📖 Concept: Trig Transformations
General Form:
\( y = A \cdot \sin(Bx - C) + D \) or \( y = A \cdot \cos(Bx - C) + D \)
A = Amplitude
B → Period
C → Phase Shift
D = Vertical Shift
Key Formulas:
🔹 Amplitude = \( |A| \)
🔹 Period = \( \dfrac{2\pi}{|B|} \)
🔹 Phase Shift = \( \dfrac{C}{B} \) (right if positive, left if negative)
🔹 Vertical Shift = \( D \) (up if positive, down if negative)
🔹 Domain: all real numbers | Range: \( [D-|A|,\; D+|A|] \)
🌟 Example: Analyze \( g(x) = 4\cos\!\left(\tfrac{1}{2}x - \tfrac{\pi}{2}\right) - 3 \)
1
Identify A, B, C, D: \(A=4,\; B=\tfrac{1}{2},\; C=\tfrac{\pi}{2},\; D=-3\)
2
Amplitude = \( |4| = 4 \) → graph stretches 4 units up/down from center
3
Period = \( \dfrac{2\pi}{1/2} = 4\pi \) → one full cycle = \(4\pi\)
4
Phase Shift = \( \dfrac{\pi/2}{1/2} = \pi \) → shifted right by \(\pi\)
5
Vertical Shift = \(-3\) → center line at \(y = -3\)
6
Range = \([-3-4,\; -3+4] = [-7,\; 1]\)
🌟 Example: Moving a coordinate point
Point \(\left(\tfrac{\pi}{6}, \tfrac{1}{2}\right)\) is on \(f(\theta) = \sin(\theta)\).
Find where it moves on \(g(\theta) = \sin\!\left(\theta - \tfrac{\pi}{3}\right) + 7\)
1
Phase shift: add \(\tfrac{\pi}{3}\) to x-coordinate → \( x_{new} = \tfrac{\pi}{6}+\tfrac{\pi}{3} = \tfrac{\pi}{2} \)
2
Vertical shift: add \(7\) to y-coordinate → \( y_{new} = \tfrac{1}{2}+7 = \tfrac{15}{2} \)
3
Answer: \(\left(\tfrac{\pi}{2},\; \tfrac{15}{2}\right)\) ✅