Trigonometry / Unit 4: Graphs of Trigonometric Functions
4.3 Phase Shift and Vertical Shift
Notes
The general form is $y = A\sin(Bx - C) + D$ (or with cosine).
- Phase shift $\displaystyle = \frac{C}{B}$: positive means right, negative means left.
- Vertical shift $D$ moves the midline to $y = D$. The range becomes $[D - |A|, D + |A|]$.
- Factoring helps: $\displaystyle \sin\left(2x - \frac{\pi}{2}\right) = \sin\left(2\left(x - \frac{\pi}{4}\right)\right)$ is shifted right $\displaystyle \frac{\pi}{4}$.
- Example: $y = 3\sin x + 2$ has midline $y = 2$ and range $[-1, 5]$.
Practice
The graph of $\displaystyle y = \sin\left(x - \frac{\pi}{3}\right)$ is the graph of $y = \sin x$ shifted:
Subtracting inside the parentheses shifts the graph right.
What is the phase shift of $y = 2\cos(2x + \pi) - 1$?
Write $\displaystyle 2x + \pi = 2\left(x + \frac{\pi}{2}\right)$. The shift is $\displaystyle -\frac{\pi}{2}$, which is left.
What is the range of $y = 3\sin x + 2$?
Midline 2, amplitude 3, so the range is $[2 - 3, 2 + 3]$.
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