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Chapter 4: Problem 53

Sketch the graph of the function. (Include two full periods.) $$y=2-\sin \frac{2 \pi x}{3}$$

Short Answer

Expert verified
The graph of the function contains two complete periods of a downward-opening sine wave, centered at y = 2, with a maximum value of y = 3, a minimum value of y = 1, and a period of 1.5 units.

Step by step solution

01

Identify the Amplitude

The amplitude of a sine function is usually the absolute value of the coefficient of the sine function. Here, there is no coefficient multiplied by the sine function, so the amplitude is 1.
02

Find the Frequency

The frequency of the sine function is given by the coefficient of x inside the sine function. In this case, the frequency is \( \frac{2 \pi }{3} \).
03

Calculate the Period

For a sine function, the period is given by \( \frac{2\pi }{frequency} \). Substituting 'frequency' from step 2, the period is \( \frac{2\pi }{ (2\pi /3) } = 1.5 \). So the function repeats every 1.5 units.
04

Identify the Phase Shift and Vertical Shift

The function is of the form \( y = A - \sin Bx \), where A is the vertical shift and B is the coefficient of x, decided the phase shift, which is 0 in this case. The value of A is 2, which means the entire function is shifted two units upwards.
05

Sketch the Graph

Now that we know all the parameters of the sine function, we can start sketching the graph. First, mark a vertical line at y=2 for the vertical shift. Then, starting at x = 0, make a point for each increment of the period until you have at least two full periods. Since the function is a negative sine function, it will start at the middle going down and returns to the middle at the end of the period. Complete each period by marking the maximum and minimum points, which are one unit above and below the middle line. Connect the points with a smooth, wavy line to complete the sketch.

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Most popular questions from this chapter

Graph the functions \(f\) and \(g .\) Use the graphs to make a conjecture about the relationship between the functions. $$f(x)=\sin x-\cos \left(x+\frac{\pi}{2}\right), \quad g(x)=2 \sin x$$

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