   SEARCH HOME Math Central Quandaries & Queries  Question from Kenzie, a parent: The graph shows an arrow going upward crossing at the -2 on the x line and crossing the 3 on the x line and the vertex on the -6 on the y line. The answer is y=x^2 -x-6 I've tried 0 in place of x and y for -6 and -2 and 3 for x! Please help, Thanks Hi Kenzie.

It sounds like you are describing a parabola that opens upward, and indeed that is what the equation is.

Here's how I would solve this problem.

1. I look at its shape and recognize it as a parabola. That means that I can write it in the following form:

y = A (x - B) (x - C)

If I can replace B, C and A with numbers, I will have the complete equation for the parabola.

2. The graph crosses the x axis when x = -2. So at that point, y = 0. That means one of the factors A, (x - B) or (x - C) must be zero. If A is zero, then y is always zero, so that's not it. So let's say x - B = 0. Since at this point, x = -2, we can write -2 - B = 0. Therefore B = -2.

3. Using the same logic, x - C = 0 when x = 3. So 3 - C = 0 and that means C = 3.

Now I have y = A (x - -2) (x - 3) which is y = A (x + 2) (x - 3).

4. I also know that the parabola passes through the vertex, whose co-ordinates are (0, -6). That means y = -6 when x = 0. So I'll substitute this into the equation:

y = A (x + 2) (x - 3)
-6 = A (0 + 2) (0 - 3)
-6 = A (-6)
A = 1

Now I have the equation of the parabola, and I can multiply it out:

y = 1 (x + 2) (x - 3)
y = x2 + 2x - 3x - 6
y = x2 - x - 6

I hope this demonstration helps,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.