Q:

Determine if the amplitude of the sinusoidal function

I know the answer is -10 because at t=0, it is 10 and the midline is 20, and therefore the amplitude is 10, but I couldnt understand the explanation or rationale behind the graph.

Can anyone elaborate the sentence boxed up in red? 

A:

To simplify the explanation, let's consider the function $f(t)=a\cos(bt)$.

Case 1: $a>0$, $b\in \mathbb{R}$.
Let's consider the specific case where $a=1$ and $b=1$. Here, I've plotted the graph for you:

Here, note where the graph starts where $t=0$: $\cos{t}=1$. At $t=0$, that is the maximum value of $\cos{t}$. This also applies for the general case, where $a>0$ and $b\in \mathbb{R}$ for $f(t)=a\cos(bt)$: there is always a maximum at $t=0$. Hence, your ferris wheel would have to initially be at its highest point for Case 1 to be true.

Case 2: $a<0$, $b \in \mathbb{R}$.
Now, let's let $a$ be negative and $b$ be a real number. Here, I choose $a=-1$ and $b=1$ for simplicity:

When $t=0$, $\cos{t}=-1$ in this case. It is the minimum value of $-\cos{t}$ at that time. Similarly, this also applies generally for $f(t)$ where $a<0$ and $b\in \mathbb{R}$. Hence, your ferris wheel would have to initially be at its lowest point for Case 2 to be true.

Now, notice for your example notice that:

At $t=0$, when she starts moving, she is at a height $10 \text{ m}$ off the ground, which is as low as she goes. After $20\pi$ seconds, she reaches her maximum height of $30 \text{ m}$.

The function $H(t)$ here starts from a minimum at $t=0$ and is followed by a maximum point. Therefore, $a<0$.

