# How do I find the equation of a sinusoidal graph?

##### 1 Answer

I will provide you with two examples.

#### Explanation:

Before we get to problems, I would like to go through a little bit of vocabulary.

*•A sinusoidal function is a function in sine or in cosine*

•The amplitude of a graph is the distance on the y axis between the normal line and the maximum/minimum. It is given by parameter

•The period of a graph is the distance on the x axis before the function repeats itself. For sinusoidal functions, it is given by evaluating

•The horizontal displacement is given by solving for

•The vertical displacement is given by

This being done, we can now look at a few applications to these particular words.

**Example 1:**

What is a **cosine** equation for the following graph?

First, let's note the amplitude. The normal line is the line that runs completely in the middle, so it is

The amplitude is given by

However, the graph of

Next, let's determine the period. Look back at the definition above of "period". It is the distance between two maximums or two minimums. In the graph above, the distance between any two maximums or minimums is

Recall the period of a sinusoidal function is given by

Solving for b:

So,

As for horizontal displacements, there are none, since the minimum is on the y axis; it hasn't been moved left or right.

In summary, we can now state that the equation of the function above is

**Example 2:**

Determine the equation of the following graph.

This is a little more complicated. We first note that a vertical displacement has occurred. The graph has been moved upwards

As for the amplitude, we find the maximum is at

This graph has undergone no reflection over the x axis, so parameter

As for the period, the distance between all two maximums and minimums is

Hence,

Finally, we need to determine the factor of the horizontal displacement. We find that it is

Hopefully this helps!