All sine functions have an amplitude and a period. The amplitude is the distance between the centerline of the function and the maximum or minimum point of the function. We can also consider the amplitude as the vertical distance between the sinusoidal axis and the maximum or minimum values of the function. With respect to sound waves, amplitude is a measure of how loud something is.

Here, we will learn to find the amplitude of sine functions and solve some practice problems.

## How to find the amplitude of sine functions?

The general form of a sine function is:

$latex f(x)=\pm A~\sin(B(x+C))+D$

In this form, the coefficient A is the “height” of the sine. If we do not have any number present, then the amplitude is assumed to be 1. We can define the amplitude using a graph. The following is the graph of the function $latex y = 2 ~ \sin(x)$, which has an amplitude of 2:

We observe that the amplitude is 2 instead of 4. In this case, the amplitude corresponds to the absolute value of the maximum value or minimum value of the function. If we had the function $latex y = -2 ~ \sin (x)$, the graph would be reflected with respect to the *x*-axis, but its amplitude would remain the same.

The sinusoidal axis is the horizontal line between the peaks and the troughs. In this function, the sinusoidal axis is simply the *x*-axis. However, if the graph were translated vertically, the sinusoidal axis would no longer be on the *x*-axis but would be located exactly in the middle of the peaks and troughs.

The larger the amplitude of the function, the taller its graph will appear. On the other hand, the smaller the amplitude of the function, the lower its graph will appear.

## Amplitude of the sine function – Examples with answers

The following examples of the amplitude of sine functions are solved using the relation of the functions with the general form. Try to solve the problems yourself before looking at the answer.

**EXAMPLE 1**

What is the amplitude of the function $latex y = 3 ~ \sin(2x)$?

##### Solution

To determine the amplitude of the function, we have to compare it with the general form $latex y = A ~ \sin(B (x + C)) + D$. Comparing the functions, we see that we have:

$latex A=3$

This means that the amplitude is equal to 3.

**EXAMPLE 2**

If we have the sine function $latex y = -4 ~ \sin(4x) +1$, what is its amplitude?

##### Solution

We use the general form $latex y = A ~ \sin(B(x+C))+D$ and compare it with the given function. When comparing them, we see that we have:

$latex A=-4$

We know that the amplitude is the absolute value of this parameter, so the amplitude is equal to 4.

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**EXAMPLE 3**

What is the amplitude of the function $latex y = \frac{1}{3} \sin(- \frac{1}{4} x-4)$?

##### Solution

Again, we have to compare the given function with the general form $latex y = A ~ \sin(B(x + C)) + D$. By doing this, we have:

$latex A=\frac{1}{3}$

The amplitude is equal to $latex \frac{1}{3}$. Therefore, the amplitude does not necessarily have to be an integer value.

**EXAMPLE 4**

If we have the function $latex y = 2 (\frac{3}{2} \sin(2x-2))$, what is its amplitude?

##### Solution

In this case, we see that the entire function is being multiplied by 2. This means that when we compare the function with the general form $latex y = A ~ \sin (B (x + C)) + D$, we have:

$latex A=2(\frac{3}{2})$

$latex A=3$

The amplitude of the function is 3.

## Amplitude of the sine – Practice problems

Practice what you have learned about the amplitude of sine functions by solving the following problems. Select an answer and check it to see if you got the correct answer.

## See also

Interested in learning more about sine of an angle? Take a look at these pages:

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