Answer
Refer to the graph below.
Work Step by Step
RECALL:
The functions $y=a \cdot \sin{bx}$ and $y=a \cdot \cos{bx}$ have:
period = $\dfrac{2\pi}{b}$
amplitude = $|a|$
The given function has $a=-\frac{2}{3}$ and $b=\frac{\pi}{4}$ thus
period = $\dfrac{2\pi}{\frac{\pi}{4}}=8$
amplitude = $|-\frac{2}{3}| = \frac{2}{3}$
To graph the function, perform the following steps:
(1) With a period of $8$, one period of the function is over the interval $[0, 8]$.
(2) Divide this interval into four parts to obtain the x-values $0, 2, 4, 6, \text{ and } 8$.
(3) Make a table of values using the x-values above. (Refer to the table below.)
(4) Plot the five points of the table of values then connect them using a sinusoidal curve whose amplitude is $\frac{2}{3}$.
(5) Repeat the cycle of the graph form one more period, which is $[8, 16]$.
(Refer to the graph in the answer part above.)
