Question

find an equation for the cosine graph, ( f(x) ):
write your answer in the form ( f(x)=a cos (b x+c)+d ), where ( a, b, c ) and numbers.

Explanation:

Step1: Find the amplitude \(A\)

The amplitude \(A=\frac{\text{max}-\text{min}}{2}\). Here, \(\text{max} = 5\), \(\text{min}=3\). So \(A=\frac{5 - 3}{2}=1\)

Step2: Find the vertical shift \(D\)

The vertical shift \(D=\frac{\text{max}+\text{min}}{2}\). So \(D=\frac{5 + 3}{2}=4\)

Step3: Find the period \(T\) and \(B\)

The period \(T = 2\pi\). Since \(T=\frac{2\pi}{B}\), and \(T = 2\pi\), then \(B = 1\)

Step4: Find the phase shift \(C\)

The graph has no phase - shift (the maximum occurs at \(x = 0\)), so \(C = 0\)

Answer:

\(f(x)=\cos(x)+4\)