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2 Tutorials that teach The Vertex Formula

# The Vertex Formula

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Author: Sophia Tutorial
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In this lesson, students will learn how to use a quadratic equation to determine the points of the vertex on a graph of the equation.

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Tutorial
This tutorial covers how to find the vertex of a parabola, using the vertex formula, through the definition and discussion of:
1. Vertex
2. The Vertex Formula
3. Finding the Vertex

## 1. Vertex

A parabola is the term used for a graph of a quadratic equation, which can generally be written as:

If the coefficient a is positive, the graph is a U shape pointing up. If the coefficient a is negative, the graph is a U shape pointing down. A parabola will have a maximum point if the graph is pointing down or a minimum point if the graph is pointing up. The maximum or minimum point of a parabola is called the vertex.

The vertex provides useful information about the highest or lowest value in a quadratic relationship. These highest or lowest values are related to optimization problems, such as maximum profit or minimum cost.

## 2. The Vertex Formula

You can use the equation for a parabola to find the vertex without looking at the graph. When doing so, it’s important to remember the following points about a parabola:

• A parabola is the graph of a quadratic equation in the form:
• When the coefficient a is positive, the parabola will be upward-facing. When the coefficient a is negative, the parabola will be downward-facing.
• The vertex is the maximum or minimum point on the parabola.

The x-coordinate of the point of the vertex can be found using the following formula, in which a and b are the coefficients in your quadratic equation.

x-coordinate of Vertex

Once you have used the formula to find the x-coordinate of the vertex, you can substitute this value in for x in the equation to determine the y-coordinate of the vertex.

## 3. Finding the Vertex

Suppose an Olympic diver is competing for a medal. Her dive can be modeled by the equation and graph shown below, where x is the time in seconds after she begins the dive and y is the height in feet above the water. In this example, the vertex of the parabola is a maximum point, so by finding the vertex using the vertex formula, you can determine the maximum height of her dive.

First, you take your values for a and b from your equation, and substitute them into the formula for the x coordinate of the vertex (the vertex formula).

This gives us:

Multiply and divide the values accordingly to provide x equals 2, so the x-coordinate of your vertex is 2.

Next, substitute 2 into your original equation for x to find the y-value of the vertex.

Simplify with your exponent, then multiply and add, according to the order of operations.

The y-coordinate of your vertex is 28, and you can see that your vertex is at the point 2, 28. This means that 2 seconds after she begins the dive, she reaches her maximum height of 28 feet above the water.

### IN CONTEXT

Natasha kicks a soccer ball during a game. How can you determine the maximum height of the ball? Well, the flight of the ball can be modeled by the equation and graph shown below, where x is the time in seconds after she kicks the ball and y is the height of the ball in feet. Note that the vertex is a maximum point.

You can start by using your formula for the x-coordinate of the vertex and substituting your values in for b and a.

Simplify the denominator and divide these values to provide the value of the x-coordinate of the vertex, which is 1.

Next, substitute 1 in for x in your original equation to find the y-coordinate of the vertex.