Question

write the equation of the trigonometric function shown in the graph. use a positive coefficient on cosine for this activity.
answer attempt 1 out of 2
you must answer all questions above in order to submit.

Explanation:

Step 1: Determine the amplitude

The amplitude \(A\) of a cosine function \(y = A\cos(B(x - C))+D\) is half the distance between the maximum and minimum values.
The maximum value \(y_{max}=1\) and the minimum value \(y_{min}=-1\). So, \(A=\frac{y_{max}-y_{min}}{2}=\frac{1 - (- 1)}{2}=1\).

Step 2: Determine the period and find \(B\)

The period \(P\) of a cosine function \(y = A\cos(B(x - C))+D\) is given by \(P=\frac{2\pi}{B}\).
Looking at the graph, the period \(P=\pi\). Then \(\pi=\frac{2\pi}{B}\), solving for \(B\) gives \(B = 2\).

Step 3: Determine the phase - shift (find \(C\))

The general form of a cosine function is \(y=A\cos(B(x - C))+D\). For the standard cosine function \(y = \cos(x)\), the maximum occurs at \(x = 0\). For the given function, the maximum occurs at \(x = 0\), so the phase - shift \(C = 0\).

Step 4: Determine the vertical shift (find \(D\))

The vertical shift \(D\) is the mid - line of the function. The mid - line \(D=\frac{y_{max}+y_{min}}{2}=\frac{1+( - 1)}{2}=0\).

Answer:

\(y = 1\cos(2(x - 0))+0\)