5.5 Four Shapes and the Period

Here are the four basic shapes of a sinusoidal curve: positive and negative sine; positive and negative cosine.

The period of the curve is the distance along the horizontal axis between one point on the curve and the same point on the next wave – eg, from crest to crest or from trough to trough.

The period informs us by how much the curve has been stretched horizontally.

Consider the value b=1. The period of the curve y=\sin(bx^{\circ}) is 360^{\circ}.

Use the applet below to see the effect of the parameter b when y=\sin(bx^{\circ})

In general, once the period has been identified the value b can be calculated as follows:

    \[b=\frac{360}{\text{period}}\]

alternatively,

    \[\text{period}=\frac{360}{b}\]

Identify the function in the following 12 graphs: