U1B 1 12B Notes Dilations — AP Precalculus | The Complete Equation

AP PRECALCULUS — UNIT 1B · POLYNOMIAL & RATIONAL FUNCTIONS

1.12B — Dilations of Functions

Notes — Stretches, Compressions, and Reflections

💡 Learning Objectives

By the end of this lesson you will be able to (AP CED 1.12.A):

Apply vertical and horizontal dilations to the graph of a function.
Interpret the effect of a negative scale factor as a reflection.
Combine dilations with translations to analyze composite transformations.

1. Vertical Dilations

Multiplying the OUTPUT by a constant a stretches or compresses the graph vertically. If |a| > 1, the graph stretches (tall). If 0 < |a| < 1, the graph compresses (short). If a < 0, the graph reflects over the x-axis in addition to any stretch or compression.

💡 Definitions

Vertical Dilation.

The graph of g(x) = a · f(x) is a vertical stretch of f by factor |a| if |a| > 1, or a compression if 0 < |a| < 1.
If a < 0, there is ALSO a reflection across the x-axis.
The point (x, y) on f's graph becomes (x, a·y) on g's graph.

2. Horizontal Dilations

Multiplying the INPUT by a constant b dilates the graph horizontally. The effect is INVERSE to the factor — multiplying the input by 2 COMPRESSES the graph horizontally by a factor of 1/2; multiplying by 1/2 STRETCHES it by factor 2. A negative factor reflects across the y-axis.

💡 Definitions

Horizontal Dilation.

The graph of g(x) = f(b · x) is a horizontal stretch of f by factor 1/|b| if 0 < |b| < 1, or a compression by factor 1/|b| if |b| > 1.
If b < 0, there is ALSO a reflection across the y-axis.
The point (x, y) on f's graph becomes (x/b, y) on g's graph.

Left: vertical dilations of f(x) = x². Right: horizontal dilations of f(x) = sin x. Note the inverse relationship for horizontal factors.

⚠️ Common Mistake

Horizontal dilations go BACKWARD relative to intuition. f(2x) COMPRESSES the graph horizontally (because you only need half the input to reach the same output). f(x/2) STRETCHES it. This matches the rule for horizontal translations — the input side of the function reverses the effect.

💡 Quick Check

Describe each transformation in words.

g(x) = 3·f(x) → vertical stretch by factor 3
g(x) = (1/2)·f(x) → vertical compression to half height
g(x) = f(4x) → horizontal compression by factor 4 (width ÷ 4)
g(x) = f(x/3) → horizontal stretch by factor 3 (width × 3)
g(x) = −f(x) → reflection across the x-axis
g(x) = f(−x) → reflection across the y-axis

3. Reflections as Dilations

Reflections are simply dilations with a negative scale factor. g(x) = −f(x) reflects f across the x-axis (vertical dilation by −1). g(x) = f(−x) reflects f across the y-axis (horizontal dilation by −1).

📘 Example

Example 1 — Combining dilations and reflections

Describe the transformation that takes y = x² to g(x) = −2x².

The factor −2 on the output can be split: first stretch vertically by 2, then reflect across the x-axis (or vice versa).

The parabola opens downward and is twice as tall at each x-value.

4. Combining Dilations and Translations

A general transformation of f has the form g(x) = a · f(b(x − h)) + k. Working outward: the input side gets dilated (by b), translated (by h), and optionally reflected; the output side gets dilated (by a), translated (by k), and optionally reflected.

📘 Example

Example 2 — Full transformation

Describe g(x) = −3(x − 1)² + 5 as a transformation of y = x².

Horizontal shift right 1 (replace x with x − 1).

Vertical stretch by 3 and reflection across x-axis (coefficient −3).

Vertical shift up 5 (add 5 at the end).

Order: shift right, stretch/reflect, shift up. The vertex is at (1, 5).

🎯 AP Tip

When applying combined transformations, work from INSIDE OUT: the innermost operation (on x) happens first, then any input-side shift, then any output-side scaling, then output-side shift. A common AP FRQ pitfall is applying them in the wrong order.

📘 Try This

Describe each transformation of the parent f(x) = |x|.

g(x) = 2|x − 1| + 3 → right 1, vertical stretch by 2, up 3. Vertex at (1, 3).
g(x) = −|x + 2| → left 2, reflect over x-axis. Vertex at (−2, 0), opens downward.
g(x) = (1/2)|x| → vertical compression by 1/2. Vertex still at (0, 0).

5. Summary

Vertical dilation: g(x) = a · f(x). Stretches (|a|>1), compresses (|a|<1), reflects across x-axis (a<0).
Horizontal dilation: g(x) = f(bx). Stretches (|b|<1), compresses (|b|>1), reflects across y-axis (b<0).
Horizontal effects are INVERSE to the factor — opposite of what seems intuitive.
General transformation: g(x) = a · f(b(x − h)) + k — apply from inside out.

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