Questions: rite the equation of the trigonometric graph.

Solution Steps

The amplitude is the distance from the midline to the maximum or minimum value of the function. In this graph, the midline is \(y=1\) and the maximum value is \(y=4\). The amplitude is \(4 - 1 = 3\).

The vertical shift is the midline of the function. In this case, the midline is \(y = 1\), so the vertical shift is 1.

The period is the horizontal distance it takes for the graph to complete one full cycle. Observe that the graph starts at a maximum at \(x=0\), reaches a minimum, and then returns to a maximum at \(x=4\pi\). Thus the period is \(4\pi\).

The period is related to _b_ by the equation \( \text{Period} = \frac{2\pi}{b} \). We found that the period is \(4\pi\). Thus, \( 4\pi = \frac{2\pi}{b} \). Solving for _b_, we get \( b = \frac{2\pi}{4\pi} = \frac{1}{2} \).

The graph resembles a cosine function since it starts at a maximum value. There is no horizontal shift, so the phase shift is 0.

The general equation for a cosine function is \( y = A\cos(b(x - c)) + d \), where A is the amplitude, _b_ is related to the period, _c_ is the phase shift, and _d_ is the vertical shift.

In this case, we have \(A = 3\), \(b = \frac{1}{2}\), \(c = 0\), and \(d = 1\).

Final Answer