Problem 60 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The function \(y=-3 \sin 2 \pi x+2\) represents an inverted sine wave with an amplitude of 3, period of \( \frac{1}{2 \pi}\), and vertical shift of 2 units upwards.

Step 1: Identify amplitude, frequency, period and vertical shift

The sine function in this case is made up of different components. The amplitude is defined by the absolute value of the coefficient before the sine function, thus the amplitude of this function is 3. The frequency of the function is determined by the coefficient before \(x\), in this case it is \(2 \pi\), and as the relationship between frequency and period is reciprocal, the period of the function is \(T=\frac{1}{f} = \frac{1}{2 \pi}\). The vertical shift is defined by the constant added to the function, thus the function is shifted 2 units upwards.

Step 2: Plot the function

To plot the function, start by setting a period from 0 to \( \frac{1}{2 \pi}\). Given the standard formation of the sine wave, plot the points at the start, peak, middle, valley and end of the period (they should occur at 0, \( \frac{1}{4(2\pi)}\), \( \frac{1}{2(2\pi)}\), \( \frac{3}{4(2\pi)}\) and \( \frac{1}{2\pi}\)). Remember that our function is inverted due to the negative amplitude, and vertically shifted by 2. Thus, the start and end points of the period are at \(y=2\), peak is at \(y=2-3=-1\), middle point at \(y=2\), and valley at \(y=2+3=5\).

Step 3: Draw the function

Now draw the sine wave y=-3 sin (2 π x + 2) within the plotted points for one period, making sure to reflect the correct amplitude and vertical shift.