Question

graph at least one full cycle of the of the following trig function label the amplitude midline minimum and the intervalsf(x)=1/3cps 2(x+pi)

Answer 1

Given the function

$f(x)=\frac{1}{3}\cos (2(x+\pi))$

Which is equivalent to

$\Leftrightarrow f(x)=\frac{1}{3}\cos (2x+2\pi))$

In general, a trigonometric equation has the following structure

$g(x)=A\cos (B(x-C))+D$

Where A is the amplitude.

Therefore, the amplitude of our function is 1/3.

As for the minimum and maximum of the function, remember that the range of the cosine function is [-1,1]; therefore,

$\begin{gathered} \text{minimum(f(x))}=\frac{1}{3}(-1)=-\frac{1}{3} \\ \text{maximum(f(x))}=\frac{1}{3}(1)=\frac{1}{3} \end{gathered}$

Furthermore, the midline of the graph is a parallel line to the x-axis that crosses the midpoint between the maximum and the minimum; in our case,

$\begin{gathered} \frac{\frac{1}{3}+(-\frac{1}{3})}{2}=0 \\ \Rightarrow\text{midline is y=0} \end{gathered}$

Finally, the graph of the function in the [0,2pi] interval is