Function Transformations
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Function Transformation Rules
Basic Transformation Formula:
\[ g(x) = a \cdot f(b(x - h)) + k \]
where:
• \( h \) = horizontal shift (translation left/right)
• \( k \) = vertical shift (translation up/down)
• \( a \) = vertical stretch/compression (dilation)
• \( b \) = horizontal stretch/compression (dilation)
• Negative \( a \) = reflection over x-axis
• Negative \( b \) = reflection over y-axis
Quick Reference Table:
| Transformation | Formula | Effect |
|---|---|---|
| Vertical Shift Up | \( f(x) + k \) | Moves graph k units up |
| Vertical Shift Down | \( f(x) - k \) | Moves graph k units down |
| Horizontal Shift Left | \( f(x + h) \) | Moves graph h units LEFT |
| Horizontal Shift Right | \( f(x - h) \) | Moves graph h units RIGHT |
| Reflection over x-axis | \( -f(x) \) | Flips graph upside down |
| Reflection over y-axis | \( f(-x) \) | Flips graph left to right |
| Vertical Stretch | \( a \cdot f(x) \), \( a > 1 \) | Stretches graph vertically |
| Vertical Compression | \( a \cdot f(x) \), \( 0 < a < 1 \) | Compresses graph vertically |
| Horizontal Compression | \( f(b \cdot x) \), \( b > 1 \) | Compresses graph horizontally |
| Horizontal Stretch | \( f(b \cdot x) \), \( 0 < b < 1 \) | Stretches graph horizontally |
2. Translations of Functions (Shifts)
Vertical Translations:
Formula:
\[ g(x) = f(x) + k \]
Rule:
• If \( k > 0 \): Shift UP k units
• If \( k < 0 \): Shift DOWN |k| units
• Every point \( (x, y) \) moves to \( (x, y + k) \)
• The shape of the graph doesn't change
Horizontal Translations:
Formula:
\[ g(x) = f(x - h) \]
Rule (OPPOSITE of what you expect!):
• If \( h > 0 \) in \( f(x - h) \): Shift RIGHT h units
• If \( h < 0 \) in \( f(x - h) \): Shift LEFT |h| units
• Or think: \( f(x + h) \) shifts LEFT h units
• Every point \( (x, y) \) moves to \( (x + h, y) \)
Examples:
Example 1: \( g(x) = x^2 + 3 \)
This is \( f(x) = x^2 \) shifted UP 3 units
Example 2: \( g(x) = (x - 4)^2 \)
This is \( f(x) = x^2 \) shifted RIGHT 4 units
Example 3: \( g(x) = (x + 2)^2 - 5 \)
This is \( f(x) = x^2 \) shifted LEFT 2 units and DOWN 5 units
3. Reflections of Functions
Reflection over x-axis:
\[ g(x) = -f(x) \]
Effect:
• Flips the graph upside down
• Every point \( (x, y) \) becomes \( (x, -y) \)
• Multiplies all y-values by -1
• Points above x-axis move below, and vice versa
Reflection over y-axis:
\[ g(x) = f(-x) \]
Effect:
• Flips the graph left to right
• Every point \( (x, y) \) becomes \( (-x, y) \)
• Replaces x with -x in the function
• Points on right move to left, and vice versa
Examples:
Example 1: \( g(x) = -x^2 \)
This is \( f(x) = x^2 \) reflected over x-axis (opens downward instead of upward)
Example 2: \( g(x) = \sqrt{-x} \)
This is \( f(x) = \sqrt{x} \) reflected over y-axis
Example 3: \( g(x) = -|x| \)
This is \( f(x) = |x| \) reflected over x-axis (V opens downward)
4. Dilations of Functions (Stretches & Compressions)
Vertical Stretches and Compressions:
\[ g(x) = a \cdot f(x) \]
Rules:
If \( |a| > 1 \): VERTICAL STRETCH
Graph becomes taller/narrower; y-values multiply by a
If \( 0 < |a| < 1 \): VERTICAL COMPRESSION
Graph becomes shorter/wider; y-values multiply by a (fraction)
If \( a < 0 \): ALSO reflects over x-axis
Combines stretch/compression with reflection
Horizontal Stretches and Compressions:
\[ g(x) = f(b \cdot x) \]
Rules (OPPOSITE of vertical!):
If \( |b| > 1 \): HORIZONTAL COMPRESSION
Graph becomes narrower; divide x-values by b
If \( 0 < |b| < 1 \): HORIZONTAL STRETCH
Graph becomes wider; divide x-values by b (fraction)
If \( b < 0 \): ALSO reflects over y-axis
Combines stretch/compression with reflection
Examples:
Example 1: \( g(x) = 3x^2 \)
Vertical stretch by factor of 3 (graph is 3× taller)
Example 2: \( g(x) = \frac{1}{2}x^2 \)
Vertical compression by factor of 1/2 (graph is half as tall)
Example 3: \( g(x) = (2x)^2 \)
Horizontal compression by factor of 2 (graph is half as wide)
Example 4: \( g(x) = (\frac{1}{2}x)^2 \)
Horizontal stretch by factor of 2 (graph is twice as wide)
5. Transformations of Functions (Multiple Transformations)
Order of Operations for Transformations:
General Form:
\[ g(x) = a \cdot f(b(x - h)) + k \]
Apply in this order:
1. Horizontal Translation (shift by h)
Move graph left or right
2. Horizontal Stretch/Compression & Reflection (factor b)
Change width and/or flip horizontally
3. Vertical Stretch/Compression & Reflection (factor a)
Change height and/or flip vertically
4. Vertical Translation (shift by k)
Move graph up or down
Complete Example:
Transform \( f(x) = x^2 \) to \( g(x) = -2(x + 3)^2 + 5 \)
Step-by-step transformations:
1. Start with \( f(x) = x^2 \)
2. Shift LEFT 3 units: \( f(x + 3) = (x + 3)^2 \)
3. Vertical stretch by 2: \( 2(x + 3)^2 \)
4. Reflect over x-axis: \( -2(x + 3)^2 \)
5. Shift UP 5 units: \( -2(x + 3)^2 + 5 \)
Result: Parabola opening downward, vertex at (-3, 5), narrower than original
6. Describe Function Transformations
How to Describe Transformations:
Steps:
1. Identify the parent function
2. Look for changes INSIDE the function (horizontal changes)
3. Look for changes OUTSIDE the function (vertical changes)
4. Describe each transformation in order
5. Be specific about direction and magnitude
Practice Examples:
Example 1: \( g(x) = 3|x - 2| + 1 \)
Description:
• Parent function: \( f(x) = |x| \)
• Shift RIGHT 2 units (x - 2)
• Vertical stretch by factor of 3
• Shift UP 1 unit
Example 2: \( g(x) = -\frac{1}{2}(x + 4)^2 - 3 \)
Description:
• Parent function: \( f(x) = x^2 \)
• Shift LEFT 4 units (x + 4)
• Vertical compression by factor of 1/2
• Reflection over x-axis (negative sign)
• Shift DOWN 3 units
Example 3: \( g(x) = \sqrt{-2x} + 4 \)
Description:
• Parent function: \( f(x) = \sqrt{x} \)
• Horizontal compression by factor of 2
• Reflection over y-axis (negative inside)
• Shift UP 4 units
7. Quick Reference Summary
Master Formula:
\[ g(x) = a \cdot f(b(x - h)) + k \]
| Parameter | Effect |
|---|---|
| h (inside) | Horizontal shift: RIGHT if +, LEFT if − |
| k (outside) | Vertical shift: UP if +, DOWN if − |
| a (outside) | Vertical stretch/compression; negative = reflect x-axis |
| b (inside) | Horizontal compression/stretch; negative = reflect y-axis |
⚠️ Key Reminders:
• INSIDE changes (with x) work OPPOSITE to what you expect
• OUTSIDE changes (separate from x) work as expected
• Vertical stretch/compression: |a| > 1 stretches, 0 < |a| < 1 compresses
• Horizontal stretch/compression: |b| > 1 compresses, 0 < |b| < 1 stretches (opposite!)
📚 Study Tips
✓ Remember: horizontal changes work OPPOSITE to intuition (inside paradox)
✓ Apply transformations in order: horizontal shift → h/v stretch → vertical shift
✓ Negative coefficients mean reflections PLUS stretch/compression
✓ Practice identifying parent functions first, then describe transformations
✓ Graph transformations step-by-step to visualize each change