2. consider the functions f(x)=cos x and g(x)=cos 4x. 2b complete the table of values for g(x). 2c state the…

Explanation:

Step1: Recall cosine - function properties

The general form of a cosine function is $y = A\cos(Bx - C)+D$. For $g(x)=\cos(4x)$, $A = 1$, $B = 4$, $C = 0$, $D = 0$.

Step2: Find the period formula

The period of a cosine function $y=\cos(Bx)$ is given by $T=\frac{360^{\circ}}{|B|}$.

Step3: Calculate the period of $g(x)$

Since $B = 4$, then $T=\frac{360^{\circ}}{4}=90^{\circ}$.

Step4: Analyze graph - transformation

For the functions $f(x)=\cos(x)$ and $g(x)=\cos(4x)$, when we go from $y = \cos(x)$ to $y=\cos(4x)$, we replace $x$ with $4x$. This is a horizontal dilation. If we have $y = f(bx)$ where $b>1$, the graph of $y = f(x)$ is horizontally dilated by a factor of $\frac{1}{b}$. Here $b = 4$, so the graph of $f(x)$ is horizontally dilated by a factor of $\frac{1}{4}$ to get the graph of $g(x)$.

Answer:

The period of $g(x)$ is $90$ degrees. The transformation from the graph of $f(x)$ to the graph of $g(x)$ is a horizontal dilation by a factor of $\frac{1}{4}$.