BetterGrades Precalculus · Unit 3 · Lesson

Horizontal translations

Analyze how replacing x by x-h changes input locations, domain, intercepts, extrema, and vertical asymptotes.

Opening

Start with the situation

Replacing $x$ by x-h relocates every input feature $h$ units to the right.

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

Solve the inside equation for the new x-coordinate; translate domain endpoints, zeros, extrema, and vertical asymptotes.

The range remains unchanged because the same outputs occur at new inputs.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through input relocation, domain interval translation, 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: analyze how replacing $x$ by x-h changes input locations, domain, intercepts, extrema, and vertical asymptotes. 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

Solve the inside equation for the new x-coordinate; translate domain endpoints.
Zeros.
Extrema.
Vertical asymptotes.

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

Describe $sqrt(x-4)$ .

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Solve the inside equation for the new x-coordinate; translate domain endpoints, zeros, extrema, and vertical asymptotes.
Conclusion: Endpoint $(4,0),$ domain $[4,∞)$ .
Why the check works: The root graph shifts right $4$ .

Worked examples

See the idea in three forms

foundation example

Describe $sqrt(x-4)$ .

SolutionEndpoint $(4,0),$ domain $[4,∞)$ .

The root graph shifts right $4$ .

representation example

Vertex of $(x-7)^2$ .

Solution $(7,0)$

This example expresses horizontal translations in a second form.

transfer example

Move $(-2,4]$ right $3$ .

Solution $(1,7]$

The range remains unchanged because the same outputs occur at new inputs.

Input relocation. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The root graph shifts right 4. — **Anchor figure · Input relocation**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The root graph shifts right $4$ .

Domain interval translation. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for horizontal translations. — **Mechanism figure · Domain interval translation**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for horizontal translations.

Delay interpretation. Compare the valid path with the tempting shortcut. The figure shows why reading the inside sign literally without solving for the old input leads to a false conclusion. — **Comparison and error figure · Delay interpretation**
Compare the valid path with the tempting shortcut. The figure shows why reading the inside sign literally without solving for the old input leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is reading the inside sign literally without solving for the old input.

Check yourself

Endpoint of $sqrt(x+5)$ .

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

Endpoint of $sqrt(x+5)$ .

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

Vertex of $(x-7)^2$ .

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 303

Move $(-2,4]$ right $3$ .

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 404

Map $(5,-2)$ under $f(x+3)$ .

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 505

Explain why this conclusion is valid: Endpoint $(4,0),$ domain $[4,∞)$ . Use the foundation problem as evidence: Describe $sqrt(x-4)$ .

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 606

Solve the representation example, then name the feature of horizontal translations that it illustrates: Vertex of $(x-7)^2$

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 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is reading the inside sign literally without solving for the old input.

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 808

Connect two representations for this example: Describe $sqrt(x-4)$ . Describe what a graph, table, mapping, or algebraic form would have to show.

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 909

Create a nearby example by changing one number or condition in this prompt: Move $(-2,4]$ right $3$ . Predict the effect, solve your new example, and compare it with the original.

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 1010

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

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.

Lesson close

Connect forward

The next lesson, Reflections and symmetry, 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.

Describe sqrt(x−4)sqrt(x-4)sqrt(x−4).

Describe sqrt(x−4)sqrt(x-4)sqrt(x−4).

Vertex of (x−7)2(x-7)^2(x−7)2.

Move (−2,4](-2,4](−2,4] right 333.

Original BetterGrades manuscript, rights-separated references.

Describe $sqrt(x-4)$ .

Describe $sqrt(x-4)$ .

Vertex of $(x-7)^2$ .

Move $(-2,4]$ right $3$ .