Question

a passenger is riding a ferris wheel. the graph shows the height, in feet, of the passenger as a function of time.
height above the ground
(feet)
time (minutes)
move values to the boxes to create a function, h(t), that models the height, in feet, of the passenger after t minutes on the ferris wheel
$h(t)=\square\cos(\square\pi t)+\square$
$\frac{1}{3}$
3
5
20
25
40

Explanation:

Step1: Determine the amplitude

The amplitude \( A \) of a cosine function \( y = A\cos(Bx)+C \) is half the distance between the maximum and minimum values. From the graph, the maximum value is \( 3 \) and the minimum value is \( 1 \). So, \( A=\frac{3 - 1}{2}=1 \). But since the function is \( h(t)=-A\cos(Bt)+C \), \( A = 1 \).

Step2: Determine the period

The general form of the period of \( y=\cos(Bx) \) is \( T=\frac{2\pi}{B} \). From the graph, the period \( T = 2 \) minutes. Using \( T=\frac{2\pi}{B} \), we substitute \( T = 2 \): \( 2=\frac{2\pi}{B}\), so \( B=\pi \).

Step3: Determine the vertical shift

The vertical shift \( C \) is the mid - line of the function. The mid - line \( C=\frac{3 + 1}{2}=2 \).

Answer:

\( h(t)=-1\cos(\pi t)+2 \)