Question

write the equation of the trigonometric function shown in the graph.
$y = \cos( x ) + $
answer

Explanation:

Step1: Find the amplitude

The amplitude \(A\) of a cosine function \(y = A\cos(Bx)+C\) is half the distance between the maximum and minimum values. For the solid - line graph (the function we are interested in), the maximum value \(y_{max}=1\) and the minimum value \(y_{min}=-1\). So, \(A=\frac{y_{max}-y_{min}}{2}=\frac{1 - (-1)}{2}=1\).

Step2: Find the period and \(B\)

The general formula for the period of a cosine function \(y = A\cos(Bx)+C\) is \(T=\frac{2\pi}{|B|}\). For the solid - line graph, the period \(T = 2\pi\). Using the formula \(T=\frac{2\pi}{|B|}\), and since \(T = 2\pi\), we have \(2\pi=\frac{2\pi}{|B|}\), which gives \(|B| = 1\).

Step3: Find the vertical shift \(C\)

The vertical shift \(C\) of a cosine function \(y=A\cos(Bx)+C\) is the mid - value between the maximum and minimum values. \(C=\frac{y_{max}+y_{min}}{2}=\frac{1+( - 1)}{2}=0\).

Answer:

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