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Chapter 4: Problem 54

Sketch the graph of the function. (Include two full periods.) $$y=-3+5 \cos \frac{\pi t}{12}$$

Short Answer

Expert verified
The graph of the function \(y=-3+5 \cos \frac{\pi t}{12}\) starts at its maximum value of 2, at t = 0. It oscillates down to a minimum value of -8, at t = 12, and back to the maximum at t = 24. This completes one cycle, and the function repeats this cycle for the second period until t = 48. The function oscillates around the midline y = -3.

Step by step solution

01

Determine the Midline, Amplitude, Period and Frequency of the Function

The midline of the function is given by the constant -3. This means that the function oscillates about the line y = -3. The amplitude of the function is 5, which is the absolute value of the coefficient of the cosine function. Since there is no horizontal shift in the function, the graph starts from the maximum value of the function. The period of the function is given by \(\frac{2\pi}{\frac{\pi}{12}}\), which simplifies to 24. This means the function repeats every 24 units. The function is a cosine function, and it has not been reflected, so it starts at its maximum value.
02

Sketch the Function Using the Determined Characteristics

Start by drawing the midline on y = -3. Since the period is 24, mark the increments along the t-axis as a multiple of 6, since the cosine function has its critical points at each quarter of its period. The maximum value would be -3+5 = 2, and the minimum value would be -3-5 = -8. At t = 0, the value of y is at its maximum. At t = 6, it hits the midline; at t = 12, it gets to the minimum; at t = 18, it returns to the midline; and at t = 24, it returns to the maximum completing one cycle. For two cycles, this continues till t = 48.
03

Highlighting the Key Points on the Graph

Highlight the key points on the graph which would include the maximums, minimums and midline intersections. Note the symmetry of the curve about the midline and be sure to clearly label the maximum, minimum and period on the graph.

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