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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 xx by cx divides horizontal coordinates by cc.

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 (ac,b)(\frac{a}{c},b) for f(cx)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

  1. Use (a,b) to (a,cb) for cf(x) and (ac,b)(\frac{a}{c},b) for f(cx)f(cx).
  2. Does the operation act on input or output?.
  3. 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)(6,4) under −2f(3x)-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 (ac,b)(\frac{a}{c},b) for f(cx)f(cx).
Conclusion
(2,−8)(2,-8)
Why the check works
Inputs divide by 33; outputs multiply by −2-2.
Worked examples

See the idea in three forms

foundation example

Map (6,4)(6,4) under −2f(3x)-2f(3x).

Solution(2,−8)(2,-8)

Inputs divide by 33; outputs multiply by −2-2.

representation example

Describe f(x3)f(\frac{x}{3}).

SolutionHorizontal stretch 33.

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

transfer example

Map (8,−1)(8,-1) under f(4x)f(4x).

Solution(2,−1)(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.
Read this graph as text

Vertical and horizontal scaling · 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. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Distinguish output scaling from input scaling and determine the resulting stretches, compressions, domains, and ranges.

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 33; outputs multiply by −2-2.

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.
Read this graph as text

Vertical and horizontal scaling · 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. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Distinguish output scaling from input scaling and determine the resulting stretches, compressions, domains, and ranges.

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.
Read this graph as text

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. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Distinguish output scaling from input scaling and determine the resulting stretches, compressions, domains, and ranges.

Comparison and error figure · Boundary scaling

Compare the valid path with the tempting shortcut. The figure shows why multiplying horizontal coordinates by cc instead of dividing leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is multiplying horizontal coordinates by cc instead of dividing.

Check yourself

Describe 5∣x∣5|x|.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

Describe 5∣x∣5|x|.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 202

Describe f(x3)f(\frac{x}{3}).

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

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Practice 303

Map (8,−1)(8,-1) under f(4x)f(4x).

Write a complete attempt before opening the exact answer.

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Practice 404

Range [1,4][1,4] under −2f-2f.

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Practice 505

Explain why this conclusion is valid: (2,−8)(2,-8). Use the foundation problem as evidence: Map (6,4)(6,4) under −2f(3x)-2f(3x).

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

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Practice 606

Solve the representation example, then name the feature of vertical and horizontal scaling that it illustrates: Describef(x3)f(\frac{x}{3})

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Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is multiplying horizontal coordinates by cc instead of dividing.

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Practice 808

Connect two representations for this example: Map (6,4)(6,4) under −2f(3x)-2f(3x). Describe what a graph, table, mapping, or algebraic form would have to show.

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Practice 909

Create a nearby example by changing one number or condition in this prompt: Map (8,−1)(8,-1) under f(4x)f(4x). Predict the effect, solve your new example, and compare it with the original.

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Practice 1010

Write a short verification checklist for vertical and horizontal scaling, then apply it to one worked example from this lesson.

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Lesson close

Connect forward

The next lesson, Combined transformations, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

  • Lippman and Rasmussen, Precalculus Volume 1
  • Utah College Algebra
  • Stitz and Zeager, Precalculus

No long source passage is reproduced.