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BetterGrades Precalculus · Unit 3 · Lesson

Vertical translations

Analyze how adding a constant outside a function changes outputs, range, intercepts, extrema, and asymptotes.

Opening

Start with the situation

Adding kk outside a function changes every output by kk and moves the graph vertically.

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

Map (a,b) to (a,b+k),(a,b+k), translate the range and horizontal asymptote, and recompute zeros.

The domain and input locations of extrema remain unchanged.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through vertical point mapping, feature ledger, 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 adding a constant outside a function changes outputs, range, intercepts, extrema, and 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

  1. Map (a,b) to (a,b+k)(a,b+k).
  2. Translate the range.
  3. Horizontal asymptote.
  4. Recompute zeros.

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 x2−5x^2-5.

Plan
Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Map (a,b) to (a,b+k),(a,b+k), translate the range and horizontal asymptote, and recompute zeros.
Conclusion
Vertex (0,−5),(0,-5), range [−5,∞),[-5,∞), zeros ±sqrt(5)\pm sqrt(5).
Why the check works
The parabola shifts down 55.
Worked examples

See the idea in three forms

foundation example

Describe x2−5x^2-5.

SolutionVertex (0,−5),(0,-5), range [−5,∞),[-5,∞), zeros ±sqrt(5)\pm sqrt(5).

The parabola shifts down 55.

representation example

Range of sqrt(x)−2sqrt(x)-2.

Solution[−2,∞)[-2,∞)

This example expresses vertical translations in a second form.

transfer example

HA of 5x+75^x+7.

Solutiony=7y=7

The domain and input locations of extrema remain unchanged.

Vertical point mapping. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The parabola shifts down 5.
Read this graph as text

Vertical translations · Vertical point mapping. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The parabola shifts down 5. 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: Analyze how adding a constant outside a function changes outputs, range, intercepts, extrema, and asymptotes.

Anchor figure · Vertical point mapping

Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The parabola shifts down 55.

Feature ledger. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for vertical translations.
Read this graph as text

Vertical translations · Feature ledger. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for vertical translations. 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: Analyze how adding a constant outside a function changes outputs, range, intercepts, extrema, and asymptotes.

Mechanism figure · Feature ledger

Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for vertical translations.

Baseline interpretation. Compare the valid path with the tempting shortcut. The figure shows why moving x-coordinates or assuming x-intercepts simply shift vertically leads to a false conclusion.
Read this graph as text

Vertical translations · Baseline interpretation. Compare the valid path with the tempting shortcut. The figure shows why moving x-coordinates or assuming x-intercepts simply shift vertically 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: Analyze how adding a constant outside a function changes outputs, range, intercepts, extrema, and asymptotes.

Comparison and error figure · Baseline interpretation

Compare the valid path with the tempting shortcut. The figure shows why moving x-coordinates or assuming x-intercepts simply shift vertically leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is moving x-coordinates or assuming x-intercepts simply shift vertically.

Check yourself

Vertex of ∣x∣+6|x|+6.

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

Vertex of ∣x∣+6|x|+6.

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

Range of sqrt(x)−2sqrt(x)-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

HA of 5x+75^x+7.

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 (−2,5)(-2,5) under f(x)−8f(x)-8.

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: Vertex (0,−5),(0,-5), range [−5,∞),[-5,∞), zeros ±sqrt(5)\pm sqrt(5). Use the foundation problem as evidence: Describe x2−5x^2-5.

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 606

Solve the representation example, then name the feature of vertical translations that it illustrates: Range ofsqrt(x)−2sqrt(x)-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 moving x-coordinates or assuming x-intercepts simply shift vertically.

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 x2−5x^2-5. Describe what a graph, table, mapping, or algebraic form would have to show.

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Attempt once to unlock the answer

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Practice 909

Create a nearby example by changing one number or condition in this prompt: HA of 5x+75^x+7. 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 vertical translations, then apply it to one worked example from this lesson.

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Attempt once to unlock the answer

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

Lesson close

Connect forward

The next lesson, Horizontal translations, 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.