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# How do I find the equation in standard form of the parabola passing through the points (2, 20), (-2,-4), and (0,8)?

I've tried to figure it out by substituting the x and y values into an equation (y=ax2+bx+c) and writing a system of equations, but I got confused.

### 3 Answers by Expert Tutors

Vivian L. | Microsoft Word/Excel/Outlook, essay composition, math; I LOVE TO TEACHMicrosoft Word/Excel/Outlook, essay comp...
3.0 3.0 (1 lesson ratings) (1)
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Hi Lillian;

y=ax2+bx+c
Always begin with the zero...(0,8)...Let's plug in...
8=a(0)2+b(0)+c
8=c
y=ax2+bx+8
Let's plug-in one set of coordinates...I randomly select (2,20)...
20=a(2)2+b(2)+8
20=4a+2b+8
12=4a+2b
Let's plug-in the other set of coordinates...(-2,-4)
-4=a(-2)2-2b+8
-12=4a-2b
Let's add the two above equations together...
0=8a
0=a
Let's subtract the two equations from each other...
24=0+4b
24=4b
6=b
y=6x+8
Kirill Z. | Physics, math tutor with great knowledge and teaching skillsPhysics, math tutor with great knowledge...
4.9 4.9 (174 lesson ratings) (174)
1
You have three unknowns: a, b, and c, as parabola coefficients. You have three points, which lie on the parabola. Plug in their coordinates to obtain three equations containing a,b, and c.

1) point (2,20)

a*22+b*2+c=20 or
4a+2b+c=20

2) point (-2, -4)
a*(-2)2-2b+c=-4 or
4a-2b+c=-4

3) point (0,8)

a*02+b*0+c=8 or

c=8

So the final system looks as follows:

4a+2b+c=20
4a-2b+c=-4
c=8

Since c is determined by the third equation already, plug its value into the first and the second equation.
We obtain:
4a+2b+8=20
4a-2b+8=-4

4a+2b=12
4a-2b=-12

Add two equations together, to obtain 8a=0; a=0; then 2b=12 or b=6.

Equation of parabola: y=6x+8.

So parabola in your case is degenerate and all three points lie on a straight line.
Robert J. | Certified High School AP Calculus and Physics TeacherCertified High School AP Calculus and Ph...
4.6 4.6 (13 lesson ratings) (13)
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y = ax2+bx+c
Plug in (0, 8),
8 = c
Plug in (2, 20), (-2,-4),
20 = 4a+2b+c ......(1)
-4 = 4a-2b+c ......(2)
(1)-(2): 24 = 4b
b = 6
From (2),
a = (1/4)(2b-c-4) = (1/4)(2*6-8-4) = 0
Answer: y = 6x+8 (This is a straight line.)