9. chloe takes a trip to an amusement park and rides a ferris wheel. the graph below shows the height, in…

Explanation:

Step1: Determine the amplitude

The amplitude (A) is half the distance between the maximum and minimum values. The maximum height (y_{max}=410) and the minimum height (y_{min} = 20). [A=\frac{y_{max}-y_{min}}{2}=\frac{410 - 20}{2}=\frac{390}{2}=195]

Step2: Determine the vertical shift

The vertical shift (D) is the mid - line of the function. (D=\frac{y_{max}+y_{min}}{2}=\frac{410+20}{2}=\frac{430}{2} = 215)

Step3: Determine the period

The period (P) is the horizontal length of one full cycle. From ((2.75,20)) to ((9.75,20)), (P=9.75 - 2.75=7). Then the frequency (B=\frac{2\pi}{P}=\frac{2\pi}{7})

Step4: Determine the phase shift

We can use the cosine function (y = A\cos(B(t - C))+D). Let's assume (C = 0) (we could also use the sine function, but for simplicity, we'll start with the cosine form). Substituting (A = 195), (B=\frac{2\pi}{7}), and (D = 215) into the formula (y=A\cos(Bt)+D)

Answer:

(y = 195\cos\left(\frac{2\pi}{7}t\right)+215)