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BetterGrades Precalculus · Unit 13 · Lesson

Hyperbolas

Analyze hyperbolas from constant-difference distance and standard equations.

Textbook reading

The problem that opens the lesson

A hyperbola has center (0,0),(0,0), vertices (±3,0),(\pm 3,0), and foci (±5,0)(\pm 5,0). Find its equation and asymptotes.

Solution

Begin by identifying the mathematical object and the information that fixes it. Determine center and orientation, locate vertices, compute c, draw the fundamental rectangle, and sketch branches approaching the asymptotes. The relevant conditions are not optional bookkeeping: The two branches are separate components of one relation. The asymptotes guide end behavior but are not part of the hyperbola. Following that structure gives a=3,c=5,b=4a=3,c=5,b=4; x29−y216=1\frac{x^2}{9}-\frac{y^2}{16}=1; y=±4x3y=\frac{\pm 4x}{3}.

Why this works

Asymptotes pass through the center and follow the diagonals of the fundamental rectangle with semiaxes a and bb. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

What this lesson is really about

A hyperbola is the locus of points whose absolute difference of distances to two foci is constant 2a2a.

Its standard forms contain one positive squared term and one negative squared term. The positive term identifies the transverse-axis direction. The focal relationship is c2=a2+b2c^2=a^2+b^2.

The point is not merely to reproduce a formula. A learner should be able to identify the quantities or geometric objects involved, explain why the relationship has its stated form, and recognize when the same idea appears in a graph, table, diagram, or model.

Textbook reading

Why the relationship works

Asymptotes pass through the center and follow the diagonals of the fundamental rectangle with semiaxes a and bb.

Textbook reading

A reliable way to work

Determine center and orientation, locate vertices, compute c, draw the fundamental rectangle, and sketch branches approaching the asymptotes.

The two branches are separate components of one relation. The asymptotes guide end behavior but are not part of the hyperbola.

After the symbolic work is complete, check the result. Depending on the lesson, this may mean substituting into an original equation, comparing coordinates, examining a graph, checking units, testing an interval, or confirming that every branch of a periodic solution has been included.

Textbook reading

What commonly goes wrong

A common error is using the ellipse relation c2=a2−b2c^2=a^2-b^2 or orienting the branches by the larger denominator rather than the positive term.

The repair is to return to the definition and identify the first step where the invalid solution stops describing the original mathematical object. Later algebra cannot rescue a first step that changed the domain, orientation, branch, or meaning of the problem.

Textbook reading

Worked examples

Worked example 1

A hyperbola has center (0,0),(0,0), vertices (±3,0),(\pm 3,0), and foci (±5,0)(\pm 5,0). Find its equation and asymptotes.

Solution

Begin by identifying the mathematical object and the information that fixes it. Determine center and orientation, locate vertices, compute c, draw the fundamental rectangle, and sketch branches approaching the asymptotes. The relevant conditions are not optional bookkeeping: The two branches are separate components of one relation. The asymptotes guide end behavior but are not part of the hyperbola. Following that structure gives a=3,c=5,b=4a=3,c=5,b=4; x29−y216=1\frac{x^2}{9}-\frac{y^2}{16}=1; y=±4x3y=\frac{\pm 4x}{3}.

Why this works

Asymptotes pass through the center and follow the diagonals of the fundamental rectangle with semiaxes a and bb. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Transfer example

Problem

Derive c2=a2+b2c^2=a^2+b^2.

Worked development

Determine center and orientation, locate vertices, compute c, draw the fundamental rectangle, and sketch branches approaching the asymptotes. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. Its standard forms contain one positive squared term and one negative squared term. The positive term identifies the transverse-axis direction. The focal relationship is c2=a2+b2c^2=a^2+b^2. Then apply the conditions explicitly: The two branches are separate components of one relation. The asymptotes guide end behavior but are not part of the hyperbola. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Hyperbolas appear in navigation differences, cooling-tower sections, optics, and inverse-product graphs after transformation.

Reasoning example

Problem

Read transverse and conjugate axes.

Worked development

Determine center and orientation, locate vertices, compute c, draw the fundamental rectangle, and sketch branches approaching the asymptotes. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. Its standard forms contain one positive squared term and one negative squared term. The positive term identifies the transverse-axis direction. The focal relationship is c2=a2+b2c^2=a^2+b^2. Then apply the conditions explicitly: The two branches are separate components of one relation. The asymptotes guide end behavior but are not part of the hyperbola. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Hyperbolas appear in navigation differences, cooling-tower sections, optics, and inverse-product graphs after transformation.

Worked example 4: quick check

Find vertices and asymptotes ofy225−x29=1\frac{y^2}{25}-\frac{x^2}{9}=1

Solution

Begin by identifying the mathematical object and the information that fixes it. Determine center and orientation, locate vertices, compute c, draw the fundamental rectangle, and sketch branches approaching the asymptotes. The relevant conditions are not optional bookkeeping: The two branches are separate components of one relation. The asymptotes guide end behavior but are not part of the hyperbola. Following that structure gives Vertices (0,±5)(0,\pm 5); asymptotes y=±(53)xy=\pm (\frac{5}{3})x.

Why this works

Asymptotes pass through the center and follow the diagonals of the fundamental rectangle with semiaxes a and bb. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Constant-difference locus. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Asymptotes pass through the center and follow the diagonals of the fundamental rectangle with semiaxes a and b. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.
Read this graph as text

Hyperbolas · Constant-difference locus. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Asymptotes pass through the center and follow the diagonals of the fundamental rectangle with semiaxes a and b. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same 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 hyperbolas from constant-difference distance and standard equations.

Anchor figure · Constant-difference locus

Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Asymptotes pass through the center and follow the diagonals of the fundamental rectangle with semiaxes a and bb. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Fundamental rectangle and asymptotes. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for hyperbolas.
Read this graph as text

Hyperbolas · Fundamental rectangle and asymptotes. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for hyperbolas. 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 hyperbolas from constant-difference distance and standard equations.

Mechanism figure · Fundamental rectangle and asymptotes

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

Horizontal-versus-vertical form comparison. Compare the valid path with the tempting shortcut. The figure shows why using the ellipse relation c^2=a^2-b^2 or orienting the branches by the larger denominator rather than the positive term leads to a false conclusion.
Read this graph as text

Hyperbolas · Horizontal-versus-vertical form comparison. Compare the valid path with the tempting shortcut. The figure shows why using the ellipse relation c^2=a^2-b^2 or orienting the branches by the larger denominator rather than the positive term 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 hyperbolas from constant-difference distance and standard equations.

Comparison and error figure · Horizontal-versus-vertical form comparison

Compare the valid path with the tempting shortcut. The figure shows why using the ellipse relation c2=a2−b2c^2=a^2-b^2 or orienting the branches by the larger denominator rather than the positive term leads to a false conclusion.

Textbook reading

Application and interpretation

Hyperbolas appear in navigation differences, cooling-tower sections, optics, and inverse-product graphs after transformation.

A contextual answer must include units, a meaningful domain, and the assumptions that make the model plausible. An exact mathematical relationship should not be diluted into a decimal unless a measurement or comparison requires it.

Check yourself

Find vertices and asymptotes ofy225−x29=1\frac{y^2}{25}-\frac{x^2}{9}=1

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Practice

16 concrete questions

Practice 1 · retrieval · foundational01

Find vertices and asymptotes ofy225−x29=1\frac{y^2}{25}-\frac{x^2}{9}=1

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Practice 2 · conceptual · foundational02

State the defining idea behind hyperbolas in one precise sentence.

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Practice 3 · verification · developing03

For hyperbolas, what condition or domain restriction must remain visible in the solution?

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Practice 4 · error analysis · developing04

For hyperbolas, describe the most likely incorrect first step and explain why it fails.

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Practice 5 · synthesis · transfer05

For hyperbolas, explain how this lesson's idea will be used later in the course.

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Practice 6 · procedural · foundational06

Solve this hyperbolas problem and state the final result: A hyperbola has center (0,0),(0,0), vertices (±3,0),(\pm 3,0), and foci (±5,0)(\pm 5,0). Find its equation and asymptotes.

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Practice 7 · procedural · developing07

In hyperbolas, for “Derive c2=a2+b2.”,c^2=a^2+b^2.”, identify the first valid mathematical step and the condition that must remain visible.

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Practice 8 · transfer · transfer08

For “Read transverse and conjugate axes.”, identify the governing definition or relationship and what a complete conclusion must include.

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Practice 9 · verification · developing09

Verify “a=3,c=5,b=4a=3,c=5,b=4; x29−y216=1\frac{x^2}{9}-\frac{y^2}{16}=1; y=±4x3y=\frac{\pm 4x}{3}.” using the required condition for hyperbolas.

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Practice 10 · explanation · developing10

Explain why “a=3,c=5,b=4a=3,c=5,b=4; x29−y216=1\frac{x^2}{9}-\frac{y^2}{16}=1; y=±4x3y=\frac{\pm 4x}{3}.” follows from this lesson’s mathematical mechanism.

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Practice 11 · conceptual · developing11

What mathematical structure is shared by the opening problem and “Read transverse and conjugate axes.”?

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Practice 12 · graphical · developing12

In “Constant-difference locus”, which mathematical objects or labels must be visible to support “a=3,c=5,b=4a=3,c=5,b=4; x29−y216=1\frac{x^2}{9}-\frac{y^2}{16}=1; y=±4x3y=\frac{\pm 4x}{3}.”?

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Practice 13 · graphical · transfer13

How should “Fundamental rectangle and asymptotes” make the governing relationship in “Derive c2=a2+b2c^2=a^2+b^2.” visible?

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Practice 14 · error analysis · transfer14

In “Horizontal-versus-vertical form comparison”, identify the first point where the misconception diverges from valid hyperbolas reasoning.

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Practice 15 · modeling · transfer15

In the application “Hyperbolas appear in navigation differences, cooling-tower sections, optics, and inverse-product graphs after transformation.”, what quantities or geometric objects must be identified, and what condition makes the model valid?

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Practice 16 · exit check · transfer16

Answer “Find vertices and asymptotes of y225−x29=1\frac{y^2}{25}-\frac{x^2}{9}=1.” and name the condition used to check the result.

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Textbook reading

Lesson summary

A hyperbola is the locus of points whose absolute difference of distances to two foci is constant 2a2a.

The central condition to remember is this: The two branches are separate components of one relation. The asymptotes guide end behavior but are not part of the hyperbola.

Connection forward

The next lesson develops the algebraic technique used to reveal translated conic forms.

The next lesson is Completing squares in two variables.

Source record

Original BetterGrades manuscript, rights-separated references.

  • Lippman & Rasmussen, Precalculus Vol. 2, Chapter 9
  • Stitz & Zeager, Precalculus, Chapter 7
  • University of Washington Precalculus, conic problem sets

No long source passage is reproduced.