review chapter 8. learning target #1: model situations with sine or cosine functions. 1. a frog is taking a…

Explanation:

Step1: Find the amplitude $a$

The total vertical distance from the lowest - point to the highest - point is $3 + 7=10$ feet. The amplitude $a$ is half of this distance, so $a=\frac{10}{2}=5$.

Step2: Find the vertical shift $k$

The mid - line of the motion is the average of the highest and lowest points. The mid - line is $\frac{- 3+7}{2}=2$ feet above the water surface, so $k = 2$.

Step3: Find the period $T$

It takes 12 seconds to go from the lowest point to the highest point. The period $T$ (time for a full cycle) is $2\times12 = 24$ seconds. Since the formula for the period of $y=a\cos(b(x - h))+k$ is $T=\frac{2\pi}{b}$, we can solve for $b$. Substituting $T = 24$ into $T=\frac{2\pi}{b}$, we get $b=\frac{2\pi}{24}=\frac{\pi}{12}$.

Step4: Find the horizontal shift $h$

The frog starts at the lowest point. For the cosine function $y = a\cos(b(x - h))+k$, when $x = 0$, we want the function to be at its minimum. The standard cosine function $y=\cos(x)$ has a minimum at $x=\pi$. So, if we set $b(0 - h)=\pi$ and $b=\frac{\pi}{12}$, then $\frac{\pi}{12}(-h)=\pi$, and $h=- 12$.

Answer:

$y = 5\cos\left(\frac{\pi}{12}(x + 12)\right)+2$