Question

write the equation of the trigonometric function shown in the graph.

Explanation:

Step1: Identify the type of trigonometric function

The graph appears to be a cosine function since it has a maximum at \( x = 0 \) (the y - axis), which is a characteristic of the cosine function \( y=\cos(x) \) (which has a maximum of 1 at \( x = 0 \)).

Step2: Determine the amplitude

The amplitude \( A \) is the maximum value of the function. From the graph, the maximum value of \( y \) is 5. So, \( A = 5 \).

Step3: Determine the period

The period of a cosine function is given by \( T=\frac{2\pi}{|B|} \) for the function \( y = A\cos(Bx) \). Let's find the period from the graph. The distance between two consecutive peaks (or troughs) is the period. Looking at the x - axis values, the distance between \( -\frac{3\pi}{2} \) and \( -\frac{\pi}{2} \) (or other consecutive peaks) is \( \pi \). So, the period \( T=\pi \).

Since \( T = \pi=\frac{2\pi}{|B|} \), we can solve for \( B \):

$$ \pi=\frac{2\pi}{|B|}\\ |B|=\frac{2\pi}{\pi}=2 $$

We can take \( B = 2 \) (since the function is not reflected over the x - axis, as the maximum is at \( x = 0 \), so the sign of \( B \) does not change the shape in terms of reflection here).

Step4: Write the equation

Putting it all together, the general form of a cosine function is \( y = A\cos(Bx) \). Substituting \( A = 5 \) and \( B = 2 \), we get the equation \( y = 5\cos(2x) \).

Answer:

\( y = 5\cos(2x) \)