Math Snap

STEP 1

What is this asking?
We need to find the equation of a trigonometric function ($y = A \sin(Bx)$ or $y = A \cos(Bx)$) that matches the given graph.
Watch out!
Don't mix up sine and cosine, and make sure you get the amplitude, period, and vertical shift right!

STEP 2

1. Analyze the graph
2. Determine the amplitude A
3. Determine the vertical shift D
4. Determine the period and B
5. Choose sine or cosine and write the equation

STEP 3

Let's look at this graph!
We see it starts at a maximum value, which hints that it might be a cosine function.
It goes up to $y = 8$ and down to $y = -2$.
The graph completes one full cycle between $x = 0$ and $x = \frac{3\pi}{4}$.

STEP 4

The amplitude is the distance from the midline to the maximum (or minimum) value.
The midline is halfway between the highest and lowest points.
We calculate this by finding the average of the maximum and minimum values: $\frac{8 + (-2)}{2} = \frac{6}{2} = 3$.
The distance from the midline ($y=3$) to the maximum ($y=8$) is $8 - 3 = 5$.
So, our amplitude $A$ is 5.

STEP 5

The vertical shift is the value of the midline.
We already found that the midline is at $y = 3$, so our vertical shift $D$ is 3.

STEP 6

The period is the length of one full cycle of the graph.
We can see that the graph completes one cycle from $x = 0$ to $x = \frac{3\pi}{4}$.
So, the period is $\frac{3\pi}{4}$.

STEP 7

Now, we know the period is related to $B$ by the formula $Period = \frac{2\pi}{B}$.
We have $\frac{3\pi}{4} = \frac{2\pi}{B}$.
To solve for $B$, we can cross-multiply: $3\pi \cdot B = 2\pi \cdot 4$, which simplifies to $3\pi B = 8\pi$.
Now, we divide both sides by $3\pi$ to isolate $B$: $B = \frac{8\pi}{3\pi} = \frac{8}{3}$.
So, $B = \frac{8}{3}$.

STEP 8

Since the graph starts at a maximum, we'll use cosine.
A standard cosine graph starts at its maximum, while sine starts at the midline.

STEP 9

We found $A = 5$, $B = \frac{8}{3}$, and $D = 3$.
Since we are using cosine, the general form is $y = A \cos(Bx) + D$.
Plugging in our values, we get $y = 5 \cos\left(\frac{8}{3}x\right) + 3$.

The equation of the function is $y = 5 \cos\left(\frac{8}{3}x\right) + 3$.