Motion**

h = 25 sin t – 15 SOLUTION 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.

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**• Shift the graph k units vertically and h units horizontally. **

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

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** Graph y = – 2 tan x + . 4 SOLUTION**

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

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** Graph y = – 2 tan x + . 4 SOLUTION**

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

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** Graph y = – 2 tan x + . 4 2 •1 4 3 4 4 •1 4 2**

Combining a Translation and a Reflection Graph y = – 2 tan x 4 Asymptotes: x = – – = – ; x = – = 2 •1 4 3 On y = k: (h, k) = – , 0 4 Halfway points: – – , 2 = – , 2 ; – , – 2 = (0, – 2) 4 •1 4 2

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