Question

amplitude =
period =
phase shift =
equation (5) = (in terms of the cosine function)

Explanation:

Step1: Find the amplitude

The amplitude \(A\) of a cosine - type function \(y = A\cos(B(x - C))+D\) is the distance from the mid - line to the maximum (or minimum) value. Looking at the graph, the maximum value is \(y = 3\) and the minimum value is \(y=-3\). The mid - line is \(y = 0\). So, \(A=\frac{3-(-3)}{2}=3\)

Step2: Find the period

The period \(T\) of a cosine function \(y = A\cos(B(x - C))+D\) is given by \(T=\frac{2\pi}{|B|}\). From the graph, we can see that one full cycle (from peak to peak or trough to trough) occurs over a horizontal distance. If we assume the standard form and count the units on the \(x\) - axis, the period \(T = 4\)

Step3: Find the phase shift

The general form of a cosine function is \(y = A\cos(B(x - C))+D\), where \(C\) is the phase shift. For a cosine function \(y=\cos(x)\), the maximum is at \(x = 0\). In our graph, if we assume the basic cosine - like shape, and rewrite the function in the form \(y = A\cos(B(x - C))+D\). Since the function has no horizontal shift (the maximum is at \(x = 0\) in the graph we are analyzing, similar to the basic \(y=\cos(x)\) function), \(C = 0\)

Step4: Write the equation

Using the formula \(y=A\cos(Bx)+D\). We know \(A = 3\), \(B=\frac{2\pi}{T}\), and since \(T = 4\), \(B=\frac{\pi}{2}\), and \(D = 0\) (because the mid - line is \(y = 0\)). So the equation \(y = 3\cos(\frac{\pi}{2}x)\)

Answer:

Amplitude \(=3\), Period \(=4\), Phase Shift \(=0\), Equation \(y = 3\cos(\frac{\pi}{2}x)\)