How To Find Equation Of A Parabola

In real-world terms, a parabola is the arc a ball makes when you throw it, or the distinctive shape of a satellite dish. In math terms, a parabola the shape you get when you slice through a solid cone at an angle that's parallel to one of its sides, which is why it's known as one of the "conic sections." The easiest way to find the equation of a parabola is by using your knowledge of a special point, called the vertex, which is located on the parabola itself.

If you see a quadratic equation in two variables, of the form ​**y = ax² + bx + c**​, where a ≠ 0, then congratulations! You've found a parabola. The quadratic equation is sometimes also known as the "standard form" formula of a parabola.

But if you're shown a graph of a parabola (or given a little information about the parabola in text or "word problem" format), you're going to want to write your parabola in what's known as vertex form, which looks like this:

​**y = a(x – h)² + k**​ (if the parabola opens vertically)

​**x = a(y – k)² + h**​ (if the parabola opens horizontally)

In either formula, the coordinates (h,k) represent the vertex of the parabola, which is the point where the parabola's axis of symmetry crosses the line of the parabola itself. Or to put it another way, if you were to fold the parabola in half right down the middle, the vertex would be the "peak" of the parabola, right where it crossed the fold of paper.

If you're being asked to find the equation of a parabola, you'll either be told the vertex of the parabola and at least one other point on it, or you'll be given enough information to figure those out. Once you have this information, you can find the equation of the parabola in three steps.

Let's do an example problem to see how it works. Imagine that you're given a parabola in graph form. You're told that the parabola's vertex is at the point (1,2), that it opens vertically and that another point on the parabola is (3,5). What is the equation of the parabola?

1. Determine Whether It's Horizontal or Vertical

Your very first priority has to be deciding which form of the vertex equation you'll use. Remember, if the parabola opens vertically (which can mean the open side of the U faces up or down), you'll use this equation:

​**y = a(x – h)² + k**​

And if the parabola opens horizontally (which can mean the open side of the U faces right or left), you'll use this equation:

​**x = a(y – k)² + h**​

Because the example parabola opens vertically, let's use the first equation.

2. Substitute in the Vertex

Next, substitute the parabola's vertex coordinates (h, k) into the formula you chose in Step 1. Since you know the vertex is at (1,2), you'll substitute in h = 1 and k = 2, which gives you the following:

​**y = a(x – 1)² + 2**​

3. Use Another Point to Find "a"

The last thing you have to do is find the value of ​a​. To do that choose any point (​x,y​) on the parabola, as long as that point is not the vertex, and substitute it into the equation.

In this case, you've already been given the coordinates for another point on the vertex: (3,5). So you'll substitute in x = 3 and y = 5, which gives you:

​**5 = a(3 – 1)² + 2**​

Now all you have to do is solve that equation for ​a​. A little simplification gets you the following:

​**5 = a(2)² + 2**​, which can be further simplified to:

​5 = a(4) + 2​, which in turn becomes:

​3 = a(4)​, and finally:

​a = 3/4​

Now that you've found the value of ​a​, substitute it into your equation to finish the example:

​**y = (3/4)(x – 1)² + 2**​ is the equation for a parabola with vertex (1,2) and containing the point (3,5).