sketch a graph of the function f(x)= - 4 cos(1/4 x)\nclear all draw:\nquestion help: video\nsubmit question

Explanation:

Step1: Identify the amplitude

The general form of a cosine - function is $y = A\cos(Bx - C)+D$. For the function $f(x)=-4\cos(\frac{1}{4}x)$, the amplitude $|A|$. Here, $A=-4$, so the amplitude is $| - 4| = 4$. This means the graph oscillates between $y = 4$ and $y=-4$.

Step2: Calculate the period

The period of a cosine function $y = A\cos(Bx - C)+D$ is given by $T=\frac{2\pi}{|B|}$. For the function $f(x)=-4\cos(\frac{1}{4}x)$, $B = \frac{1}{4}$. Then $T=\frac{2\pi}{\left|\frac{1}{4}\right|}=8\pi$.

Step3: Find key - points

• When $x = 0$, $f(0)=-4\cos(0)=-4$.
• When $x = 2\pi$, $f(2\pi)=-4\cos(\frac{\pi}{2}) = 0$.
• When $x = 4\pi$, $f(4\pi)=-4\cos(\pi)=4$.
• When $x = 6\pi$, $f(6\pi)=-4\cos(\frac{3\pi}{2}) = 0$.
• When $x = 8\pi$, $f(8\pi)=-4\cos(2\pi)=-4$.

Plot these key - points and draw a smooth cosine - like curve with an amplitude of 4 and a period of $8\pi$ on the given grid. The negative sign in front of the 4 reflects the graph of $y = 4\cos(\frac{1}{4}x)$ about the $x$ - axis.

Answer:

Sketch a cosine - like curve with amplitude 4, period $8\pi$, passing through the points $(0, - 4),(2\pi,0),(4\pi,4),(6\pi,0),(8\pi,-4)$ and reflected about the $x$ - axis.