Expert: Sherry Wallin - 1/31/2009

Question
Hi there,
I have a question which i don't really know how to categorise, I just know that it's hard for me. Here goes:
A parabola with equation y=ax^2+bx+c passes through the points (-1,-10), (2,-1), (3,-6). find the values of a,b and c.
Another similar one is:
A curve passes through the points (-1,-9), (1,-5), (2,-3) and (3,7)>.If the curve has equation y=ax^3+bx^2+cx+d, find the values for a,b,c and d.
sorry, know it's a bit long-winded. Hope you can help!
Ray

Answer
Hi Ray, how's your day going? Mine is fine.
    Remember a point is just an ordered pair (x,y), the first coordinate is just x and the second is just y. So to start, plug in each pair into the equation for the parabola and you will generate 3 equations in a,b,c:

(I will be be labeling equations by numbers and using eqn for equation)
1- a-b+c = -10
2- 4a+2b+c = -1
3- 9a+3b+c = -6

Through the use of substitution or elimination we will change 3 unknowns into 2 unknowns:

subtract eqn 2 from eqn 3 getting rid of c: 5a+b = -5 ( eqn 4)
subtract eqn 1 from eqn 2 getting rid of c: 3a+3b = 9 ( eqn 5)
multiply eqn 4 by -3 and add it to eqn 5 getting rid of b: a = -2

now "back substitute a = -2 into either of eqn 4 or eqn 5 to find b:

b = 5

using any of eqns 1, 2, or 3 put in your new found values for a and b to find c: (eqn 1 looks like a good candidate)
-2 - 5 + c = -10
-7 + c = -10
c = -3

So a = -2, b = 5, c = -3 gives us y = -2x^2+5x-3

To make sure this is correct all three given points should give you a true answer.

The 2nd problem is done similarly only you will have 4 variables that you will need to whittle down to 1 and then back substitute.

Good Luck,
Math Prof