SOLUTION: the coordinates of two point on a parabola are (7, 4) and (4, 3). Another point on the parabola is (8, 24). find the equation of the parabola F(x) = a(x-h)^2 + k.

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: the coordinates of two point on a parabola are (7, 4) and (4, 3). Another point on the parabola is (8, 24). find the equation of the parabola F(x) = a(x-h)^2 + k.      Log On


   

Question 1181358: the coordinates of two point on a parabola are (7, 4) and (4, 3). Another point on the parabola is (8, 24). find the equation of the parabola F(x) = a(x-h)^2 + k.
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

F%28x%29+=+a%28x-h%29%5E2+%2B+k+
given:
(7, 4) and (4, 3)
(8, 24)

use :
(7, 4)

4+=+a%287-h%29%5E2+%2B+k+...solve for k
4+-a%287-h%29%5E2=+k+............eq.1

use :(4, 3)
3+=+a%284-h%29%5E2+%2B+k+...solve for k
3+-+a%284-h%29%5E2+=+k+.........eq.2

use : (8, 24)
24+=+a%288-h%29%5E2+%2B+k+...solve for k
24+-+a%288-h%29%5E2+=k+..........eq.3

from eq.1 and eq.2 we have
4+-a%287-h%29%5E2=3+-+a%284-h%29%5E2+.....solve for a
4+-3=a%287-h%29%5E2+-+a%284-h%29%5E2+
1=a%28%287-h%29%5E2+-+%284-h%29%5E2%29+
a=1%2F%2833+-+6h%29+...........eq.1)


from eq.1 and eq.3 we have
4+-a%287-h%29%5E2=24+-+a%288-h%29%5E2.....solve for a
4+-24=a%287-h%29%5E2+-+a%288-h%29%5E2
+-20=a%28%287-h%29%5E2+-+%288-h%29%5E2%29
+-20=a%282h+-+15%29
+a=-20%2F%282h+-+15%29............eq.2)


from eq.1) and eq.2) we have
1%2F%2833+-+6h%29+=-20%2F%282h+-+15%29......solve for h
1%282h+-+15%29+=-20%2833+-+6h%29
2h+-+15+=-660+%2B120h
660-+15+=+120h-2h
645+=+118h
highlight%28h=645%2F118%29


go to
a=1%2F%2833+-+6h%29+...........eq.1), substitute h
a=1%2F%2833+-+6%28645%2F118%29%29+
a=1%2F%2812%2F59%29+
highlight%28a=59%2F12%29+

go to
3+-+a%284-h%29%5E2+=+k+.........eq.2, substitute a and h
3+-+%2859%2F12+%29%284-645%2F118%29%5E2+=+k+
k=3+-+%2859%2F12+%29%2829929%2F13924%29++
k=3+-+29929%2F2832++
highlight%28k=-21433%2F2832+%29+


so, your equation is:

F%28x%29+=+%2859%2F12%29%28x-645%2F118%29%5E2+-21433%2F2832


Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The process for doing this is straightforward; but the numbers are really ugly.

You won't learn anything by having us do all the ugly work for you.

The three points satisfy the quadratic equation

y+=+a%28x-h%29%5E2%2Bk

Substitute the coordinates of the three points into the equation to get three equations in the variables a, h, and k; then solve those three equations to find the values of those variables.

The work might (or might not) be easier if you use the standard form of the quadratic

y+=+ax%5E2%2Bbx%2Bc

with the coordinates of the three given points to get the three equations you need; but that might end up being more work in the end because then you would have to convert the equation in that form to the form required for the answer. That conversion of form would be relatively easy if the numbers in the problem were "nice"... but they are not.