LEARN · EXPLAIN · REVISE

Find the quadratic with vertex (2,−3) through (0,5).

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

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

TOPIC 01

Vertex form and completing the square

Explain turning points, symmetry and constrained optima through a square.

What you will be able to explain

Complete a square without changing the function.
Distinguish the extremum value, attaining input and admissible domain.

Try it. Leave your reasoning visible.

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

Examples 1–3: study a full demonstration; finish a solution from its first step; then choose a method independently before opening the solution. The remaining exercises may continue after class.

01 / Standard#Worked example

Find the quadratic with vertex (2,−3) through (0,5).

(h,k)=(2,-3)

All inputs are real unless a smaller domain is stated.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Encode the vertex first.

Hint 2

Substitute the other point to find a.

Worked solution

Use vertex form.
$y=a(x-2)^2-3$
Determine the scale from the extra point.
$5=4a-3\Rightarrow a=2$

y=2(x−2)²−3.

Checks and common pitfalls: The extra point has a different x-coordinate, so it determines a uniquely.

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

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Diagnose signs	Use both diagnostics and ask where the square vanishes.
8	Build a square	Track the added and subtracted constant in x²−6x+5.
10	Compare forms	Pairs recover a quadratic from its vertex and a point, then explain a symmetric table.
12	Defend a choice	Groups solve the pack problem and challenge arbitrary rounding.
5	Exit check	Separate the optimum value from the attaining input.

Question sequence

Why must the added square be subtracted too?
Can two distinct allowed inputs tie?

Board development

Preserve equality.
$x^2-6x+5=(x-3)^2-4$
Compare allowed distances.
$C=(n-4.5)^2+9.75$

Expected correct and incorrect responses

“The square is zero at 3, so the value is −4” connects input and output.
“Round to 5” overlooks the tied solution 4.

Follow-up questions

What changes if the leading coefficient is negative?

Remediation

For axis-sign errors, substitute both proposed inputs into the square.
For compensation errors, expand the proposed vertex form.

Exit check

For f=2(x+2)²−1 and integer x≥0, justify the minimum. Expected f(0)=7.

Assessment checklist and sample responses

2 marks: preserve equality while completing.
Full: 2[(x+2)²−4]+3; partial: missing −8.
2 marks: justify admissibility and ties.
Full: n=4 or 5, cost 10; incomplete: n=4.5.