AP Precalculus: Sine & Cosine Function Formulas & Graphing
1. General Form (Sine/Cosine)
- \( y = a\sin(bx+c)+d \)
- \( y = a\cos(bx+c)+d \)
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Where:
- Amplitude: \( |a| \)
- Period: \( \frac{2\pi}{|b|} \)
- Phase shift: \( -\frac{c}{b} \)
- Vertical shift: \( d \)
2. Properties
- Amplitude: \( |a| \)
- Period: \( \frac{2\pi}{|b|} \)
- Phase shift: \( -\frac{c}{b} \)
- Vertical shift (Midline): \( d \)
- Frequency: \( \frac{|b|}{2\pi} \)
- Max value: \( d+|a| \)
Min value: \( d-|a| \)
3. Write Equations from Graph/Properties
- Amplitude = \( \frac{\text{max}-\text{min}}{2} \)
- Midline = \( \frac{\text{max}+\text{min}}{2} \)
- Period = distance for one cycle (peak-to-peak or trough-to-trough): \( \frac{2\pi}{|b|} \)
- Phase shift = where graph starts relative to standard, \( -\frac{c}{b} \)
- Example: Amplitude 2, period \( \pi \), phase shift \( \frac{\pi}{3} \), up 1:
\( y = 2\sin(2(x-\frac{\pi}{3}))+1 \)
4. Graphs and Translations
- One cycle for \( y = \sin x \): starts at 0, up to 1, down to 0, to -1, up to 0 (for \( 0 \le x \le 2\pi \))
- For \( y = a\sin(bx + c) + d \): stretch/compress, shift horizontally/vertically as above
- Sine: starts at midline, cosine: starts at max/min
- Shift right: \( -c > 0 \); left: \( -c < 0 \); up: \( d > 0 \); down: \( d < 0 \)
5. Sine/Cosine Comparison
- \( \sin(x) = \cos(x-\frac{\pi}{2}) \)
- \( \cos(x) = \sin(x+\frac{\pi}{2}) \)
- Both have amplitude \( |a| \), period \( 2\pi/|b| \)
- Graphs are horizontally shifted by \( \frac{\pi}{2} \)