BetterGrades Precalculus · Unit 3 · Lesson

Vertical and horizontal scaling

Distinguish output scaling from input scaling and determine the resulting stretches, compressions, domains, and ranges.

Opening

Start with the situation

Multiplying outputs scales vertically; replacing $x$ by cx divides horizontal coordinates by $c$ .

Transformation language lets you read a complicated graph as a modified parent rather than a collection of disconnected points. That makes prediction possible before any calculator window is opened.

Before you begin

Prerequisite check

Recognize the parent function family.
Read domain, range, and key points.
Use coordinate mappings.

Core explanation

Explanation

Use (a,b) to (a,cb) for cf(x) and $(\frac{a}{c},b)$ for $f(cx)$ .

Negative scale factors also reflect, and domain boundaries follow the x-coordinate map.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through scaling point grid, parabola scale comparison, or another equivalent representation.

Conceptual reading

What the idea is really doing

Transformations become reliable when they are treated as coordinate mappings. Outside operations change outputs; inside operations change the inputs that produce those outputs, which is why horizontal changes often appear to work in the opposite direction.

This lesson narrows that lens to one goal: distinguish output scaling from input scaling and determine the resulting stretches, compressions, domains, and ranges. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Use (a,b) to (a,cb) for cf(x) and $(\frac{a}{c},b)$ for $f(cx)$ .
Does the operation act on input or output?.
Where do the landmark points move?.

Verification: Track at least one landmark point from the parent graph to the transformed graph, then verify the new domain, range, intercepts, or asymptotes from the formula.

Foundation walkthrough

Plan before calculating

Problem

Map $(6,4)$ under $-2f(3x)$ .

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Use (a,b) to (a,cb) for cf(x) and $(\frac{a}{c},b)$ for $f(cx)$ .
Conclusion: $(2,-8)$
Why the check works: Inputs divide by $3$ ; outputs multiply by $-2$ .

Worked examples

See the idea in three forms

foundation example

Map $(6,4)$ under $-2f(3x)$ .

Solution $(2,-8)$

Inputs divide by $3$ ; outputs multiply by $-2$ .

representation example

Describe $f(\frac{x}{3})$ .

SolutionHorizontal stretch $3$ .

This example expresses vertical and horizontal scaling in a second form.

transfer example

Map $(8,-1)$ under $f(4x)$ .

Solution $(2,-1)$

Negative scale factors also reflect, and domain boundaries follow the x-coordinate map.

Scaling point grid. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Inputs divide by 3; outputs multiply by -2. — **Anchor figure · Scaling point grid**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Inputs divide by $3$ ; outputs multiply by $-2$ .

Parabola scale comparison. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for vertical and horizontal scaling. — **Mechanism figure · Parabola scale comparison**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for vertical and horizontal scaling.

Boundary scaling. Compare the valid path with the tempting shortcut. The figure shows why multiplying horizontal coordinates by c instead of dividing leads to a false conclusion. — **Comparison and error figure · Boundary scaling**
Compare the valid path with the tempting shortcut. The figure shows why multiplying horizontal coordinates by $c$ instead of dividing leads to a false conclusion.

Common mistake