Problem 59 - Algebra and Trigonometry

Question

Use a vertical shift to graph one period of the function. $$y=-3 \cos 2 \pi x+2$$

Step-by-Step Solution

Answer

The function has amplitude 3, period 1 and vertical shift 2 and it starts at its lowest point due to the negative coefficient of cosine. By marking these points on the graph, we sketch one period.

Step 1: Identify Parameters

The given function is \(y=-3 \cos 2 \pi x +2 \). By comparing to the standard cosine function \(y=a \cos (b x) +c\), we can observe that a=-3, b=2\pi, and c=2. The value of 'a' affects the amplitude, 'b' affects the period and 'c' is the vertical shift.

Step 2: Define Amplitude and Period

Calculate the amplitude and period. The amplitude is the absolute value of 'a', so here, the amplitude is | -3 |=3. The period is given by \(\frac{2\pi}{|b|}\), so here the period is \(\frac{2\pi}{|2\pi|}\) = 1.

Step 3: Define the Vertical Shift

The vertical shift is given by the constant 'c', which is 2 in this case. This means our function shifts up by 2 units.

Step 4: Graph the function

To graph, we start by marking the period on the x-axis, which is 1. Mark the shifted midline on the y-axis, it's 2. Since it's a negative cosine function, start at the lowest point, which is 2 - amplitude = -1. The function will reach its highest point at x=0.5, so we mark the harmonious points for one complete period.