Precalculus / Unit 4: Trigonometric Functions
4.4 Graphs of Sine and Cosine
Notes
For $y = A\sin(B(x - C)) + D$ (and similarly for cosine):
- Amplitude $= |A|$. Period $\displaystyle = \frac{2\pi}{|B|}$. Phase shift $= C$ (right if positive). Midline: $y = D$.
- The range is $[D - |A|, D + |A|]$.
- Example: $\displaystyle y = 2\sin(x - \frac{\pi}{3}) + 1$ has range $[-1, 3]$.
Practice
What are the amplitude and period of $\displaystyle y = -4\sin\frac{x}{3}$?
Amplitude is $|-4| = 4$; period is $\displaystyle \frac{2\pi}{1/3} = 6\pi$.
What is the range of $\displaystyle y = 2\sin\left(x - \frac{\pi}{3}\right) + 1$?
The midline is 1 and the amplitude is 2, so the values run from $-1$ to $3$.
What is the phase shift of $\displaystyle y = \cos\left(2x - \frac{\pi}{2}\right)$?
Factor: $\displaystyle \cos\left(2\left(x - \frac{\pi}{4}\right)\right)$.
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