how do you find the equation of a parabola?

Question

its for an arch that one poing its at (0,0) and the other is (162,0)

Answer 1

The equation for the arch is y = a*(x^2 - 162x).

The best way of solving it is that this parabola is an arch, therefore it's central axis is vertical. Such parabolas have the equation y = quadratic in x. (If the axis were horizontal, then it would be x = quad in y). Therefore, our arch equation would be:

y = ax^2 + bx + c

Since it passes through origin (0,0), substitute the point coordinates:

0 = a(0^2) + b(0) + c = c

thus, we have easily eliminated 'c' & our equation reduces to:

y =ax^2 + bx

Now, as it passes through the other point (162,0), substitute the coordinates:

0 = a(162^2) + b(162)
a(162) + b = 0
b = -a*162

Thus, we have also eliminated 'b', resulting in:

y = a(x^2) - 162ax

This is the equation of our arch. The value of the arbitrary constant 'a' is decided by the height of your arch. So, for your question, you can have infinite number of possible arches passng between those two points.

Answer 2

Do you mean:

Find the equation of a parabola that passes through the points (0,0) and (162,0)

There are an infinite number of solutions.

The equation of a parabola is y = A * x^2 + B * x + C
Therefore substituting the first point 0 = A * 0 + B * 0 + C
Therefore C = 0
Substituting the second point
0 = A * 162 ^ 2 + B * 162 = 162 * (A * 162 + B)
Divide by 162 and you have
0 = 162 * A + B. Rearrange,
A = (-B) / 162

That's the best you can get for a solution. Choose any number you like, substitute for A in the above equation, work out B, and you have a parabola which passes through the two points.

For instance, pick A = 10
Then B = -1620
One parabola that passes through the two points is thus
y = 10 * x^2 - 1620 * x

Note that an arch is not a parabola. A parabolic arch would fall down because of the stress patterns in it. An arch (for instance, the Macdonald's "golden arches") is a "catenary curve."

Answer 3

You know that the x-coordinate of the vertex must be the midpoint of the two, since the two points have the same y-coordinate. I'm pretty sure that's the only restriction to the problem. If you have a third point or some other limitation you could get a definite parabola. Otherwise, there are infinitely many.

Answer 4

This website looks pretty helpful:

"Three points uniquely determine one parabola with directrix parallel to the -axis and one with directrix parallel to the -axis. If these parabola pass through the three points , , and , they are given by equations"

Answer 5

you need three points on the locus to set an equation for any two dimensional two degree curve.