Porisma

Worked solution

Read a parabola’s vertex and choose an equation

Write down one possible equation of the parabola shown in the diagram.

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A parabola opens left, with vertex labelled (5,−3). Its width is illustrative; determine one possible equation from the vertex and direction.
A parabola opens left, with vertex labelled (5,−3). Its width is illustrative; determine one possible equation from the vertex and direction.
A parabola opens left, with vertex labelled (5,−3). Its width is illustrative; determine one possible equation from the vertex and direction.

CJC · Lecture · Example 2

Read the vertex and direction

The teacher solution uses (y−k)2=a(x−h)(y-k)^2=a(x-h) with a<0a<0. The labelled vertex (5,−3)(5,-3) gives h=5,k=−3h=5,k=-3; opening left fixes the sign of aa, but the diagram does not fix its magnitude.

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Choose one admissible coefficient

Choose a=−1a=-1: one possible equation is (y+3)2=−(x−5)(y+3)^2=-(x-5). More generally, every equation (y+3)2=a(x−5)(y+3)^2=a(x-5) with a<0a<0 has the stated vertex and orientation. The teacher copy’s choice is one example, not a uniquely determined equation.

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Common mistakes

A vertex and opening direction leave infinitely many possible widths. Do not infer an exact coefficient from an illustrative sketch.

Solution notes

Question: CJC Conics Student copy, PDF pp.[5], printed pp.[117]. Working compared with the matching Teacher copy at the same pages.

No marks are printed for these examples. The original native text from both copies is retained; teacher additions are integrated into the relevant working.

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