# Precalculus : Amplitude, Period, Phase Shift of a Trig Function

## Example Questions

### Example Question #1 : Amplitude, Period, Phase Shift Of A Trig Function

What is the amplitude of the following function?

-14

-24

14

24

24

Explanation:

When you think of a trigonometric function of the form y=Asin(Bx+C)+D, the amplitude is represented by A, or the coefficient in front of the sine function. While this number is -24, we always represent amplitude as a positive number, by taking the absolute value of it. Therefore, the amplitude of this function is 24.

### Example Question #2 : Amplitude, Period, Phase Shift Of A Trig Function

Select the answer choice that correctly matches each function to its period.

Explanation:

The following matches the correct period with its corresponding trig function:

In other words, sin x, cos x, sec x, and csc x all repeat themselves every  units. However, tan x and cot x repeat themselves more frequently, every  units.

### Example Question #3 : Amplitude, Period, Phase Shift Of A Trig Function

What is the period of this sine graph?

Explanation:

The graph has 3 waves between 0 and , meaning that the length of each of the waves is  divided by 3, or .

### Example Question #4 : Amplitude, Period, Phase Shift Of A Trig Function

Write the equation for a cosine graph with a maximum at  and a minimum at .

Explanation:

In order to write this equation, it is helpful to sketch a graph:

The dotted line is at , where the maximum occurs and therefore where the graph starts. This means that the graph is shifted to the right

The distance from the maximum to the minimum is half the entire wavelength. Here it is .

Since half the wavelength is , that means the full wavelength is  so the frequency is just 1.

The amplitude is 3 because the graph goes symmetrically from -3 to 3.

The equation will be in the form  where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift.

This equation is

.

### Example Question #1 : Amplitude, Period, Phase Shift Of A Trig Function

Find the phase shift of .

-4

-2

3

2

-2

Explanation:

In the formula,

.

represents the phase shift.

Plugging in what we know gives us:

.

Simplified, the phase is then .

### Example Question #6 : Amplitude, Period, Phase Shift Of A Trig Function

Which equation would produce this sine graph?

Explanation:

The graph has an amplitude of 2 but has been shifted down 1:

In terms of the equation, this puts a 2 in front of sin, and -1 at the end.

This makes it easier to see that the graph starts [is at 0] where .

The phase shift is  to the right, or

### Example Question #7 : Amplitude, Period, Phase Shift Of A Trig Function

Which of the following equations could represent a cosine function with amplitude 3, period , and a phase shift of ?

Explanation:

The form of the equation will be

First, think about all possible values of A that could give you an amplitude of 3. Either A = -3 or A = 3 could each produce amplitude = 3. Be sure to look for answer choices that satisfy either of these.

Secondly, we know that the period is . Normally we know what B is and need to find the period, but this is the other way around. We can still use the same equation and solve:

. You can cross multiply to solve and get B = 4.

Finally, we need to find a value of C that satisfies

. Cross multiply to get:

Next, plug in B= 4 to solve for C:

Putting this all together, the equation could either be:

or

### Example Question #8 : Amplitude, Period, Phase Shift Of A Trig Function

State the amplitude, period, phase shift, and vertical shift of the function

Amplitude: 7

Period: /3

Phase Shift: /6

Vertical Shift: -4

Amplitude: -7

Period: /3

Phase Shift: -/6

Vertical Shift: 4

Amplitude: 7

Period: -/3

Phase Shift: -/6

Vertical Shift: -4

Amplitude: 7

Period: /3

Phase Shift: -/6

Vertical Shift: -4

Amplitude: 7

Period: /3

Phase Shift: -/6

Vertical Shift: -4

Explanation:

A common way to make sense of all of the transformations that can happen to a trigonometric function is the following. For the equations y = A sin(Bx + C) + D,

• amplitude is |A|
• period is 2/|B|
• phase shift is -C/B
• vertical shift is D

In our equation, A=-7, B=6, C=, and D=-4. Next, apply the above numbers to find amplitude, period, phase shift, and vertical shift.

To find amplitude, look at the coefficient in front of the sine function. A=-7, so our amplitude is equal to 7.

The period is 2/B, and in this case B=6. Therefore the period of this function is equal to 2/6 or /3.

To find the phase shift, take -C/B, or -/6. Another way to find this same value is to set the inside of the parenthesis equal to 0, then solve for x.
6x+=0
6x=-
x=-/6
Either way, our phase shift is equal to -/6.

The vertical shift is equal to D, which is -4.

y=-7\sin(6x+\pi)-4

### Example Question #9 : Amplitude, Period, Phase Shift Of A Trig Function

State the amplitude, period, phase shift, and vertical shift of the function

Amplitude: 1

Period:

Phase Shift:

Vertical Shift: 0

Amplitude: 1

Period:

Phase Shift:

Vertical Shift: 3

Amplitude: 1

Period:

Phase Shift:

Vertical Shift: 3

Amplitude: 1

Period:

Phase Shift:

Vertical Shift: 3

Amplitude: 1

Period:

Phase Shift:

Vertical Shift: 3

Amplitude: 1

Period:

Phase Shift:

Vertical Shift: 3

Explanation:

A common way to make sense of all of the transformations that can happen to a trigonometric function is the following. For the equations y = A sin(Bx + C) + D,

• amplitude is |A|
• period is 2/|B|
• phase shift is -C/B
• vertical shift is D

In our equation, A=-1, B=1, C=-, and D=3. Next, apply the above numbers to find amplitude, period, phase shift, and vertical shift.

To find amplitude, look at the coefficient in front of the sine function. A=-1, so our amplitude is equal to 1.

The period is 2/B, and in this case B=1. Therefore the period of this function is equal to 2.

To find the phase shift, take -C/B, or . Another way to find this same value is to set the inside of the parenthesis equal to 0, then solve for x.
x-=0
x=
Either way, our phase shift is equal to .

The vertical shift is equal to D, which is 3.

### Example Question #10 : Amplitude, Period, Phase Shift Of A Trig Function

State the amplitude, period, phase shift, and vertical shift of the function

Amplitude: 1

Period: 3/2

Phase Shift:

Vertical Shift: 2

Amplitude: 1

Period:

Phase Shift: -3/2

Vertical Shift: -2

Amplitude: 1

Period:

Phase Shift: -3/2

Vertical Shift: 2

Amplitude: 1

Period:

Phase Shift: 3/2

Vertical Shift: 2

Amplitude: 1

Period:

Phase Shift: 3/2

Vertical Shift: 2

Explanation:

A common way to make sense of all of the transformations that can happen to a trigonometric function is the following. For the equations y = A sin(Bx + C) + D,

• amplitude is |A|
• period is 2/|B|
• phase shift is -C/B
• vertical shift is D

In our equation, A=1, B=2, C=-3, and D=2. Next, apply the above numbers to find amplitude, period, phase shift, and vertical shift.

To find amplitude, look at the coefficient in front of the sine function. A=1, so our amplitude is equal to 1.

The period is 2/B, and in this case B=2. Therefore the period of this function is equal to .

To find the phase shift, take -C/B, or 3/2. Another way to find this same value is to set the inside of the parenthesis equal to 0, then solve for x.
2x-3=0
2x=3
x=3/2
Either way, our phase shift is equal to 3/2.

The vertical shift is equal to D, which is 2.