### Online College Courses for Credit

#### FREE EDUCATIONAL RESOURCES PROVIDED by SOPHIA

##### Are you a student?
Free Professional Development
+
2 Tutorials that teach Identifying Points on Parabola

# Identifying Points on Parabola

##### Rating:
(0)
Author: Sophia Tutorial
##### Description:

In this lesson, students will learn how to determine the vertex and solutions of a quadratic equation when given its parabolic graph.

(more)

Sophia’s self-paced online courses are a great way to save time and money as you earn credits eligible for transfer to many different colleges and universities.*

No credit card required

29 Sophia partners guarantee credit transfer.

312 Institutions have accepted or given pre-approval for credit transfer.

* The American Council on Education's College Credit Recommendation Service (ACE Credit®) has evaluated and recommended college credit for 27 of Sophia’s online courses. Many different colleges and universities consider ACE CREDIT recommendations in determining the applicability to their course and degree programs.

Tutorial
This tutorial covers how to identify points on a parabola, through the definition and discussion of:
1. Parabolas
2. Vertex of a Parabola
3. x-Intercepts of a Parabola
4. Solutions to a Quadratic Equation on a Graph

## 1. Parabolas

A parabola is the shape of the graph of a quadratic equation. Parabolas have a general U shape to them, either opening up or opening down. In the general quadratic equation below, the coefficient a determines the upward or downward shape of the parabola.

• When the coefficient a is positive, the parabola will open upward.
• When the coefficient a is negative, the parabola will open downward.

Even though we don’t know the equation to the graphs above, you can see if the value of a is positive or negative. Graphing parabolas can be used to model the path of objects in motion, solve problems involving area, and solve optimization problems.

## 2. Vertex of a Parabola

Every parabola has either a low point or a high point on the graph called the vertex. In the graph below, the parabola has a low, or minimum, point. The minimum point has the lowest y-value on the parabola. The second parabola below has a high, or maximum, point, which has the highest y-value on the parabola.

Vertex (of a Parabola)
The maximum or minimum point of a parabola
Remember, when looking at the equation of a quadratic graph, if the a coefficient of a quadratic equation is positive, the parabola opens upward and the vertex is a minimum point. If the a coefficient is negative, the parabola opens downward and the vertex is a maximum point.

## 3. x-Intercepts of a Parabola

The x-intercept of a graph is a point where that graph intersects the x-axis and when y equals 0. The y-intercept of a graph is a point where the graph intersects the y-axis and when x equals 0. On a parabola, the x-intercepts are the x-values that make y equal to 0, and they also correspond to the solutions of the quadratic equation.

For example, in the graph above, how can you solve the following quadratic equation?

You solve the equation by graphing the equation below and identifying the x-intercepts of the graph.

You can see in the graph above that your x-intercepts are (3, 0) and (5, 0). Therefore, the solutions to the quadratic equation are:

## 4. Solutions to a Quadratic Equation on a Graph

You can determine the solutions to a quadratic equation by looking at a graph. Suppose you have another example of a parabola, with the equation:

The vertex of this graph is a maximum point and is at the point (-2, 2) on the graph. The x-intercepts of the graph are at the points (- 4, 0) and (0, 0).

Again, the x-intercepts are the solution to this quadratic equation.

Therefore, the solutions to this equation are:

Today you learned that a parabola is the shape of the graph of a quadratic equation, and that it has a general U shape, either opening up or opening down. You learned that in the general quadratic equation, the coefficient a determines the upward or downward shape of the graph. You also learned that every parabola has either a low point or a high point on the graph called the vertex, and that if the coefficient of a quadratic equation is positive, the vertex is a minimum point, whereas if the a coefficient is negative, the vertex is a maximum point. Lastly, you learned that the x-intercepts of a parabola are the x-values that make y equal 0 and also correspond to the solutions of the quadratic equation on a graph.

Source: This work is adapted from Sophia author Colleen Atakpu.

Terms to Know
Vertex (of a Parabola)

The maximum or minimum point of a parabola.