11 Graph two full periods of the function $f(x)=\cos (6 x)$ and state the amplitude and period. Enter the exact answers. For the number $\pi$, either choose $\pi$ from the drop-down menu or type in Pi (with a capital P). Amplitude: $A=$ Number $a^{b}$ $\frac{a}{b}$ $\sqrt{a}$ $|a|$ $\sin (a)$ Period: $P=$
Solution
Step 1 ：The amplitude of a function \(f(x) = A\cos(Bx)\) is given by the absolute value of \(A\). In this case, \(A = 1\), so the amplitude of the function is \(\boxed{1}\).
Step 2 ：The period of a function \(f(x) = A\cos(Bx)\) is given by \(\frac{2\pi}{|B|}\). In this case, \(B = 6\), so the period of the function is \(\frac{2\pi}{6} = \boxed{\frac{\pi}{3}}\).
Step 3 ：To graph two full periods of the function, we start at \(x = 0\) and end at \(x = 2P = 2\cdot\frac{\pi}{3} = \frac{2\pi}{3}\) for the first period, and then continue to \(x = 4P = 4\cdot\frac{\pi}{3} = \frac{4\pi}{3}\) for the second period.
Step 4 ：The graph starts at the maximum value (the amplitude), decreases to the minimum value (the negative of the amplitude), and then increases back to the maximum value. This pattern repeats for the second period.
