How to find the equation of a parabola given two points

To find the vertex of a parabola, you first need to know how to graph quadratic equations. When graphing these, remember that every quadratic function can be put into a standard form (more on this later). This allows you to find the leading coefficient and solve for the x-intercepts. The x-intercept and y-intercept are points on the graph where the parabola intersects the x-axis or y-axis.

Putting the quadratic function into standard form will also let you find the axis of symmetry, the line that runs through the vertex and divides the parabola in half. You can then find the x-coordinate and y-coordinate of the vertex, which is the highest or lowest point on a parabola.

Definition of a Parabola

A parabola is a set of points that are equal distances from both a focus (a fixed point) and a directrix (a fixed line). It’s the “u” shape that forms when one graphs a quadratic equation or quadratic function.

Depending on the coefficients of the original equation, the parabola opens to the right side, to the left side, upwards, or downwards.

The Axis of Symmetry of a Parabola

Before we find the vertex of a parabola, let’s review the axis of symmetry.

Remember, in a parabola, every point represents an x and a y that solves the quadratic function.

The axis of symmetry is the vertical line that goes through the vertex of a parabola. The vertex of the parabola is the maximum or minimum point on the graph of the quadratic function.

Remember that every quadratic function can be written in the standard form

The equation for the axis of symmetry of a parabola can be expressed as:

Finding the Vertex of the Parabola

To find the coordinates for the vertex of the parabola, you should first use the equation to find the axis of symmetry. Then, substitute the x value that you find back into the original question to get the y-value.

Let’s solve for the axis of symmetry when a = 1 and b = —4.

Now, we know that x = 2. Now we substitute that back into the original quadratic equation.

Solving it gives us y = -1. We now know that the vertex of the parabola is the coordinate (2, -1). Finding the vertex of a parabola couldn’t be easier once you know these steps!

Find the Vertex of a Parabola in No Time

To find the vertex of a parabola, you first need to find x (or y, if your parabola is sideways) through the formula for the axis of symmetry. Then, you’ll use that value to solve for y (or x if your parabola opens to the side) by using the quadratic equation. Those two coordinates are your parabola’s vertex.

More Math Homework Help

• How To Use the Leading Coefficient Test To Graph End Behavior
• One-to-One Functions: The Exceptional Geometry Rule
• How To Master Quadratic Regression
• How To Find the Axis of Symmetry 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.

Recognizing a Parabola Formula

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)

What's the Vertex of the Parabola?

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.

Finding the Equation of a Parabola

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?

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.

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​

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).

Tips

• 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.

How do I find a quadratic equation given 2 points and no vertex?

How do you find the vertex of a parabola given two points?