Kenneth ChengMathematical modeling
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Quadratic functions, equations and inequalities

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

高一必修 第一册(A版).pdf · 2.3 · PDF 57 / printed page 50

Revisit first: Basic inequality: AM–GM

TOPIC 01

Quadratic functions, equations and inequalities

Build understanding of quadratic functions, equations and inequalities through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Apply quadratic functions, equations and inequalities with explicit conditions.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Signs and roots

Roots divide the number line into intervals of fixed sign.

Vertex and discriminant

Complete the square for extrema and use the discriminant to count roots.

Δ=b2−4ac\Delta=b^2-4ac

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in quadratic functions, equations and inequalities changes. Record a reason.

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 one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

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

Solve the quadratic inequality.

(x−2)(x−5)≤0(x-2)(x-5)\le0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Find the roots or vertex, then check signs and endpoints.
Hint 2
The upward-opening quadratic is nonpositive between its roots.
Worked solution

  1. Find the roots or vertex, then check signs and endpoints.

  2. Calculate or simplify this relation.

    2≤x≤52\le x\le5

  3. The non-strict comparison includes both roots.

x∈[2,5].

Checks and common pitfalls: The non-strict comparison includes both roots.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

02 / Standard#Worked example

Count the integer solutions.

(x+2)(x−4)<0(x+2)(x-4)<0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Find the roots or vertex, then check signs and endpoints.
Hint 2
Exclude the two integer roots.
Worked solution

  1. Find the roots or vertex, then check signs and endpoints.

  2. Calculate or simplify this relation.

    −2<x<4;(3)−(−1)+1=5-2<x<4;\quad (3)-(-1)+1=5

  3. Strict comparisons exclude equality.

The requested value is 5.

Checks and common pitfalls: Strict comparisons exclude equality.

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

03 / Transfer#Worked example

Solve the case a=0 separately.

ax2+2x−6≥0ax^2+2x-6\ge0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Find the roots or vertex, then check signs and endpoints.
Hint 2
Substitute before applying a quadratic formula.
Worked solution

  1. Find the roots or vertex, then check signs and endpoints.

  2. Calculate or simplify this relation.

    2x≥6⇒x≥32x\ge6\Rightarrow x\ge3

  3. A zero leading coefficient makes the expression linear here.

x≥3.

Checks and common pitfalls: A zero leading coefficient makes the expression linear here.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

04 / Foundation#Your turn

Solve the quadratic inequality.

(x−3)(x−6)≤0(x-3)(x-6)\le0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Find the roots or vertex, then check signs and endpoints.
Hint 2
The upward-opening quadratic is nonpositive between its roots.
Worked solution

  1. Find the roots or vertex, then check signs and endpoints.

  2. Calculate or simplify this relation.

    3≤x≤63\le x\le6

  3. The non-strict comparison includes both roots.

x∈[3,6].

Checks and common pitfalls: The non-strict comparison includes both roots.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

05 / Foundation#Your turn

Count the integer solutions.

(x+3)(x−5)<0(x+3)(x-5)<0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Find the roots or vertex, then check signs and endpoints.
Hint 2
Exclude the two integer roots.
Worked solution

  1. Find the roots or vertex, then check signs and endpoints.

  2. Calculate or simplify this relation.

    −3<x<5;(4)−(−2)+1=7-3<x<5;\quad (4)-(-2)+1=7

  3. Strict comparisons exclude equality.

The requested value is 7.

Checks and common pitfalls: Strict comparisons exclude equality.

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

06 / Foundation#Your turn

Calculate the discriminant and interpret its sign.

x2−6x+10=0x^2-6x+10=0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Find the roots or vertex, then check signs and endpoints.
Hint 2
Use the full coefficient of x.
Worked solution

  1. Find the roots or vertex, then check signs and endpoints.

  2. Calculate or simplify this relation.

    Δ=(−6)2−4(10)=−4\Delta=(-6)^2-4(10)=-4

  3. A negative discriminant means no real root.

The requested value is -4.

Checks and common pitfalls: A negative discriminant means no real root.

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

07 / Foundation#Your turn

Solve the inequality with a negative leading coefficient.

−(x−3)(x−5)>0-(x-3)(x-5)>0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Find the roots or vertex, then check signs and endpoints.
Hint 2
Multiply by −1 and reverse the comparison.
Worked solution

  1. Find the roots or vertex, then check signs and endpoints.

  2. Calculate or simplify this relation.

    (x−3)(x−5)<0(x-3)(x-5)<0

  3. The sign outside the roots depends on the leading coefficient.

x∈(3,5).

Checks and common pitfalls: The sign outside the roots depends on the leading coefficient.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

08 / Standard#Your turn

For which m is this positive for every real x?

x2−6x+mx^2-6x+m
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Find the roots or vertex, then check signs and endpoints.
Hint 2
Its minimum occurs when the square vanishes.
Worked solution

  1. Find the roots or vertex, then check signs and endpoints.

  2. Calculate or simplify this relation.

    x2−6x+m=(x−3)2+m−9x^2-6x+m=(x-3)^2+m-9

  3. Strict positivity requires a strictly positive minimum.

m>9.

Checks and common pitfalls: Strict positivity requires a strictly positive minimum.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

09 / Standard#Your turn

Find the length of the x-interval where the line is above or on the parabola.

y=(10)x−21,y=x2y=(10)x-21,\quad y=x^2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Find the roots or vertex, then check signs and endpoints.
Hint 2
Compare their vertical difference.
Worked solution

  1. Find the roots or vertex, then check signs and endpoints.

  2. Calculate or simplify this relation.

    (x−3)(x−7)≤0;(7)−3=4(x-3)(x-7)\le0;\quad (7)-3=4

  3. Interval length is not the count of integer points.

The requested value is 4.

Checks and common pitfalls: Interval length is not the count of integer points.

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

10 / Standard#Your turn

Solve the case a=0 separately.

ax2+3x−12≥0ax^2+3x-12\ge0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Find the roots or vertex, then check signs and endpoints.
Hint 2
Substitute before applying a quadratic formula.
Worked solution

  1. Find the roots or vertex, then check signs and endpoints.

  2. Calculate or simplify this relation.

    3x≥12⇒x≥43x\ge12\Rightarrow x\ge4

  3. A zero leading coefficient makes the expression linear here.

x≥4.

Checks and common pitfalls: A zero leading coefficient makes the expression linear here.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

11 / Standard#Your turn

Solve the quadratic inequality.

(x−4)(x−7)≤0(x-4)(x-7)\le0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Find the roots or vertex, then check signs and endpoints.
Hint 2
The upward-opening quadratic is nonpositive between its roots.
Worked solution

  1. Find the roots or vertex, then check signs and endpoints.

  2. Calculate or simplify this relation.

    4≤x≤74\le x\le7

  3. The non-strict comparison includes both roots.

x∈[4,7].

Checks and common pitfalls: The non-strict comparison includes both roots.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

12 / Transfer#Your turn

Count the integer solutions.

(x+4)(x−6)<0(x+4)(x-6)<0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Find the roots or vertex, then check signs and endpoints.
Hint 2
Exclude the two integer roots.
Worked solution

  1. Find the roots or vertex, then check signs and endpoints.

  2. Calculate or simplify this relation.

    −4<x<6;(5)−(−3)+1=9-4<x<6;\quad (5)-(-3)+1=9

  3. Strict comparisons exclude equality.

The requested value is 9.

Checks and common pitfalls: Strict comparisons exclude equality.

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

13 / Transfer#Your turn

Calculate the discriminant and interpret its sign.

x2−8x+17=0x^2-8x+17=0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Find the roots or vertex, then check signs and endpoints.
Hint 2
Use the full coefficient of x.
Worked solution

  1. Find the roots or vertex, then check signs and endpoints.

  2. Calculate or simplify this relation.

    Δ=(−8)2−4(17)=−4\Delta=(-8)^2-4(17)=-4

  3. A negative discriminant means no real root.

The requested value is -4.

Checks and common pitfalls: A negative discriminant means no real root.

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

    • What must be true before using the main rule for quadratic functions, equations and inequalities?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Apply quadratic functions, equations and inequalities with explicit conditions.
    • Signs and roots: Roots divide the number line into intervals of fixed sign.
    • Vertex and discriminant: Complete the square for extrema and use the discriminant to count roots.
      Δ=b2−4ac\Delta=b^2-4ac
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Roots divide the number line into intervals of fixed sign.
    • Expected reasoning: Complete the square for extrema and use the discriminant to count roots.
    • Expected correction: Treat a=0 separately.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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    Curriculum and source notes ↗