Question

17. Identify the amplitude, period, and vertical shift for the

Answer 1

Answer:

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

Step-by-step explanation:

Because the function is symmetric about the y-axis, using the cosine function is most appropriate.

Refer to the equation for a cosine function:

$f(x)=acos(bx+c)+d$

Amplitude: $|a|$

Period: $\frac{2\pi}{|b|}$

Phase shift: $-\frac{c}{b}$

Midline: $y=d$

The amplitude would be the average of the maximum and minimum y-values of the function, which would be $|a|=|\frac{5-1}{2}| =|\frac{4}{2}| =|2|=2$.

The value of $b$ in $\frac{2\pi}{|b|}$ represents the length of the period, so since the length of the period is $2\pi$, this means that $b=\frac{2\pi}{|2\pi|} =1$.

The phase shift, $-\frac{c}{b}$, describes the horizontal shift of a function. Because the phase shift is $-\frac{\pi}{2}$, then we can set up the equation $-\frac{\pi}{2}=-\frac{c}{1}$ where we determine $c=\frac{\pi}{2}$.

The midline (or vertical shift), $d$, is the horizontal line that passes through between the maximum and minimum points, which the function oscillates. In this case, the midline would be located at the line $y=3$, therefore, $d=3$.

Putting all our information together, your final equation is:

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