Investigation

## Things you might have noticed

How would we describe it?

Can we describe it using a function we know?

The graph looks like a sine graph, so we could try and describe it using a transformation of the sine curve, $y=R\sin(x+\alpha)$ where $R>0$ and $0<\alpha<\frac{\pi}{2}$.

Why can we choose $0<\alpha<\frac{\pi}{2}$?

Looking at the graph we can see that if this was given by a transformation of a sine graph the closest way to get to this graph is by a translation of a sine graph to the left, so $\alpha$ will be positive. However, since the peak is between the $x=0$ and $x=\frac{\pi}{2}$ lines we can choose $0<\alpha<\frac{\pi}{2}$ as our smallest amount we need to translate the curve.

Once we find this value for $\alpha$, we can create similar graphs using $\alpha+2\pi$ instead of $\alpha$, since sine has a period of $2\pi$.

We can also find the value of $R$. Since the function has a maximum value of $5$ and a minimum of $-5$, we can conclude that $R=5$.

Why can we choose $R>0$?

When trying to find $\alpha$, we cannot obtain an exact value by looking at the graph, but we can see that $\frac{\pi}{4}<\alpha<\frac{3\pi}{8}$.

Is there any way we could find out what $\alpha$ is precisely?

### An alternative to sine

We could have also have described this as a translated cosine graph, writing it in the form $y=R\cos(x-\beta)$ where $R>0$ and $0<\beta<\frac{\pi}{2}$.

Why can we choose $0<\beta<\frac{\pi}{2}$ this time?

Again, $R=5$ by looking at the maximum $y$ coordinate the graph has, but this time we can see that the translation of the cosine curve will mean that $\frac{\pi}{8}<\beta<\frac{\pi}{4}$.

Is there any way we could find out what $\beta$ is precisely?

Is there any connection between the value of $\alpha$ and the value of $\beta$?