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### Equation of a Parabola

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Date: 12/20/2001 at 17:22:18
From: Richard DeVries
Subject: Equation of a parabola

Given several points, how do you determine the equation? Assuming
that the points appear to be a parabola, how do you approximate the
equation that would give the similar graph? Is there a way to be
exact?
```

```
Date: 12/21/2001 at 05:15:27
From: Doctor Jeremiah
Subject: Re: Equation of a parabola

Hi Richard,

The equation for a parabola is y = ax^2 + bx + c

So if you have three points, you can calculate the equation of the
parabola. But you need three points.

Let's say the points are (-4,15) (2,3) and (1,-5). Notice that these
points have an x value and a y value, and the equation has a place
for x and a place for y.

So for each point you could plug in the the x and y values.

(-4,15): y = ax^2 + bx + c   <===   x=-4, y=15
15 = a(-4)^2 + b(-4) + c
15 = 16a - 4b + c

(2,3):  y = ax^2 + bx + c   <===   x=2, y=3
3 = a(2)^2 + b(2) + c
3 = 4a + 2b + c

(1,-5):  y = ax^2 + bx + c   <===   x=1, y=-5
-5 = a(1)^2 + b(1) + c
-5 = a + b + c

Now we have these three equations:

15 = 16a - 4b + c
3 = 4a + 2b + c
-5 = a + b + c

There are three equations and three unknowns, so if we can solve these
equations for a, b, and c, we will be able to turn y = ax^2 + bx + c
into the equation for the graph.

Step 1:  pick an equation at random and solve for one unknown

-5 = a + b + c
c = -5 - a - b

Step 2:  plug that into both of the other equations

15 = 16a - 4b + c   <===   c = -5 - a - b
15 = 16a - 4b + (-5 - a - b)
15 = 15a - 5b - 5
20 = 15a - 5b

3 = 4a + 2b + c    <===   c = -5 - a - b
3 = 4a + 2b + (-5 - a - b)
3 = 3a + b - 5
8 = 3a + b

Now we have only two equations with two unknowns:

20 = 15a - 5b
8 = 3a + b

Step 3:  pick an equation at random and solve for one unknown

8 = 3a + b
b = 8 - 3a

Step 4:  plug that into the other equation

20 = 15a - 5b   <===   b = 8 - 3a
20 = 15a - 5(8 - 3a)
20 = 15a - 40 + 15a
60 = 15a + 15a
60 = 30a
60/30 = a
a = 2

Step 4:  plug that into the answer for step 3

b = 8 - 3a   <===   a=2
b = 8 - 3(2)
b = 2

Step 5:  plug both of those into the answer from Step 1

c = -5 - a - b   <===   a=2, b=2
c = -5 - (2) - (2)
c = -9

The equation for a parabola is y = ax^2 + bx + c and we have values
for a, b, and c, so:

y = ax^2 + bx + c   <===   a=2, b=2, c=-9
y = 2x^2 + 2x - 9

And that's the equation of our parabola.

- Doctor Jeremiah, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Conic Sections/Circles
High School Equations, Graphs, Translations
High School Geometry

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