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Parabola through three points worksheets

Three points fix a parabola, and finding it is a system of three equations in disguise. These free printable worksheets give three points and ask for the a, b and c in y = ax² + bx + c, with an answer sheet that shows the working.

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Parabola through three points
Parabola through three points worksheet — Whole a, b and c, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

Each point becomes one equation. Put x = 2 and y = 11 into y = ax² + bx + c and the result is 4a + 2b + c = 11. The letters that were unknowns in the usual sense, x and y, are now just numbers, and a, b and c are what is being solved for. Seeing that switch is the main idea on this page.

Why this sheet works

From there it is the same elimination as any other three-equation system, and it has a helpful feature: c has a coefficient of 1 on every line, so it is the obvious letter to remove first. If one of the points has x = 0, c can be read off straight away.

How this one works

y = ax² + bx + c through (−2, 7), (0, 1), (1, 4)

(1) c − 2b + 4a = 7
(2) c = 1
(3) c + b + a = 4
(4) [(1) − (2)] ÷ 2: −b + 2a = 3
(5) [(1) − (3)] ÷ 3: −b + a = 1
(6) (4) − (5): a = 2
b = 1
c = 1
  1. Put each point into y = ax² + bx + c. Each one gives an equation in a, b and c.
  2. Take c out of two different pairs. That leaves two equations with only b and a in them.
  3. Now take b out of those two. Only a is left, so its value is one division away.
  4. Put a = 2 into (4) to find b, then both into (1) to find c.

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