Write the equation of a parabola

Learn how to write the equation of a parabola in vertex form using the graph of the parabola.

Example #1

Use the graph of the parabola below to write the equation of the parabola

The vertex form of a parabola is y = a(x - h)² + k, (h, k) is the vertex

Looking at the parabola above, the vertex is located at (h, k) = (-4, -2)

Substitute h = -4 and k = -2 into the vertex form.

y = a(x - h)² + k

y = a(x - -4)² + -2

y = a(x + 4)² + - 2

Pick any point on the graph (except the vertex) and substitute the point into y = a(x + 4)² + - 2 in order to find a.

Let us pick (-2, 10)

10 = a(-2 + 4)² + - 2

10 = a(2)² + - 2

10 = a(4) + - 2

Add 2 to each side of the equation

10 + 2 = a(4) + - 2 + 2

12 = 4a

Divide each side by 4

12/4 = 4a/4

a = 3

The equation of the parabola is y = 3(x + 4)² + - 2

1 = a(-3 + 4)² + - 2

1 = a(1)² + - 2

1 = a + - 2

a = 3

Example #2

Use the graph of the parabola below to write the equation of the parabola

Looking at the parabola above, the vertex is located at (h, k) = (2, 3)

Substitute h = 2 and k = 3 into the vertex form.

y = a(x - h)² + k

y = a(x - 2)² + 3

Pick any point on the graph and substitute the point into y = a(x - 2)² + 3 in order to find a.

Let us pick (-2, -5)

-5 = a(-2 - 2)² + 3

-5 = a(-4)² + 3

-5 = a(16) + 3

Subtract 3 from each side of the equation

-5 - 3 = a(16) + 3 - 3

-8 = a(16) 

Divide each side by 16

-8/16 = a(16)/16

a = -1/2 = -0.5

The equation of the parabola is y = -0.5(x - 2)² + 3

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