In order to examine other sine graphs, I will vary a, b and c individually and discuss the results. I will begin with a, by substituting 2, 3, -1, -2, and -1/2.
The graphs indicate that the numeric value of a will affect the
height of the sine wave. For a=2, the sine wave increases to 2 and
decreases to -2. For a=3, it increases to 3 and decreases to -3,
and so on. If a is negative, this creates a reflection about the
x-axis. By definition, a is the amplitude of y = a sin (bx + c).
I will do the same to b, by substituting 2, 3, 1/2, -1 and -2.
Changing b increases the intensity of the sine wave. By restricting the domain of our graph to [-pi,pi], we notice that as b increases, we increase the number of complete waves within this region. As b decreases (as in y = sin(1/2x+1)), we decrease the number of complete waves within this region. This happens because b directly affects the period of the sine wave. The period of a trigonometric function is the distance of one complete wave. The sign of b shifts the graph horizontally. This brings us to c . I will substitute 2, 3, -1, and -1/2 for c.
The value of c yields a horizontal phase shift of the graph.
If c is positive, the graph shifts c units to the left. If
c is negative, the graph shifts c units to the right. Recall,
y = sin(-x+1) gave us a horizontal shift (by substituting b= -1)
to the right. This happened since y = sin (-x+1) is equivalent to y = sin
- (x - 1), written in standard form. This allows us to see that c
= -1. This is why we have a shift to the right 1 unit.