Mathfolis

Sinusoidal Function Transformations

Unit 3 · Trigonometric and Polar Functions

What AP Precalc asks here

Each parameter in f(x)=Asin⁡(B(x−h))+kf(x) = A \sin(B(x - h)) + k controls one geometric feature. Doubling AA doubles the amplitude. Doubling BB halves the period. Increasing hh shifts right. Increasing kk shifts up. The College Board tests both forward (predict the graph from parameters) and reverse (recover parameters from a graph description).

Transformation effects

Multiply A by cc
amplitude scales by ∣c∣; negative c reflects vertically\text{amplitude scales by } |c|; \text{ negative } c \text{ reflects vertically}
Multiply B by cc
period divided by ∣c∣\text{period divided by } |c|
Add to h
horizontal shift right by the increase\text{horizontal shift right by the increase}
Add to k
vertical shift up by the increase\text{vertical shift up by the increase}
Caution: If B=2B = 2, the period is π\pi, not 2⋅2π2 \cdot 2\pi. Larger BB means a faster oscillation, not a slower one.
Type 1

Apply a single transformation

Identify which parameter changes and apply its rule.

Example 1
If g(x)=5sin⁡xg(x) = 5 \sin x, what are the maximum and minimum values? (A) Max 1, min −1-1 (B) Max 5, min −5-5 (C) Max 5, min 0 (D) Max 10, min −10-10

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Type 2

Combined transformations

Combine all four parameters; predict the new graph's amplitude, period, and midline.

Example 2
For g(x)=2sin⁡(πx)−1g(x) = 2 \sin(\pi x) - 1, what are the period and midline? (A) Period 2, midline y=−1y = -1 (B) Period 2π2\pi, midline y=−1y = -1 (C) Period π\pi, midline y=1y = 1 (D) Period π/2\pi/2, midline y=−1y = -1

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Type 3

Reverse-engineer the equation from features

Match each feature to its parameter; write the resulting equation.

Example 3
A sinusoid has amplitude 3, period π\pi, midline y=2y = 2, no phase shift. Which equation matches? (A) f(x)=3sin⁡(2x)+2f(x) = 3 \sin(2x) + 2 (B) f(x)=3sin⁡(πx)+2f(x) = 3 \sin(\pi x) + 2 (C) f(x)=2sin⁡(3x)+2f(x) = 2 \sin(3x) + 2 (D) f(x)=3sin⁡x+2f(x) = 3 \sin x + 2

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