Kenneth ChengMathematical modeling
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Find the parabola with vertex O and focus (0,−t).

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

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TOPIC 01

Conic sections: mixed review

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

  • Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

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

01 / Foundation#Your turn

Find the parabola with vertex O and focus (0,−t).

t=22t=22
  • a must be nonzero; a parabola has a vertex and no center of symmetry.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
x2=4ay,a=−tx^2=4ay,\quad a=-t
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    x2=4(−22)yx^2=4(-22)y

  3. The focus below the vertex gives a downward opening.

The requested relation or conclusion is shown below.

x2=−88yx^2=-88y

Checks and common pitfalls: The focus below the vertex gives a downward opening.

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

Teacher preparation and assessment

Question sequence

  • Ask students to name the relevant condition before calculating.

Board plan

  • Compare valid methods and annotate their conditions.

Anticipated thinking

  • A correct final value may still hide a missing assumption.

Assessment checklist

  • Check the method, conditions, reasoning and interpretation separately.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗