What is the Phase Shift of a Sine Function?

What is the pahse shift of a sine function? An interactive tutorial using an html 5 applet to investigate the phase shift of a sine functionof the form \[ f(x) = a \sin(bx + c) \] where \( a \), \( b \) and \( c \) are real numbers and \(a \) not equal to zero, 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 \( h(x) = a \sin(bx) \) and a blue one corresponding to the function \( f(x) = a \sin(bx + c) \). The function \( h(x) = a \sin(bx) \) has no shift and is the graph of reference.
2. Use the slider to change the value of \( c \) by small steps of 0.1 (you may also enter values in the box for \( c \)), starting from \( c = 0 \) value of \( c \) for which the two graphs are the same. Only the blue graph is shifted to the left or to the right depending whether \( c \) is positive or negative.
3. Keep \( a = 2 \) and \( b = 1 \) (for easy calculations) and change \( c \) using the slider or entering values in the box for parameter \( c \) then press the button 'draw'. Verify that the measure of the shift is equal to \[ -\frac{c}{b} \] and that shift is to the left for \( -\frac{c}{b} < 0 \) and that shift is to the right for \( -\frac{c}{b} > 0 \).
4. Keep \( a = 2 \) and \( b = 2 \) (for easy calculations again) and change \( c \) then press the button 'draw'. Verify that the measure of the shift is equal to \[ -\frac{c}{b} \] and that shift is to the left for \( -\frac{c}{b} < 0 \) and that shift is to the right for \( -\frac{c}{b} > 0 \).