Question

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

$y=squarecos(square x)+square$

Explanation:

Step1: Find the amplitude

The general form of a cosine function is \(y = A\cos(Bx)+C\). The amplitude \(A\) is half the distance between the maximum and minimum values.
The maximum value of the dotted - line (auxiliary cosine - like function) is \(y = 5\) and the minimum is \(y = 1\). So, \(A=\frac{5 - 1}{2}=2\).

Step2: Find the period

The period \(T\) of a cosine function \(y=\cos(Bx)\) is given by \(T=\frac{2\pi}{B}\).
For the dotted - line (auxiliary cosine - like function), from \(x = 0\) to \(x = 2\pi\), the function completes one full cycle. So, \(T = 2\pi\). Using the formula \(T=\frac{2\pi}{B}\), we substitute \(T = 2\pi\) and get \(B = 1\).

Step3: Find the vertical shift

The vertical shift \(C\) is the mid - value of the maximum and minimum of the function. \(C=\frac{5 + 1}{2}=3\).

Answer:

\(y = 2\cos(x)+3\)