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A graph with four items. The x-axis ranges from -6pi to 6pi. The y-axis ranges from -4 to 4. The first item is the graph of sin(x), which has an amplitude of 1. The second is a graph of 2sin(x), which has amplitude of 2. The third is a graph of 3sin(x), which has an amplitude of 3. The fourth is a graph of 4 sin(x) with an amplitude of 4.

Amplitude of sinusoidal functions

If we let C = 0 and D = 0 in the general form equations of the sine and cosine functions, we obtain the forms

y = A sin ( B x )  and  y = A cos ( B x )

The amplitude    is A , and the vertical height from the midline    is | A | . In addition, notice in the example that

| A |  = amplitude =  1 2 | maximum   minimum |

Identifying the amplitude of a sine or cosine function

What is the amplitude of the sinusoidal function f ( x ) = −4 sin ( x ) ? Is the function stretched or compressed vertically?

Let’s begin by comparing the function to the simplified form y = A sin ( B x ) .

In the given function, A = −4 , so the amplitude is | A | = | −4 | = 4. The function is stretched.

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What is the amplitude of the sinusoidal function f ( x ) = 1 2 sin ( x ) ? Is the function stretched or compressed vertically?

1 2 compressed

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Analyzing graphs of variations of y = sin x And y = cos x

Now that we understand how A and B relate to the general form equation for the sine and cosine functions, we will explore the variables C and D . Recall the general form:

y = A sin ( B x C ) + D  and  y = A cos ( B x C ) + D o r y = A sin ( B ( x C B ) ) + D  and  y = A cos ( B ( x C B ) ) + D

The value C B for a sinusoidal function is called the phase shift , or the horizontal displacement of the basic sine or cosine function    . If C > 0 , the graph shifts to the right. If C < 0 , the graph shifts to the left. The greater the value of | C | , the more the graph is shifted. [link] shows that the graph of f ( x ) = sin ( x π ) shifts to the right by π units, which is more than we see in the graph of f ( x ) = sin ( x π 4 ) , which shifts to the right by π 4 units.

A graph with three items. The first item is a graph of sin(x). The second item is a graph of sin(x-pi/4), which is the same as sin(x) except shifted to the right by pi/4. The third item is a graph of sin(x-pi), which is the same as sin(x) except shifted to the right by pi.

While C relates to the horizontal shift, D indicates the vertical shift from the midline in the general formula for a sinusoidal function. See [link] . The function y = cos ( x ) + D has its midline at y = D .

A graph of y=Asin(x)+D. Graph shows the midline of the function at y=D.

Any value of D other than zero shifts the graph up or down. [link] compares f ( x ) = sin x with f ( x ) = sin x + 2 , which is shifted 2 units up on a graph.

A graph with two items. The first item is a graph of sin(x). The second item is a graph of sin(x)+2, which is the same as sin(x) except shifted up by 2.

Variations of sine and cosine functions

Given an equation in the form f ( x ) = A sin ( B x C ) + D or f ( x ) = A cos ( B x C ) + D , C B is the phase shift    and D is the vertical shift    .

Identifying the phase shift of a function

Determine the direction and magnitude of the phase shift for f ( x ) = sin ( x + π 6 ) 2.

Let’s begin by comparing the equation to the general form y = A sin ( B x C ) + D .

In the given equation, notice that B = 1 and C = π 6 . So the phase shift is

C B = π 6 1     = π 6

or π 6 units to the left.

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Determine the direction and magnitude of the phase shift for f ( x ) = 3 cos ( x π 2 ) .

π 2 ; right

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Identifying the vertical shift of a function

Determine the direction and magnitude of the vertical shift for f ( x ) = cos ( x ) 3.

Let’s begin by comparing the equation to the general form y = A cos ( B x C ) + D .

In the given equation, D = −3 so the shift is 3 units downward.

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Determine the direction and magnitude of the vertical shift for f ( x ) = 3 sin ( x ) + 2.

2 units up

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Given a sinusoidal function in the form f ( x ) = A sin ( B x C ) + D , identify the midline, amplitude, period, and phase shift.

  1. Determine the amplitude as | A | .
  2. Determine the period as P = 2 π | B | .
  3. Determine the phase shift as C B .
  4. Determine the midline as y = D .

Questions & Answers

sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
f(x)= 1350. 2^(t/20); where t is in hours.
Merkeb
Practice Key Terms 5

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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