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High School Algebra II: Help and Review26 chapters | 296 lessons

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Lesson Transcript

Instructor:
*Sarah Spitzig*

Sarah has taught secondary math and English in three states, and is currently living and working in Ontario, Canada. She has recently earned a Master's degree.

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.

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

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.

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.

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.

In this case, since the *a* value is -1/2, the graph opens down and the vertex is a maximum.

Once again, the **intercept form of a parabola** is *y* = *a* (*x* - *r*)(*x* - *s*), where *r* and *s* are the *x*-intercepts, or where the graph passes through the *x*-axis. The benefit of using the intercept form is that you can easily find the *x*-intercepts without factoring or using the quadratic formula. Remember that the intercepts will always be the opposite sign, since the formula is *x* - *r* and *x* - *s*.

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 |

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

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High School Algebra II: Help and Review26 chapters | 296 lessons

- Using Tables and Graphs in the Real World 5:50
- Scatterplots and Line Graphs: Definitions and Uses 7:17
- What is a Parabola? 4:36
- Parabolas in Standard, Intercept, and Vertex Form 6:15
- Multiplying Binomials Using FOIL and the Area Method 7:26
- Multiplying Binomials Using FOIL & the Area Method: Practice Problems 5:46
- How to Factor Quadratic Equations: FOIL in Reverse 8:50
- Factoring Quadratic Equations: Polynomial Problems with a Non-1 Leading Coefficient 7:35
- Solving Quadratic Trinomials by Factoring 7:53
- How to Complete the Square 8:43
- Completing the Square Practice Problems 7:31
- How to Solve a Quadratic Equation by Factoring 7:53
- Parabola Intercept Form: Definition & Explanation 3:46
- Go to Algebra II Quadratic Equations: Help and Review

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