4.3 vertical and horizontal translations

4.3 Vertical and Horizontal Translations

OBJ: Graph sine and cosine with vertical and horizontal

translations

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.

5 EX: Graph y = 2 – 2 sin x

0 π π 3π 2π

2 2

6 EX: Graph y = – 3 + 2 sin x

0 π π 3π 2π

2 2

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.

7 EX: Graph y = sin (x – π/3)

8 EX: Graph y = 3cos (x + π/4)

9 EX: Graph y = 4 – sin (x – π/3)

10 EX: Graph y =-3 + 3cos(x+π/4)

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

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)

11 EX: • Graph y = sin 2x

12 EX: • Graph y = -2cos 3x

13 EX: • Graph y = 3 – 2cos 3x

14 EX: Graph y = –2cos(3x+π)

15 EX: • Graph y = cos(2x/3)

16 EX: Graph y = –2 sin 3x

17 EX: Graph y = 3 cos ½ x

GRAPHING SINE AND COSINE FUNCTIONS

In previous chapters you learned that the graph ofy = a • f (x – h) + k is related to the graph of y = | a | • f (x) by horizontal and vertical translationsand by a reflection when a is negative. This also applies to sine, cosine, and tangent functions.


TRANSFORMATIONS OF SINE AND COSINE GRAPHS

To obtain the graph of

Transform the graph of y = | a | sin bx ory = | a | cos bx as follows.

y = a sin b (x – h) + k or y = a cos b (x – h) + k



VERTICAL SHIFT

Shift the graph k units vertically.

y = a • sin bx

y = a • sin bx + k

k



HORIZONTAL SHIFT

Shift the graph h units Vertically.

y = a • sin b(x – h)

h

y = a • sin bx



REFLECTION

If a < 0, reflect the graph in the liney = k after anyvertical and horizontalshifts have been performed.

y = a • sin bx + k

y = – a • sin bx + k

38

8

4

2

Graphing a Vertical Translation

Graph y = – 2 + 3 sin 4 x.

SOLUTION

By comparing the given equationto 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 = . 24

2

38

8

4

2


The graph oscillates 3 units up and down from its centerline y = – 2.

SOLUTION

38

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.

38

8

4

2


The five key points are:

On y = k : (0, 2); , – 2 ; , – 2 4

2

Maximum: , 1 8

Minimum: , – 5 38

SOLUTION

Graph y = – 2 + 3 sin 4 x.

38

8

4

2


CHECK

You can check your graph with a graphing calculator. Use theMaximum, Minimum and Intersect features to check the key points.

Graph y = – 2 + 3 sin 4 x.


Graph y = 2 cos x – .4

23

SOLUTION

3

Because the graph is a transformation of the graph of

y = 2 cos x, the amplitude is 2 and the period

is = 3 .

23

22


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.

23

4

Graph y = 2 cos x – .π4

23

SOLUTION

Notice that the maximum

occurs unit to the right of

the y-axis.

4

Graphing a Horizontal Translation

The five key points are:

Graph y = 2 cos x – .4

23

SOLUTION

4

14

On y = k : • 3 + , 0 = (, 0);

• 3 + , 0 = , 052

34

4

Minimum: • 3 + , – 2 = , – 2 74

4

12

Maximum: 0 + , 2 = , 2 ;

134

3 + , 2 = , 2 ;

4

4

4

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 theintercepts are the same as they are for the graph of y = 3 sin x.

SOLUTION


However, when the graph is reflected in the x-axis, themaximum becomes a minimum and the minimum becomes a maximum.


SOLUTION


On y = k : (0, 0); (2, 0);

12

• 2, 0 = (, 0)

Minimum: • 2, – 3 = , – 3 14

2

Maximum: • 2, 3 = , 3 34

32


SOLUTION The five key points are:

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 – 7.5 + 3015

The Ferris wheel turns for 135 seconds before it stopsto 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?

SOLUTION


The amplitude is 25 and the period is = 30.2

15

h = 25 sin t – 7.5 + 3015

The wheel turns = 4.5 times in 135 seconds,

so the graph shows 4.5 cycles.

13030


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 + 3015

SOLUTION


Since the amplitude is 25 and the graph is shifted up 30units, the maximum height is 30 + 25 = 55 feet.

The minimum height is 30 – 25 = 5 feet.

h = 25 sin t – 7.5 + 3015

SOLUTION

GRAPHING TANGENT FUNCTIONS

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. ||

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 .


Therefore translate the graph of

y = 2 tan x left unit and then

reflect it in the x-axis.

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


Asymptotes:

On y = k:

Halfway points:

x = – – = – ; x = – = 2 •1

4

34

4

4

2 •1

(h, k) = – , 04

– – , 2 = – , 2 ; – , – 2 = (0, – 2) 4 •1

4

2 4 •1

4


4.3 vertical and horizontal translations

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