How to Find the Vertex of a Quadratic Equation

Steps to Solve

We are going to learn how to find the vertex of a quadratic equation. As you may know, the graph of a quadratic equation, y = ax2 + bx + c, is the shape of a parabola. A parabola looks like a U or an upside-down U. The vertex of a quadratic equation is the maximum or minimum point on the equation’s parabola.

See why this point is so important? It’s always incredibly useful to know the maximum or minimum value of an equation, especially when that equation represents real world phenomena. What’s really nice about the vertex of a quadratic equation is that it’s pretty easy to find. To do so, we use the following steps:

1. Get the equation in the form y = ax2 + bx + c.
2. Calculate –b / 2a. This is the x-coordinate of the vertex.
3. To find the y-coordinate of the vertex, simply plug the value of –b / 2a into the equation for x and solve for y. This is the y-coordinate of the vertex.

After following these steps, you have your x and y coordinates of the vertex, so you have your vertex. Not too bad, huh? We see that the vertex of a quadratic equation, y = ax2 + bx + c, is the following point.

(-b / 2a, a(-b / 2a)2 + b(-b / 2a) + c)

Solution

To find the vertex of a quadratic equation, y = ax2 + bx + c, we find the point (-b / 2a, a(-b / 2a)2 + b(-b / 2a) + c), by following these steps.

1. Get the equation in the form y = ax2 + bx + c.
2. Calculate –b / 2a. This is the x-coordinate of the vertex.
3. To find the y-coordinate of the vertex, simply plug the value of –b / 2a into the equation for x and solve for y. This is the y-coordinate of the vertex.

Application

We saw that this process doesn’t look so bad, so let’s go ahead and give it a go with an application. Suppose you and a few of your friends decide to open a math tutoring business. Taking time and travel expenses into account, your business’s monthly profit can be modeled by the quadratic equation:

y = -0.7x2 + 140x

where y is the profit when you sell x tutoring packages. We see that this is the quadratic equation, and its graph is shown here:

Now, when it comes to the profit of your business, what is the number one question you want to know about it? It’s what is the maximum amount of profit you can make, and how many tutoring packages do you have to sell to get that maximum profit?