SOLUTION: Find the quadratic function {{{f(x)=ax^2+bx+c}}} for which f(1)=6 , f(-2)=18 , and f(2)=6. I need help with this one. I am having a hard time figuring it out!

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons -> SOLUTION: Find the quadratic function {{{f(x)=ax^2+bx+c}}} for which f(1)=6 , f(-2)=18 , and f(2)=6. I need help with this one. I am having a hard time figuring it out!       Log On


   

Question 675648: Find the quadratic function f%28x%29=ax%5E2%2Bbx%2Bc for which f(1)=6 , f(-2)=18 , and f(2)=6. I need help with this one. I am having a hard time figuring it out!
Found 2 solutions by stanbon, MathLover1:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Find the quadratic function f%28x%29=ax%5E2%2Bbx%2Bc for which
You have 3 points. Substitute into the quadratic form and solve the system:
----------------------------------
f(1)=6 ::: a + b + c = 6
f(-2)=18::4a -2b + c = 18
f(2)=6::::4a +2b + c = 6
-------
I used a matrix method on a TI-84 to get:
a = 1
b = -3
c = 1
==========
Equation:
y = x^2 -3x + 1
====================
Cheers,
Stan H.

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=ax%5E2%2Bbx%2Bc
for f(1)=6
6=a%2A1%5E2%2Bb%2A1%2Bc
6=a%2Bb%2Bc..............1
f(-2)=18
18=a%2A%28-2%29%5E2%2Bb%28-2%29%2Bc
18=4a-2b%2Bc..............2
and f(2)=6
6=a%2A2%5E2%2Bb%2A2%2Bc
6=4a%2B2b%2Bc.................3
solve this system:
6=a%2Bb%2Bc..............1
18=4a-2b%2Bc..............2
6=4a%2B2b%2Bc.................3
______________________________
6=a%2Bb%2Bc..............1...solve for a
a=6-b-c........substitute in 2

18=4%286-b-c%29-2b%2Bc..............2
18=24-4b-4c-2b%2Bc
18=24-6b-3c....both sides divide by 3
6=8-2b-c.....solve for c
c=8-2b-6
c=2-2b..........substitute cin a=6-b-c
a=6-b-%282-2b%29
a=6-b-2%2B2b
a=4%2Bb

substitute c and a in 3

6=4a%2B2b%2Bc.................3
6=4%284%2Bb%29%2Bcross%282b%29%2B2-cross%282b%29.....solve for b
6=16%2B4b%2B2
6=18%2B4b
6-18=4b
-12=4b
highlight%28b=-3%29
now find a
a=4%2Bb
a=4-3
highlight%28a=1%29
and c
c=2-2b
c=2-2%28-3%29
c=2%2B6
highlight%28c=8%29

so. your function is:
f%28x%29=x%5E2-3x%2B8

+graph%28+600%2C+600%2C+-6%2C+10%2C+-5%2C+15%2C+x%5E2-3x%2B8%29+

let's see if all given points lie on this parabola

points are:f(1)=6 , f(-2)=18 , and f(2)=6, or (1,6),(2,6),(-2,18)