I am trying to find the equation of a parabola using standard form. It's not going so well. My points (-1, 9) (5,7)(-6,6)

Question

My points (-1, 9) (5,7)(-6,6).

Answer 1

For a parabola, the standard form is as follows: y = Ax² + Bx + C, where A, B, C are coefficients to be determined. Since we have three unknown coefficients, we need three points to describe our parabola. We are given exactly three points, from which we can determine the values of our coefficients using a system of equations after plugging in the provided values for x and y. The equations for our system are as follows:

 

1. (9) = A(-1)² + B(-1) + C

2. (7) = A(5)² + B(5) + C

3. (6) = A(-6)² + B(-6) + C

 

When evaluating, this simplifies to: 

 

1. 9 = A - B + C

2. 7 = 25A + 5B + C

3. 6 = 36A -6B + C

 

You can now determine the coefficients by solving the system of equations, using your favorite method (elimination, substitution, matrix algebra, etc.) 

 

I prefer the matrix algebra method for more than 2 equations with numbers that I suspect look ugly. (Here, I assume you have a graphing calculator and have been exposed to basic matrix algebra to be able to solve this system quickly, usually taught at the Algebra II level). Here I describe how to do this method so that this can be applied to any similar questions you may have. Under the matrix section of your graphing calculator, you can construct the following 3x4 matrix to represent the system:

 

1    -1   1    9

25   5   1    7

36  -6   1    6

 

Using the rref command (reduced row echelon form) on our matrix, we obtain the matrix 

 

1   0   0  -14/165

0   1   0    1/ 165

0   0   1   100/11 

 

Thus,

A = -14/165

B = 1/165

C = 100/11

 

Substituting the coefficients,

y = -14/165x² + 1/165x + 100/11 

 

In order to reduce into standard form, we also do not want fractions. We can multiply the entire equation by 165 in order to get rid of all fractions (since 165 is perfectly divisible by 11).

 

Therefore, the standard form equation is:

 

165y = -14x² + x + 1500