Question

graph the following function. show at least two cycles. use the graph to determine the domain and range of the function.
y = cos(8x)
use the graphing tool to graph the equation. type pi to insert π as needed.
click to enlarge graph

Explanation:

Step1: Recall the general form of cosine function

The general form of a cosine function is $y = A\cos(Bx - C)+D$. For $y=\cos(8x)$, $A = 1$, $B = 8$, $C=0$, $D = 0$. The period of the cosine - function $y = A\cos(Bx - C)+D$ is given by $T=\frac{2\pi}{|B|}$.

Step2: Calculate the period

Substitute $B = 8$ into the period formula $T=\frac{2\pi}{|B|}$. We get $T=\frac{2\pi}{8}=\frac{\pi}{4}$.

Step3: Find key - points for one cycle

For the cosine function $y=\cos(u)$, when $u = 0$, $y = 1$; when $u=\frac{\pi}{2}$, $y = 0$; when $u=\pi$, $y=-1$; when $u=\frac{3\pi}{2}$, $y = 0$; when $u = 2\pi$, $y = 1$. For $y=\cos(8x)$, when $8x = 0$, $x = 0$ and $y = 1$; when $8x=\frac{\pi}{2}$, $x=\frac{\pi}{16}$ and $y = 0$; when $8x=\pi$, $x=\frac{\pi}{8}$ and $y=-1$; when $8x=\frac{3\pi}{2}$, $x=\frac{3\pi}{16}$ and $y = 0$; when $8x = 2\pi$, $x=\frac{\pi}{4}$ and $y = 1$.

Step4: Graph two cycles

To graph two cycles, we start from $x = 0$ and go to $x=\frac{\pi}{2}$ (since the period is $\frac{\pi}{4}$ and two - cycles have a length of $2\times\frac{\pi}{4}=\frac{\pi}{2}$). Plot the key - points and draw a smooth cosine - curve.

Step5: Determine the domain

The domain of the cosine function $y=\cos(8x)$ is all real numbers. In interval notation, the domain is $(-\infty,\infty)$.

Step6: Determine the range

Since $A = 1$ and $D = 0$ for $y=\cos(8x)$, the maximum value of $y$ is $A + D=1$ and the minimum value of $y$ is $-A + D=-1$. So the range is $[-1,1]$.

Answer:

Domain: $(-\infty,\infty)$; Range: $[-1,1]$