Updated November 19, 2018
By Lisa Maloney
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?
With all those letters and numbers floating around, it can be hard to know when you're "done" finding a formula! As a general rule, when you're working with problems in two dimensions, you're done when you have only two variables left. These variables are usually written as x and y, especially when you're dealing with "standardized" shapes such as a parabola.
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:
And if the parabola opens horizontally (which can mean the open side of the U faces right or left), you'll use this equation:
Because the example parabola opens vertically, let's use the first equation.
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:
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:
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:
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).
Tips
This calculator will find either the equation of the parabola from the given parameters or the vertex, focus, directrix, axis of symmetry, latus rectum, length of the latus rectum, focal parameter, focal length (distance), eccentricity, x-intercepts, y-intercepts, domain, and range of the entered parabola. Also, it will graph the parabola. Steps are available.
Related calculators: Circle Calculator, Ellipse Calculator, Hyperbola Calculator, Conic Section Calculator
Your Input
Find the vertex, focus, directrix, axis of symmetry, latus rectum, length of the latus rectum, focal parameter, focal length, eccentricity, x-intercepts, y-intercepts, domain, and range of the parabola $$$y = \left(x - 2\right)^{2} + 5$$$.
Solution
The equation of a parabola is $$$y = \frac{1}{4 \left(f - k\right)} \left(x - h\right)^{2} + k$$$, where $$$\left(h, k\right)$$$ is the vertex and $$$\left(h, f\right)$$$ is the focus.
Our parabola in this form is $$$y = \frac{1}{4 \left(\frac{21}{4} - 5\right)} \left(x - 2\right)^{2} + 5$$$.
Thus, $$$h = 2$$$, $$$k = 5$$$, $$$f = \frac{21}{4}$$$.
The standard form is $$$y = x^{2} - 4 x + 9$$$.
The general form is $$$x^{2} - 4 x - y + 9 = 0$$$.
The vertex form is $$$y = \left(x - 2\right)^{2} + 5$$$.
The directrix is $$$y = d$$$.
To find $$$d$$$, use the fact that the distance from the focus to the vertex is the same as the distance from the vertex to the directrix: $$$5 - \frac{21}{4} = d - 5$$$.
Thus, the directrix is $$$y = \frac{19}{4}$$$.
The axis of symmetry is the line perpendicular to the directrix that passes through the vertex and the focus: $$$x = 2$$$.
The focal length is the distance between the focus and the vertex: $$$\frac{1}{4}$$$.
The focal parameter is the distance between the focus and the directrix: $$$\frac{1}{2}$$$.
The latus rectum is parallel to the directrix and passes through the focus: $$$y = \frac{21}{4}$$$.
The length of the latus rectum is four times the distance between the vertex and the focus: $$$1$$$.
The eccentricity of a parabola is always $$$1$$$.
The x-intercepts can be found by setting $$$y = 0$$$ in the equation and solving for $$$x$$$ (for steps, see intercepts calculator).
Since there are no real solutions, there are no x-intercepts.
The y-intercepts can be found by setting $$$x = 0$$$ in the equation and solving for $$$y$$$: (for steps, see intercepts calculator).
y-intercept: $$$\left(0, 9\right)$$$.
Answer
Standard form: $$$y = x^{2} - 4 x + 9$$$A.
General form: $$$x^{2} - 4 x - y + 9 = 0$$$A.
Vertex form: $$$y = \left(x - 2\right)^{2} + 5$$$A.
Focus-directrix form: $$$\left(x - 2\right)^{2} + \left(y - \frac{21}{4}\right)^{2} = \left(y - \frac{19}{4}\right)^{2}$$$A.
Graph: see the graphing calculator.
Vertex: $$$\left(2, 5\right)$$$A.
Focus: $$$\left(2, \frac{21}{4}\right) = \left(2, 5.25\right)$$$A.
Directrix: $$$y = \frac{19}{4} = 4.75$$$A.
Axis of symmetry: $$$x = 2$$$A.
Latus rectum: $$$y = \frac{21}{4} = 5.25$$$A.
Length of the latus rectum: $$$1$$$A.
Focal parameter: $$$\frac{1}{2} = 0.5$$$A.
Focal length: $$$\frac{1}{4} = 0.25$$$A.
Eccentricity: $$$1$$$A.
x-intercepts: no x-intercepts.
y-intercept: $$$\left(0, 9\right)$$$A.
Domain: $$$\left(-\infty, \infty\right)$$$A.
Range: $$$\left[5, \infty\right)$$$A.