Lesson Objectives

- Learn how to find the equation of a parabola

## How to Find the Equation of a Parabola Given Three Points

In some cases, we may be asked to find the equation of a parabola given three points on the parabola. To accomplish this task, we set up and solve a system of equations with three equations and three unknowns. Let's look at an example.

Example #1: Find the equation of the parabola that goes through the given points. $$y=ax^2 + bx + c$$ $$(0, 4)$$ $$(2, -2)$$ $$(3, 10)$$ We need to find a, b, and c. These are the three unknowns. Let's set up a system of equations. For each equation, we can plug in for x and y:

Plug in a 0 for x and a 4 for y: $$4=c$$ Plug in a 2 for x and a -2 for y: $$-2=4a + 2b + c$$ Plug in a 3 for x and a 10 for y: $$10=9a + 3b + c$$ Now that we have our system, let's solve for a, b, and c:

Since c = 4, we can plug into the bottom two equations: $$-2=4a + 2b + 4$$ $$10=9a + 3b + 4$$ If we solve this system, we end up with: $$a=5, b=-13, c=4$$ Which gives us a parabola: $$y=5x^2 - 13x + 4$$

Example #1: Find the equation of the parabola that goes through the given points. $$y=ax^2 + bx + c$$ $$(0, 4)$$ $$(2, -2)$$ $$(3, 10)$$ We need to find a, b, and c. These are the three unknowns. Let's set up a system of equations. For each equation, we can plug in for x and y:

Plug in a 0 for x and a 4 for y: $$4=c$$ Plug in a 2 for x and a -2 for y: $$-2=4a + 2b + c$$ Plug in a 3 for x and a 10 for y: $$10=9a + 3b + c$$ Now that we have our system, let's solve for a, b, and c:

Since c = 4, we can plug into the bottom two equations: $$-2=4a + 2b + 4$$ $$10=9a + 3b + 4$$ If we solve this system, we end up with: $$a=5, b=-13, c=4$$ Which gives us a parabola: $$y=5x^2 - 13x + 4$$

#### Skills Check:

Example #1

Find the equation of the parabola. $$y=ax^2 + bx + c$$ $$(2, 10)$$ $$\left(1, \frac{9}{2}\right)$$ $$\left(-1, \frac{11}{2}\right)$$

Please choose the best answer.

A

$$y=3x^2 + 7x + 15$$

B

$$y=2x^2 - \frac{1}{2}x + 3$$

C

$$y=9x^2 + 5x + 7$$

D

$$y=x^2 + \frac{1}{5}x + 3$$

E

$$y=\frac{1}{3}x^2 + 2x + 7$$

Example #2

Find the equation of the parabola. $$y=ax^2 + bx + c$$ $$(0, -2)$$ $$(2, 6)$$ $$(-8, 6)$$

Please choose the best answer.

A

$$y=x^2 + \frac{3}{2}x - 2$$

B

$$y=8x^2 + x + 3$$

C

$$y=6x^2 + 11x + 4$$

D

$$y=3x^2 + 8x + 1$$

E

$$y=\frac{1}{2}x^2 + 3x - 2$$

Example #3

Find the equation of the parabola. $$x=ay^2 + by + c$$ $$(1, 0)$$ $$(4, 1)$$ $$(8, -1)$$

Please choose the best answer.

A

$$x=y^2 + 3y + 8$$

B

$$x=y^2 + 4y + 1$$

C

$$x=3y^2 + 8y + 4$$

D

$$x=5y^2 - 2y + 1$$

E

$$x=y^2 + \frac{3}{2}y + 5$$

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