Vertex: Definition & Function

Instructor: Kimberlee Davison

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

Many shapes have a vertex. On a parabola, the vertex is the maximum or minimum point. Find out how to find the vertex using either a graph or an equation.

Definition of the Vertex

The vertex of a parabola is the low or high point of the curve, sometimes called the maximum or minimum.

Suppose you are playing catch with your football in the living room. As you throw the ball across the room to your mother, you notice the ball makes a nice arc shape as it rises and then gently falls into her arms. Like all objects that are thrown or launched at an upward angle, the object travels both horizontally toward the target (your mother, in this case) and vertically (upward until gravity pulls it back downward).

After your last toss of the ball, you worry that maybe you threw it a little too high - it looks like it is going to scrape the ceiling you spent the morning freshly painting. Being a fan of all things mathematical, you wonder whether it would be possible to map the path your football follows and even calculate the ball's maximum height.

The long, arching path your ball travels is a parabola, and it has some very specific mathematical requirements. One of those is that it has a clear maximum or minimum, called the vertex. In figure 1, each of the parabolas has a vertex.

Figure 1
Several parabolas

Finding the Vertex

If you have the graph of your parabola, you can sometimes find the vertex by inspection. In other words, find the minimum or maximum of the curve using the grid on the graph. In figure 1, the vertex of the purple parabola is at the point (1,2).

Using the Equation of the Parabola

The equation, or function, of a parabola can be written in standard form, as shown here:

equation of parabola

In this equation, the a tells you how steep the parabola is. The h and k tell you where the vertex is. For example, suppose you have the following equation:

Equation of a parabola

The vertex is at the point (1,4). Be careful, because the '-1' inside the parentheses may make you think the vertex is really at (-1,4).

Sometimes, however, the equation isn't written so nicely for you. Instead of being in standard form, it may look more like this:


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