which graph represents $f(x)=6cos(4pi x)$?

Explanation:

Step1: Recall the general form of cosine - function

The general form of a cosine function is $y = A\cos(Bx - C)+D$. For the function $f(x)=6\cos(4\pi x)$, we have $A = 6$, $B = 4\pi$, $C = 0$, and $D = 0$.

Step2: Determine the amplitude

The amplitude of the cosine - function $y = A\cos(Bx - C)+D$ is given by $|A|$. Here, $|A|=|6| = 6$, so the maximum value of the function is $y = 6$ and the minimum value is $y=-6$.

Step3: Determine the period

The period of the cosine - function $y = A\cos(Bx - C)+D$ is given by $T=\frac{2\pi}{|B|}$. Here, $B = 4\pi$, so $T=\frac{2\pi}{4\pi}=\frac{1}{2}$.

The cosine function $y = \cos(x)$ has a maximum value at $x = 0$. For $y = 6\cos(4\pi x)$, when $x = 0$, $y=6\cos(0)=6$.

The graph that has an amplitude of 6 (maximum value of 6 and minimum value of - 6) and a period of $\frac{1}{2}$ and starts at the maximum value ($y = 6$ when $x = 0$) is the correct one.

Answer:

The graph that has a maximum value of 6, a minimum value of - 6, a period of $\frac{1}{2}$, and starts at the point $(0,6)$. Without specific labels for the graphs, we can't point out a specific letter - named graph, but based on the above characteristics, you can identify the correct one among the given options.