Determining Period From A Graph

I'm having trouble understanding what exactly the period is in a graph.

My understanding is that the period is horizontal distance between 2 curves in the graph. Like in the following picture which shows the period of the graph is 10:

Now what I'm confused at is the following graphs:

In the first graph, the $15$ is stated and I added the $30$, $45$ and $60$. I thought the period would be $15$ because the horizontal distance between each curve is $15$. The period though (according to the book) is in fact $60$ because $4 * 15 = 60$?

The next graph is even more foreign to me and the period is supposed to be $2 * 6.28 = 12.56$.

Can somebody please explain how they are getting the period for these graphs? I'm a bit lost.

• In the second graph, the first maximum value to the right of the $y$-axis is at $x = 15$, so the first $x$-intercept to the right of the origin is at $x = 30$. Therefore, the first minimum value to the right of the $y$-axis is at $x = 45$ and the second $x$-intercept to the right of the origin is at $x = 60$. Commented Nov 6, 2014 at 3:15

1 Answer

A period of a function $f$ is $b$ such that for all $x,f(x)=f(x+b)$. Geometrically you can just observe when the function repeats itself.

For the second example you said that the period might be 15, which is not a bad guess, but at the point 0, your function proceeds to increase, and at the point 15, your function takes on a whole different value. Maybe you could guess the period is 30, double 15, since at 0 your function is zero and at 30 your function is zero, but immediately after these points the function does different things so this guess is also wrong.

Graphically, imagine cuttings you graph into segments of size 15, do they all look the same? No. So what about segments of size 30, we get vee's and upside down vee's, again not the same. Try cutting our graph into pieces of size 60, do these pieces all look the same?

The period that your book states is correct in both cases, but can you see this more clearly now?

(Also you have labelled the graph incorrectly, you should have 45, 75, 105 where you have 30 45 and 60!)