Integrated Math 3 / Unit 5: Trigonometric Functions
5.4 Midline and Phase Shift
Notes
The general form is $y = A\sin(B(x - C)) + D$ (or cosine).
- $|A|$ is the amplitude and $\displaystyle \frac{2\pi}{|B|}$ is the period.
- $C$ is the phase shift (horizontal shift right by $C$).
- $D$ is the vertical shift, and the midline is $y = D$.
- Maximum $= D + |A|$ and minimum $= D - |A|$.
- Factor out $B$ to find the shift: $\displaystyle y = \cos(2x - \pi) = \cos\left(2\left(x - \frac{\pi}{2}\right)\right)$ is shifted right $\displaystyle \frac{\pi}{2}$.
Practice
What are the midline and range of $y = -3\sin(2x) - 4$?
$D = -4$ and the amplitude is 3, so the values go from $-7$ to $-1$.
What is the phase shift of $y = \cos(2x - \pi)$?
Factor out 2: $\displaystyle \cos(2(x - \frac{\pi}{2}))$.
The graph of $\displaystyle y = \sin\left(x - \frac{\pi}{4}\right)$ is the graph of $y = \sin x$:
Subtracting inside the function shifts the graph right.
Ready for more? Take the Unit 5 test.