In this lesson, we will define a parabola, discover its everyday uses, determine the functions of the intercept form of the equation, and find the x-intercepts of a parabola.

## What Is a Parabola?

Before we can understand parabola intercept form, we first need to go through a few other definitions, starting with a parabola. A **parabola** is a U-shaped graph that always has an *x*-squared term in its equation. The graph will open either up, like a smiley face, or down, like a sad face, and the vertex will be the lowest point if it opens up and the highest point if it opens down. In this graph, the vertex is (1,-4) and is the lowest point. When the vertex is the lowest point, it’s called a minimum, and when it’s the highest point, it’s called a maximum.

The next definition we need to know about is the line of symmetry. The **line of symmetry** is a vertical line that passes through the vertex and cuts the parabola in half.

Any image on one side of the parabola has a mirror image on the other side of the line of symmetry. The *x*-intercepts are always equidistant from the line of symmetry. Going back to our previous graph, you can see the line of symmetry is *x* = 1.Next, you should have an understanding of the *x*– and *y*-intercepts. The ** y-intercept** is the point where the graph passes through the

*y*-axis. In our graph, the

*y*-intercept is (0, -3).

The ** x-intercepts** are where the graph passes through the

*x*-axis. In our graph, the

*x*-intercepts are (-1, 0) and (3, 0).

## Everyday Uses of a Parabola

The graph of a parabola has many uses in the real world. It can approximate the path of a projectile, such as a ball, plane, or rocket. It can also demonstrate the shape of a bridge, roller coaster, arch, or even the path of water in a drinking fountain.

## Intercept Form of a Parabola

There are many forms of the parabola, such as vertex form, standard form, and intercept form. Each form has special qualities that help tell us specific things about the graph. For example, vertex form helps us determine the vertex by looking at the equation, standard form helps us see the *y*-intercept without graphing, and intercept form helps us find the *x*-intercepts of the graph without factoring or using the quadratic formula.The **intercept form** is *y* = *a*(*x* – *r*)(*x* – *s*), where *r* and *s* are the *x*-intercepts on the graph. The intercept form will tell us if there are two *x*-intercepts, one *x*-intercept or no *x*-intercepts.

## Examples

Let’s go over some examples.Look at the graph of the equation *y* = 1(*x* + 1)(*x* – 5).

Notice that the *x*-intercepts are at -1 and 5, and the equation shows *x* + 1 and *x* – 5. This happens in the intercept form because the equation is *x* – *r* and *x* – *s*, where *r* and *s* are the *x*-intercepts. Since we’re subtracting *r* and *s*, it would make sense that the actual intercepts are the opposite sign. Since the *a* value is 1, the graph opens up and the vertex is a minimum.In this graph of the equation *y* = -1/2(*x* – 0)(*x* – 4), the *x*-intercepts are 0 and 4.

Terms | Definitions |
---|---|

Parabola | a U-shaped graph that always has an x-squared term in its equation |

Line of symmetry | a vertical line that passes through the vertex and cuts the parabola in half |

y-intercept |
the point where the graph passes through the y-axis |

x-intercepts |
where the graph passes through the x-axis |

Intercept form of a parabola | y = a (x – r)(x – s), where r and s are the x-intercepts, or where the graph passes through the x-axis |

## Learning Outcomes

As soon as your lesson on parabola intercept form ends, try to:

- Illustrate a parabola
- Identify the
*x*and*y*axis - Find the intercept form of a parabola