There is a unique parabola that goes through the points

Question

P = (−1, 1), Q = (1, 1), and R = (5, 6) that can be described by y = ax^2 + bx + c. Find coefficients a, b, and c.

Answer 1

I find the easiest way to find the unique parabola is to plug in the values to find 3 unique equations.

(-1,1) gives

1 = a - b + c

(1,1) gives

1 = a + b + c

(5,6) gives

6 = 25a + 5b + c

Simultaneously solve these equations

looking at the first two you can see

1 = a + b + c

1 = a - b + c

This is only true is b = 0 since everything else is the same.

Now we can use this to solve for the other two variables

6 = 25a + 5* 0 + c Will be our new third equation.

Solve 1 = a + c

and 6 = 25a + c

Since c = 1 - a

substitute that into the second equations to get

6 = 25a + 1 - a

Thus a = 5/24

Since a + c = 1 c = 19/24

In the end

a = 5/24

b = 0

c = 19/24

A parabola must be described by three points.