Find an equation for the parabola that has a vertical axis and passes through the given points. P(2, 13), Q(−2, −11), R(1, 10)

Question

Find an equation for the parabola that has a vertical axis and passes through the given points.

P(2, 13),    Q(−2, −11),    R(1, 10)

Answer 1

The parabola has a vertical axis, which means it is of the form y = a(x-b)^2 + c.

The easiest way to solve: guess the solution is at the integers, note the curve rises much less from 1 to 2 than from -2 to 1, so the axis is either at 2 or, if still rising, at 3, but if it were at 2, the value at -2 would be much less. Let's try b = 3. Then, c would have to be 14 and a would be -1. Does this work?

If f(x) = -(x-3)^2 + 14

f(2) = -(2-3)^2 + 14 = -1 + 14 = 13

f(1) = -(1-3)^2 + 14 = -4 + 14 = 10

f(-2) = -(-2-3)^2 + 14 = -25 + 14 = -11

Yep, our guess method worked.

We may write y=-(x-3)^2 + 14 as -(x^2-6x+9) + 14 = -x^2 + 6x + 5

Now, for the method you're supposed to use.

Write f(x) = ax^2 + bx + c

Then, we get,

f(-2) = 4a - 2b + c = -11

f(1) = a + b + c = 10

f(2) = 4a + 2b + c = 13

I leave it to you to solve these 3 simultaneous equations. You will get a solution of a = -1, b = 6, and c = 5

Guessing and checking was sure easier this time!