Kenneth ChengMathematical modeling
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Parametric curves and preserved domains

Read the idea, work independently, then explain what changed.

T04高二理組數學思維本(2026).pdf · V. Parametric equations (PDF27–29) · PDF 27 / printed page 26

Revisit first: Parabola

TOPIC 01

Parametric curves and preserved domains

Eliminate parameters while retaining ranges, missing points and the geometry of the locus.

What you will be able to explain

  • Eliminate parameters while retaining ranges, missing points and the geometry of the locus.
  • Justify the method and check the conditions in a new situation.

Model or definition

Eliminate parameters while retaining ranges, missing points and the geometry of the locus.

x=f(s),y=g(s)x=f(s),\quad y=g(s)

Conditions

An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Do x=s²,y=s⁴ and x=s,y=s² trace the same set?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Quadratic graph: horizontal shift 0, vertical shift 0. Input is replaced by x−0.

Quadratic graph: horizontal shift 0, vertical shift 0. Input is replaced by x−0.

Explain: Compare two admissible cases and explain their different results using the stated model.

Transfer: Compare restrictions on x after elimination; then build another missing-range example.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

Eliminate s from the line representation.

x=s+2,y=2s+1x=s+2,\quad y=2s+1
  • An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
s=x−ts=x−t
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    y=2(x−2)+1=2x−3y=2(x-2)+1=2x-3

  3. An unrestricted real parameter traces the entire line.

The requested relation or conclusion is shown below.

y=2x−3y=2x-3

Checks and common pitfalls: An unrestricted real parameter traces the entire line.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

A point on x=t cos s,y=2sin s maximises x+ty. Find the maximum.

t=3t=3
  • An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
x+ty=t(coss+2sins)x+ty=t(cos s+2sin s)
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    max⁡=t12+22=35\max=t\sqrt{1^2+2^2}=3\sqrt5

  3. The full angle range permits the amplitude maximum.

The requested value is 6.7082039325.

Checks and common pitfalls: The full angle range permits the amplitude maximum.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Eliminate s from x=t+s,y=t−s and find the range if 0≤s≤2.

t=4t=4
  • An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Add equations and translate the parameter interval.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    x+y=8x+y=8

  3. Apply the stated relation and retain its conditions.

    4≤t+s≤64\le t+s\le6

  4. The result is a segment, not the entire supporting line.

The requested relation or conclusion is shown below.

x+y=8,4≤x≤6x+y=8,\quad 4\le x\le6

Checks and common pitfalls: The result is a segment, not the entire supporting line.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Eliminate s from the line representation.

x=s+5,y=2s+1x=s+5,\quad y=2s+1
  • An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
s=x−ts=x−t
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    y=2(x−5)+1=2x−9y=2(x-5)+1=2x-9

  3. An unrestricted real parameter traces the entire line.

The requested relation or conclusion is shown below.

y=2x−9y=2x-9

Checks and common pitfalls: An unrestricted real parameter traces the entire line.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Eliminate s and retain the range.

x=s2,y=s4+6x=s^2,\quad y=s^4+6
  • An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
x=s2≥0x=s²≥0
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    y=(s2)2+6=x2+6y=(s^2)^2+6=x^2+6

  3. Only the right half of the parabola is traced.

The requested relation or conclusion is shown below.

y=x2+6,x≥0y=x^2+6,\quad x\ge0

Checks and common pitfalls: Only the right half of the parabola is traced.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Find the Cartesian equation of the parametrised circle.

x=1+7cos⁡s,y=2+7sin⁡sx=1+7\cos s,\quad y=2+7\sin s
  • An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
cos2s+sin2s=1cos²s+sin²s=1
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    (x−1)2+(y−2)2=49(cos⁡2s+sin⁡2s)(x-1)^2+(y-2)^2=49(\cos^2s+\sin^2s)

  3. An unrestricted angle traces the complete circle.

The requested relation or conclusion is shown below.

(x−1)2+(y−2)2=49(x-1)^2+(y-2)^2=49

Checks and common pitfalls: An unrestricted angle traces the complete circle.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

Eliminate s, retaining the excluded point.

x=1/s,y=8/s,s≠0x=1/s,\quad y=8/s,\quad s\ne0
  • An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
x=1/s cannot equal zero.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    y=8xy=8x

  3. The origin satisfies the line equation but is never reached by this parametrisation.

The requested relation or conclusion is shown below.

y=8x,x≠0y=8x,\quad x\ne0

Checks and common pitfalls: The origin satisfies the line equation but is never reached by this parametrisation.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Eliminate s, retaining the excluded point.

x=1/s,y=9/s,s≠0x=1/s,\quad y=9/s,\quad s\ne0
  • An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
x=1/s cannot equal zero.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    y=9xy=9x

  3. The origin satisfies the line equation but is never reached by this parametrisation.

The requested relation or conclusion is shown below.

y=9x,x≠0y=9x,\quad x\ne0

Checks and common pitfalls: The origin satisfies the line equation but is never reached by this parametrisation.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

A point on x=t cos s,y=2sin s maximises x+ty. Find the maximum.

t=10t=10
  • An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
x+ty=t(coss+2sins)x+ty=t(cos s+2sin s)
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    max⁡=t12+22=105\max=t\sqrt{1^2+2^2}=10\sqrt5

  3. The full angle range permits the amplitude maximum.

The requested value is 22.360679775.

Checks and common pitfalls: The full angle range permits the amplitude maximum.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

Describe the locus when x=t cos s,y=t sin s and 0≤s≤π.

t=11t=11
  • An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
sin s≥0 on the stated interval.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    x2+y2=121x^2+y^2=121

  3. Apply the stated relation and retain its conditions.

    y=11sin⁡s≥0y=11\sin s\ge0

  4. The angle restriction traces the upper semicircle only.

The requested relation or conclusion is shown below.

x2+y2=121,y≥0x^2+y^2=121,\quad y\ge0

Checks and common pitfalls: The angle restriction traces the upper semicircle only.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Eliminate s from x=t+s,y=t−s and find the range if 0≤s≤2.

t=12t=12
  • An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Add equations and translate the parameter interval.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    x+y=24x+y=24

  3. Apply the stated relation and retain its conditions.

    12≤t+s≤1412\le t+s\le14

  4. The result is a segment, not the entire supporting line.

The requested relation or conclusion is shown below.

x+y=24,12≤x≤14x+y=24,\quad 12\le x\le14

Checks and common pitfalls: The result is a segment, not the entire supporting line.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

Describe the locus when x=t cos s,y=t sin s and 0≤s≤π.

t=13t=13
  • An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
sin s≥0 on the stated interval.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    x2+y2=169x^2+y^2=169

  3. Apply the stated relation and retain its conditions.

    y=13sin⁡s≥0y=13\sin s\ge0

  4. The angle restriction traces the upper semicircle only.

The requested relation or conclusion is shown below.

x2+y2=169,y≥0x^2+y^2=169,\quad y\ge0

Checks and common pitfalls: The angle restriction traces the upper semicircle only.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Eliminate s from x=t+s,y=t−s and find the range if 0≤s≤2.

t=14t=14
  • An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Add equations and translate the parameter interval.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    x+y=28x+y=28

  3. Apply the stated relation and retain its conditions.

    14≤t+s≤1614\le t+s\le16

  4. The result is a segment, not the entire supporting line.

The requested relation or conclusion is shown below.

x+y=28,14≤x≤16x+y=28,\quad 14\le x\le16

Checks and common pitfalls: The result is a segment, not the entire supporting line.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • Eliminate parameters while retaining ranges, missing points and the geometry of the locus.
    • Which condition is essential in parametric curves and preserved domains?
    • Do x=s²,y=s⁴ and x=s,y=s² trace the same set?

    Board plan

    • Model or definition: Eliminate parameters while retaining ranges, missing points and the geometry of the locus.
      x=f(s),y=g(s)x=f(s),\quad y=g(s)
    • Conditions: An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.

    Anticipated thinking

    • The same Cartesian equation does not always define the same parametric locus.

    Assessment checklist

    • 1 mark: choose the correct representation and conditions.
    • 1 mark: establish the intermediate relation.
    • 1 mark: complete a connected calculation or proof.
    • 1 mark: interpret and check the conclusion.

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    Curriculum and source notes ↗