Question

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

Explanation:

Step1: Calculate the amplitude

The amplitude is the distance from the mid - line to the maximum (or minimum) value. The mid - line is \(y = 0\). The maximum value is \(y=- 1\) and the minimum value is \(y = - 7\). The amplitude \(A=\frac{\vert-1-(-7)\vert}{2}=\frac{\vert6\vert}{2}=3\)

Step2: Calculate the period

The period is the length of one full cycle. From the graph, the distance between two consecutive maxima (or minima) is \(4\) units. So the period \(T = 4\). Using the formula \(T=\frac{2\pi}{B}\), we can solve for \(B\). Since \(T = 4\), then \(B=\frac{\pi}{2}\)

Step3: Determine the phase shift

The graph is a cosine function (starts at a maximum when \(x = 0\) for \(y = A\cos(Bx - C)+D\)). For \(y=\cos(x)\), the maximum is at \(x = 0\). In our case, the function \(y=-3\cos(\frac{\pi}{2}x)-4\) (we will get to the vertical shift \(D\) in the next step). The phase shift \(C = 0\) (because there is no horizontal shift from the standard cosine function's maximum at \(x = 0\) in terms of the basic shape)

Step4: Find the vertical shift and write the equation

The mid - line \(y = D\). Since the maximum \(y=-1\) and \(A = 3\), using \(y=A\cos(Bx)+D\), we substitute \(y=-1\), \(A = 3\) and \(x = 0\) (for the maximum of the cosine function \(\cos(0)=1\)). Then \(-1=3\times1+D\), so \(D=-4\)

The general form of a cosine function is \(y = A\cos(Bx)+D\). Substituting \(A=- 3\) (because the graph is reflected over the \(x\) - axis compared to \(y = 3\cos(\frac{\pi}{2}x)-4\)), \(B=\frac{\pi}{2}\) and \(D=-4\), the equation is \(y=-3\cos(\frac{\pi}{2}x)-4\)

Answer:

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