Transformations of Trigonometric Functions

Case 2:amath

y=sin b x

If b < 0, for instance when b=-1, we have a reflection across the $y$endamath-axis, since amath (x,y) gets mapped to (-x,y).
Here is the animation demonstrating the transformation of y=sin x to y=sin(-x).
sinx to sin(-x)
Note that one can categorize this transformation in two other ways as well:
  1. Reflection across the $x$endamath-axis.amath Since the sine function is an odd periodic function, sin(-x)=-sin x. Thus, for sin x, the reflection across the $y$endamath-axis amath is the same as the reflection about the $x$endamath-axis.amath.

  2. Horizontal translation to the right by $\pi$. This results from the identity, sin($\pi$-x)=sin x.
To illustrate the reflection across the $y$endamath-axis amath more clearly, please look at the following animation which divides the y=sin(x) graph into two parts:
-$\pi$<= x <= 0 and 0 <= x <= $\pi$
sinx to sin(-x) in two parts

We have seen that b < 0 causes a reflection across the $y$endamath-axis.amath
The absolute value of b changes the period of the sine function. For |b|>1, we have a horizontal shrink, as you can view in the following animation as b oscillates between 1,2, 3 and 4.
For |b|<1, we have a horizontal stretch, as you can see in the following animation where b oscillates between 1,1/2 and 1/4.

The period of y=sin x is 2$\pi$.

y = sin b x has period = (2 pi)/ |b|.

Click here for case endamath 3: amath y = sin (x + c) + d. endamath