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 \(\beta\)?