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:

Step1: 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. From the graph, the maximum value is \(1\) and the minimum value is \(- 1\). So, \(A=\frac{1 - (-1)}{2}=1\).

Step2: Determine the period

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

Step3: Determine the phase - shift

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

Step4: Determine the vertical - shift

The vertical - shift \(D\) of a cosine function \(y = A\cos(B(x - C))+D\) is the mid - line. The mid - line is \(y=\frac{1+(-1)}{2}=0\).

Answer:

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