Question

The equation for a parabola has the form y = ax^2 + bz + c, where a, b, and c are constants and aメ0. Find an equation for the parabola that passes through the points (-1,14), (2,-7), and (5, 8) Answer: y-

Answer 1

a = 2, b = -9, c = 3

Explanation:

Replacing x, y values of the points in the equation y = a*x^2 + b*x +c give the following:

(-1,14)

14 = a*(-1)^2 + b*(-1) + c

(2,-7)

-7 = a*2^2 + b*2 + c

(5, 8)

8 = a*5^2 + b*5 + c

Rearranging:

a - b + c = 14

4*a + 2*b + c = -7

25*a + 5*b + c = 8

This is a linear system of equations with 3 equations and 3 unknows. In matrix notation the system is A*x = b whith:

A =

1 -1 1

4 2 1

25 5 1

x =

a

b

c

b =

14

-7

8

Solving A*x = b gives x = Inv(A)*b, where Inv(A) is the inverse matrix of A. From calculation software (I used Excel) you get:

inv(A) =

0.055555556 -0.111111111 0.055555556

-0.388888889 0.444444444 -0.055555556

0.555555556 0.555555556 -0.111111111

inv(A)*b

2

-9

3

So, a = 2, b = -9, c = 3