Vertical Shift of a Sine Function

What is the vertical shift of a sine function? An interactive tutorial using an html 5 applet to investigate the vertical shift of a sine function of the form \[ f(x) = a \sin(bx + c) + d \] 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 \[ h(x) = a \sin(bx + c) \] and a blue one corresponding to the function \[ f(x) = a \sin(bx + c) + d \].
2. Use the slider to change the value of \( d \) (you may also enter values in the box for \( d \)). Only the blue graph is shifted up or down depending on whether \( d \) is positive or negative.
3. Keep \( a \), \( b \), and \( c \) constant and change \( d \) using the slider or entering values in the box for parameter \( d \), then press the button 'draw'. For each value of \( d \) selected, verify that the maximum value of the function \[ f(x) = a \sin(bx + c) + d \] (blue graph) is equal to \( d + |a| \) and its minimum value is equal to \( d - |a| \).