AP Precalculus: Function Transformations
Master translations, reflections, stretches, and compressions with the complete transformation formula
📚 Understanding Function Transformations
Function transformations allow you to modify the graph of any parent function by shifting, stretching, compressing, or reflecting it. These concepts are essential for AP Precalculus and provide the foundation for graphing complex functions quickly and accurately.
🎯 The General Transformation Formula
\(h > 0\): shift right
\(h < 0\): shift left
\(k > 0\): shift up
\(k < 0\): shift down
\(|a| > 1\): vertical stretch
\(0 < |a| < 1\): vertical
compression
\(a < 0\): reflect over x-axis
\(|b| > 1\): horizontal compression
\(0 < |b| < 1\): horizontal
stretch
\(b < 0\): reflect over y-axis
1 Translations (Shifts)
A translation moves every point on a graph the same distance in the same direction. The shape of the graph stays exactly the same — it just changes position.
Horizontal Translations
Vertical Translations
Horizontal shifts work opposite to what you might expect! \(f(x - 3)\) shifts the graph right (not left), because you need \(x = 3\) to get the same output as \(f(0)\).
\(g(x) = f(x - 3) + 2\)
Transformation: Shift the graph of \(f(x)\) right 3 units and up 2 units
If \(f(x) = x^2\), then \(g(x) = (x-3)^2 + 2\) — the vertex moves from \((0, 0)\) to \((3, 2)\)
2 Reflections
A reflection creates a mirror image of the graph across an axis. The graph flips over the specified axis while maintaining its shape.
Two Types of Reflections
(negates all y-values)
(negates all x-values)
Negative on the outside (\(-f(x)\)) → reflects over x-axis (affects outputs)
Negative on the inside (\(f(-x)\)) → reflects over y-axis (affects inputs)
Given: \(f(x) = x^2\)
\(-f(x) = -x^2\) → Parabola opens downward instead of upward
\(f(-x) = (-x)^2 = x^2\) → Same graph! (because \(x^2\) is symmetric about y-axis)
Given: \(f(x) = \sqrt{x}\)
\(-f(x) = -\sqrt{x}\) → Reflects below the x-axis
\(f(-x) = \sqrt{-x}\) → Reflects to left side of y-axis (domain: \(x \leq 0\))
3 Dilations (Stretches & Compressions)
A dilation changes the size of the graph by stretching or compressing it either vertically or horizontally. The graph maintains its basic shape but becomes taller/shorter or wider/narrower.
Vertical Dilations: \(a \cdot f(x)\)
- \(|a| > 1\): Vertical stretch — graph becomes taller, pulled away from x-axis
- \(0 < |a| < 1\): Vertical compression — graph becomes shorter, pushed toward x-axis
- \(a < 0\): Also includes reflection over x-axis
Horizontal Dilations: \(f(bx)\)
- \(|b| > 1\): Horizontal compression — graph becomes narrower (factor of \(\frac{1}{b}\))
- \(0 < |b| < 1\): Horizontal stretch — graph becomes wider (factor of \(\frac{1}{b}\))
- \(b < 0\): Also includes reflection over y-axis
\(f(2x)\) makes the graph narrower (compressed by factor of \(\frac{1}{2}\)), not wider. Think of it as the graph reaching its key points twice as fast.
Vertical Dilations:
\(g(x) = 2f(x)\) → Graph is twice as tall (stretched vertically by factor 2)
\(g(x) = \frac{1}{2}f(x)\) → Graph is half as tall (compressed vertically by factor \(\frac{1}{2}\))
Horizontal Dilations:
\(g(x) = f(2x)\) → Graph is half as wide (compressed horizontally by factor \(\frac{1}{2}\))
\(g(x) = f(\frac{1}{2}x)\) → Graph is twice as wide (stretched horizontally by factor 2)
4 Combining Transformations
When multiple transformations are applied, the order matters. Follow the correct sequence to accurately describe and apply transformations.
1️⃣ Horizontal shift (inside parentheses)
2️⃣ Horizontal dilation/reflection (coefficient of \(x\))
3️⃣ Vertical dilation/reflection (coefficient \(a\))
4️⃣ Vertical shift (added constant \(k\))
Given: \(g(x) = -3f(2(x + 4)) - 5\)
Step-by-step breakdown:
- Start with the parent function \(f(x)\)
- Horizontal shift left 4 — replace \(x\) with \((x + 4)\)
- Horizontal compression by \(\frac{1}{2}\) — the factor of 2 inside
- Vertical stretch by 3 and reflect over x-axis — the \(-3\) outside
- Vertical shift down 5 — the \(-5\) at the end
When describing transformations in words, use this format: "[Direction] by [amount] units" for shifts, "[stretched/compressed] vertically/horizontally by a factor of [value]" for dilations, and "reflected over the [x/y]-axis" for reflections.
5 Mapping Points Under Transformations
The mapping rule tells you exactly where each point on the original graph moves after transformation. This is crucial for quickly sketching transformed graphs.
\[\left(\frac{x_0}{b} + h, \; ay_0 + k\right)\]
Understanding the Formula
- New x-coordinate: Divide by \(b\) (undo horizontal dilation), then add \(h\) (shift)
- New y-coordinate: Multiply by \(a\) (apply vertical dilation/reflection), then add \(k\) (shift)
- This works for any point — use key points like intercepts, vertices, and endpoints
Given: \(g(x) = 2f(3(x - 1)) + 4\)
Parameters: \(a = 2\), \(b = 3\), \(h = 1\), \(k = 4\)
If \((6, 3)\) is on \(f(x)\), find the corresponding point on \(g(x)\):
New x: \(\frac{6}{3} + 1 = 2 + 1 = 3\)
New y: \(2(3) + 4 = 6 + 4 = 10\)
Answer: The point \((6, 3)\) maps to \((3, 10)\)
📋 Quick Reference: All Transformations
| Transformation | Notation | Effect on Graph |
|---|---|---|
| Shift Right \(h\) | \(f(x - h)\) | All points move right \(h\) units |
| Shift Left \(h\) | \(f(x + h)\) | All points move left \(h\) units |
| Shift Up \(k\) | \(f(x) + k\) | All points move up \(k\) units |
| Shift Down \(k\) | \(f(x) - k\) | All points move down \(k\) units |
| Reflect over x-axis | \(-f(x)\) | Flip upside down (negate y) |
| Reflect over y-axis | \(f(-x)\) | Flip left-to-right (negate x) |
| Vertical Stretch | \(af(x)\), \(|a| > 1\) | Graph becomes taller |
| Vertical Compression | \(af(x)\), \(0 < |a| < 1\) | Graph becomes shorter |
| Horizontal Compression | \(f(bx)\), \(|b| > 1\) | Graph becomes narrower |
| Horizontal Stretch | \(f(bx)\), \(0 < |b| < 1\) | Graph becomes wider |