Algebra and Trigonometry, 5th Edition

6.6 Graphs of Transformed Sine and Cosine Functions

• Graph transformations of $y=\sin x$ and $y=\cos x$ in the form

  $y=A\text{sin}\left(Bx-C\right)+D$

  and

  $y=Acos\left(Bx-C\right)+D$

  and determine the amplitude, the period, and the phase shift.

• Graph sums of functions.

• Graph functions (damped oscillations) found by multiplying trigonometric functions by other functions.

Variations of Basic Graphs

In Section 6.5, we graphed all six trigonometric functions. In this section, we will consider variations of the graphs of the sine and the cosine functions. For example, we will graph equations like the following:

$$\begin{array}{llll}y=5\sin {\displaystyle \frac{1}{2}}x,\hfill & y=\cos (2x-\pi ),\hfill & \text{and}\hfill & y={\displaystyle \frac{1}{2}}\sin x-3.\hfill \end{array}$$

In particular, we are interested in graphs of functions in the form

$$y=A\text{sin}\left(Bx-C\right)+D$$

and

$$y=A\text{cos}\left(Bx-C\right)+D$$

where A, B, C, and ...