Can I get some help please? Thanks

Sep 22, 2017

Part a.

Please observe that the cosine function starts at the value of 4 at $x = 0$:

$4 = A \cos \left(B \left(0\right)\right)$

and then repeats the value of 4:

$4 = A \cos \left(B \left(\pi\right)\right)$

Therefore, the period must be $\pi$

We can double-check this by observing that it repeats the value of 4, again, $\pi$ units later.

Because the value of the function varies between 4 and -4, the amplitude must be 4.

End part a.

Part b.

By observation:

The maximum values occur at $0 , \pi , \mathmr{and} 2 \pi$

The minimum values occur at $\frac{\pi}{2} \mathmr{and} \frac{3 \pi}{2}$

The zeros occur at $\frac{\pi}{4} , \frac{3 \pi}{4} , \frac{5 \pi}{4} , \mathmr{and} \frac{7 \pi}{4}$

Sep 22, 2017

$\text{see explanation}$

Explanation:

• " Amplitude is the distance from the "theta" axis to the"
$\text{peak of the wave}$

$\text{from the origin } O \to 4$

$\Rightarrow \text{amplitude } = 4$

• " period is the distance covered by 1 complete wave"

$\text{from 1 point on the wave to the same point along the wave}$

$\text{consider the peak of 4 at the origin, then the next peak}$
$\text{of 4 occurs at "pi" on the "theta" axis}$

$\Rightarrow \text{period } = \pi$

$\text{maximum values of + 4 at } \theta = 0 , \theta = \pi , \theta = 2 \pi$

$\text{minimum values of - 4 at } \theta = \frac{\pi}{2} , \theta = \frac{3 \pi}{2}$

$\textcolor{b l u e}{\text{zeros"" are where the wave intersects the "theta" axis}}$

$\text{zeros occur at } \theta = \frac{\pi}{4} , \theta = \frac{3 \pi}{4} , \theta = \frac{5 \pi}{4} , \theta = \frac{7 \pi}{4}$

Sep 22, 2017

See below.

Explanation:

a.) The period is the distance from one peak to the next peak, or from one trough to the next trough. From the graph this can be seen as being from $\theta = 0$ to $\theta = \pi$;

So the period is $\pi$

The amplitude is the distance from the horizontal axis to the top of a peak. From the graph this can be seen as 4:

So amplitude is 4:

b.)
Maximum value of 4 between $0 \le \theta \le 2 \pi$ are the peaks, this can be seen where:

$\theta = 0$
$\theta = \pi$
$\theta = 2 \pi$

Minimum value -4 between $0 \le \theta \le 2 \pi$ are the lowest part of the troughs. From graph these can be seen to occur at:

$\theta = \frac{\pi}{2}$

$\theta = \frac{3 \pi}{2}$

The zeros occur where the curve cuts the horizontal axis. From graph these can be seen to occur at:

$\theta = \frac{\pi}{4}$

$\theta = \frac{3 \pi}{4}$

$\theta = \frac{5 \pi}{4}$

$\theta = \frac{7 \pi}{4}$