Period of a Sine Function

How does the period of a sine function of the form \[f(x) = a \sin(b x) \] change in terms of \( b \) ? An interactive tutorial using an html 5 applet to investigate the priod of a sine function is presented.

Interactive Tutorial

1. Press the button 'draw' on the left panel below. Two graphs are shown: a red one corresponding to the function \( h(x) = a \sin(x) \) and a blue one corresponding to the function \(f(x) = a \sin(b x) \).
2. Use the slider to change the value of \( b \) (you may also enter values in the box for b). The period of the blue graph with equation \( y = a \sin(b x) \) changes, that of the red is constant and equal to \( 2\pi \).
3. Keep \( a = 2 \) (it has no effects on the period) and change \( b \) using the slider or entering values in the box for parameter \( b \) then press the button 'draw'. Use integer numbers: \( \pm 2, \pm 3, ... for \( b \) so that it is easier to check the formula for the period : \( \dfrac{2\pi}{| b |} \). (In fact for \( | b | > 1 \) and \( b \) integer, \( b | \) is given by the number of cycles of the blue graph within one cycle of the red graph).