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

Jul 12, 2016

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 $a$ in function $y = a \sin b \left(x - c\right) + d \mathmr{and} y = a \cos b \left(x - c\right) + d$

•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 $\left(2 p\right) \frac{i}{b}$ in $y = a \cos b \left(x - c\right) + d \mathmr{and} y = a \sin b \left(x - c\right) + d$

•The horizontal displacement is given by solving for $x$ in $x - c = 0$ in $y = a \cos b \left(x - c\right) + d \mathmr{and} y = a \sin b \left(x - c\right) + d$. The horizontal displacement means the number of units right or left in from the x axis

•The vertical displacement is given by $d$ in $y = a \cos b \left(x - c\right) + d \mathmr{and} y = a \sin b \left(x - c\right) + d$. The vertical displacement is the displacement up or down from the y axis.

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 $x = 0$. This also signifies that there is no vertical displacement, or $d = 0$ in $y = a \cos b \left(x - c\right) + d$.

The amplitude is given by $\text{equation of max" - "equation of normal}$. In this case, the equation of the maximum is $y = 2$ while the equation of the normal is $y = 0$. Hence, the amplitude is $2 - 0 = 2$.

However, the graph of $y = \cos x$ has a maximum on the y axis, not a minimum like in our graph. What does this signify? It signifies there has been a reflection over the x axis, which means parameter $a$ is negative. Hence, parameter $a$ is $- 2$. Note that the amplitude can never be negative, so it's given by $| a |$.

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 $\pi$. We know the period now, all that remains is to find the value of $b$.

Recall the period of a sinusoidal function is given by $\frac{2 \pi}{b}$. Hence, we can state that $\frac{2 \pi}{b} = \pi$

Solving for b:

$2 \pi = b \pi$

$\frac{2 \pi}{\pi} = b$

$b = 2$

So, $b = 2$.

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 $y = - 2 \cos \left(2 x\right)$.

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 $3$ units relative to that of $y = \sin x$ (the normal line has equation $y = 3$). We can also conclude that this is a sine function, because the graph meets the $y$ axis at the normal line, and not at a maximum/minimum.

As for the amplitude, we find the maximum is at $y = 5$ while the normal line is $y = 3$. Hence, the amplitude is $5 - 3 = 2$.

This graph has undergone no reflection over the x axis, so parameter $a$ is positive in this scenario.

As for the period, the distance between all two maximums and minimums is $1$, so the period is $1$. We must determine the value of $b$:

$\frac{2 \pi}{b} = 1$

$2 \pi = b$

Hence, $b = 2 \pi$.

Finally, we need to determine the factor of the horizontal displacement. We find that it is $1$ unit to the right. Hence, our equation is $y = 2 \sin \left(2 \pi \left(x - 1\right)\right) + 3$.

Hopefully this helps!