AP Precalculus: Function Transformations Formulas

General Transformation Formula

\( g(x) = a\cdot f(b(x - h)) + k \)

• \(h\): horizontal translation (right if \(h>0\), left if \(h<0\))
• \(k\): vertical translation (up if \(k>0\), down if \(k<0\))
• \(a\): vertical stretch/compression (\(|a|>1\) stretches, \(0<|a|<1\) compresses); \(a<0\) also reflects over x-axis
• \(b\): horizontal stretch/compression (\(|b|>1\) compresses, \(0<|b|<1\) stretches); \(b<0\) also reflects over y-axis

1. Translations (Shifts)

• Horizontal Shift: \( f(x-h) \): right by \(h\) units; \( f(x+h) \): left by \(h\) units
• Vertical Shift: \( f(x) + k \): up by \(k\) units;
  \( f(x) - k \): down by \(k\) units

2. Reflections

• Over x-axis: \( -f(x) \) (flips output up/down)
• Over y-axis: \( f(-x) \) (flips input left/right)

3. Dilations (Stretches/Compressions)

• Vertical Dilation: \( a \cdot f(x) \)
    \(|a|>1\): vertical stretch (taller)
    \(0<|a|<1\): vertical compression (shorter)
    \(a<0\): includes reflection over x-axis
• Horizontal Dilation: \( f(bx) \)
    \(|b|>1\): horizontal compression (narrower)
    \(0<|b|<1\): horizontal stretch (wider)
    \(b<0\): includes reflection over y-axis

4. Describe/Combine Transformations

To break down \( g(x) = -3f(2(x+4)) - 5 \):

1. Start with parent function \(f(x)\)
2. Shift left by 4 (replace x with x+4)
3. Compress horizontally by factor 1/2 (\(f(2x)\))
4. Stretch vertically by 3, reflect over x-axis (\(-3f(...)\))
5. Shift down by 5

Mapping Points Rule

If point \((x_0, y_0)\) is on \( f(x) \), then on \( g(x) = a f(b(x-h)) + k \), its image is:
\[ \left( x = \frac{x_0}{b} + h,\quad y = a y_0 + k \right) \]
Use this to quickly find and predict new key points!