# 12.12: Understanding the Graphs of a Parabola

**At Grade**Created by: CK-12

**Practice**Graphs of Quadratic Functions in Intercept Form

Have you ever seen a parabola? Take a look at this dilemma.

The students in Mr. Nelson's class went on a tour of downtown. While they were seeing the sights, they were also learning about the city that they live in. The students reached the park and stopped at the entrance. The entrance to the park was a beautiful arc decorated with ivy and flowering vines.

“This is beautiful,” Kelsey said looking at the entrance.

“It sure is. It looks like a parabola,” Kenny commented.

“A what?” Kelsey exclaimed.

“A parabola. Don’t you know what a parabola is?” Kenny asked.

**Do you know what a parabola is? This Concept is all about parabolas. Pay close attention because at the end of the Concept, you will need to define the term parabola.**

### Guidance

Graphs and equations are important parts of mathematics. Equations can show relationships between different variables, like \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. Graphs can show all of the pairs of numbers, \begin{align*}x\end{align*} and \begin{align*}y\end{align*}, that make an equation true—they show all of the solutions. Oftentimes, graphs represent infinite pairs of such numbers.

You have studied the relationships of linear equations—their graphs were straight lines. But not all graphs will be linear.

**Some graphs are shaped like U's. These U shaped graphs are called** *parabolas***.**

Here you'll learn about parabolas.

When we graphed linear equations, we made tables of value, or *t-tables*, using the equation that was given. For instance, for the equation \begin{align*}y = x + 4\end{align*}, we use the \begin{align*}x\end{align*}-values in the t-table to find the corresponding \begin{align*}y\end{align*}-values.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

\begin{align*}-4\end{align*} | |

\begin{align*}-3\end{align*} | \begin{align*}1\end{align*} |

\begin{align*}-2\end{align*} | \begin{align*}2\end{align*} |

\begin{align*}-1\end{align*} | \begin{align*}3\end{align*} |

\begin{align*}{\color{white}-}0\end{align*} | \begin{align*}4\end{align*} |

\begin{align*}{\color{white}-}1\end{align*} | \begin{align*}5\end{align*} |

\begin{align*}{\color{white}-}2\end{align*} | \begin{align*}6\end{align*} |

We then used the t-table to create a graph on a coordinate plane.

Now that you’ve seen so many exponents, you know that the power of \begin{align*}x\end{align*} in the equation above is

But what happens if we change the power to 2?

Let’s try.

**Start with an equation: \begin{align*}y = x^2\end{align*}**

**Make a t-table just like with linear equations.**

**Be sure to include negative numbers, zero, and positive numbers. Then, substitute the values of \begin{align*}x\end{align*} in the equation to find the values of \begin{align*}y\end{align*}.**

**Start with -3.**

**So for the \begin{align*}x\end{align*} value of -3, the \begin{align*}y\end{align*} value is 9. Use the same process to complete your t-table.**

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

\begin{align*}-3\end{align*} | \begin{align*}9\end{align*} |

\begin{align*} -2\end{align*} | \begin{align*}4\end{align*} |

\begin{align*} -1\end{align*} | \begin{align*}1\end{align*} |

\begin{align*}{\color{white}-}0\end{align*} | |

\begin{align*}{\color{white}-}1\end{align*} | \begin{align*}1\end{align*} |

\begin{align*}{\color{white}-}2\end{align*} | \begin{align*}4\end{align*} |

\begin{align*}{\color{white}-}3\end{align*} | \begin{align*}9\end{align*} |

**Now you can graph the values on the coordinate plane.**

Look at the difference in the graph.

This shape is called a ** parabola**.

**Equations to the \begin{align*}2^{nd}\end{align*} power are called** *quadratic equations***and their graphs are always parabolas.**

Compare the graph of the equation \begin{align*}y=x^2-2\end{align*} to the one above.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

\begin{align*}-3\end{align*} | \begin{align*}{\color{white}-} 7\end{align*} |

\begin{align*} -2\end{align*} | \begin{align*}{\color{white}-} 2\end{align*} |

\begin{align*} -1\end{align*} | \begin{align*} -1\end{align*} |

\begin{align*}{\color{white}-} 0\end{align*} | \begin{align*} -2\end{align*} |

\begin{align*}{\color{white}-} 1\end{align*} | \begin{align*} -1\end{align*} |

\begin{align*}{\color{white}-} 2\end{align*} | \begin{align*}{\color{white}-} 2\end{align*} |

\begin{align*}{\color{white}-} 3\end{align*} | \begin{align*}{\color{white}-} 7\end{align*} |

**Here we can see that the shape is the same but it moves down two points on the \begin{align*}y\end{align*}-axis. This is because of the constant -2.**

It’s your turn to graph.

**First, use the given equations to complete a t-table.** Be sure to include positive numbers, zero, and negative numbers in your t-table. Also, be careful with your order of operations calculate the \begin{align*}y\end{align*}-values.

**Then graph your points on a coordinate plane.** You can check your graphs using the knowledge that you have about the \begin{align*}a, b,\end{align*} and \begin{align*}c\end{align*} values. We know that the parabolas must be symmetrical, too.

If your graph does not fit the knowledge that you have learned about parabolas then it may indicate an error in your calculations or in graphing.

Graph the equation \begin{align*}y=x^2-4\end{align*}.

Start with a t-table. Enter your \begin{align*}x\end{align*}-values.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

\begin{align*}-3\end{align*} | |

\begin{align*} -2\end{align*} | |

\begin{align*} -1\end{align*} | |

\begin{align*}{\color{white}-} 0\end{align*} | |

\begin{align*}{\color{white}-} 1\end{align*} | |

\begin{align*}{\color{white}-} 2\end{align*} | |

\begin{align*}{\color{white}-} 3\end{align*} |

Then, substitute your first value of \begin{align*}x\end{align*} in the equation.

\begin{align*}y&=(-3)^2-4\\ y&=9-4\\ y&=5\end{align*}

Do this with all of the \begin{align*}x\end{align*} values so that you can fill in the appropriate \begin{align*}y\end{align*} values.

Place your \begin{align*}y\end{align*}-value in the t-table next to its \begin{align*}x\end{align*}-value.

Your completed t-table should look like this.

Use the points from your t-table to create a graph. Your first point is (-3, 5).

Graph all of the points and connect them with a smooth parabolic shape.

Graph the equation \begin{align*}y=-\frac{1}{2}x^2+5\end{align*}.

#### Example A

First, create the t- table.

**Solution:**

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

\begin{align*}-3\end{align*} | \begin{align*}0.5\end{align*} |

\begin{align*} -2\end{align*} | \begin{align*}3\end{align*} |

\begin{align*} -1\end{align*} | \begin{align*}4.5\end{align*} |

\begin{align*}{\color{white}-} 0\end{align*} | \begin{align*}5\end{align*} |

\begin{align*}{\color{white}-} 1\end{align*} | \begin{align*}4.5\end{align*} |

\begin{align*}{\color{white}-} 2\end{align*} | \begin{align*}3\end{align*} |

\begin{align*}{\color{white}-} 3\end{align*} | \begin{align*}0.5\end{align*} |

#### Example B

Now graph the parabola.

**Solution:**

#### Example C

Where is the vertex of this parabola?

**Solution: \begin{align*}5\end{align*}**

Now let's go back to the dilemma from the beginning of the Concept.

**A parabola is the shape that is created by a quadratic equation. It forms an arc. A parabola has a vertex that is either a maximum point or a minimum point. If the squared value is positive then the parabola opens upward. If the squared value is negative then the parabola opens downward.**

### Vocabulary

- Parabola
- a U shaped graph that is non – linear.

- Quadratic Equations
- equations to the second power that will always graph as a parabola.

- Vertex of a Parabola
- the bottom or top point of a parabola.

### Guided Practice

Here is one for you to try on your own.

Create the graph of \begin{align*}y=-x^2\end{align*}.

**Solution**

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

\begin{align*}-3\end{align*} | -9 |

-2 | -4 |

-1 | -1 |

0 | 0 |

1 | -1 |

2 | -4 |

3 | -9 |

**This time the graph is inverted. Instead of opening upwards, it opens downward. This is because of the coefficient -1.**

### Video Review

### Practice

Directions: Answer the following questions about parabolas.

- True or false. A parabola is always formed by a quadratic equation.
- True or false. A parabola can have a vertex that is positive or negative.
- True or false. If the vertex is positive, then the parabola will be located above the x-axis.
- True or false. If the vertex is negative, then the vertex will be below the x-axis.
- True or false. All parabolas are symmetrical.

Directions: Match the three graphs to their equations.

- \begin{align*}y=3x^2-2\end{align*}
- \begin{align*}y=x^2+x-3\end{align*}
- \begin{align*}y=-\frac{1}{2}x^2+2\end{align*}

**Graph A**

**Graph B**

**Graph C**

Now answer these questions about the graphs.

- What is the vertex of graph A?
- What is the vertex of graph B?
- What is the vertex of graph C?
- Which graph is the narrowest graph?

Graph the following equations using a t-table:

- \begin{align*}y=x^2-1\end{align*}
- \begin{align*}y=-x^2+x\end{align*}
- \begin{align*}y=\frac{1}{2}x^2+1\end{align*}

### Notes/Highlights Having trouble? Report an issue.

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Term | Definition |
---|---|

Intercepts |
The intercepts of a curve are the locations where the curve intersects the and axes. An intercept is a point at which the curve intersects the -axis. A intercept is a point at which the curve intersects the -axis. |

Line of Symmetry |
A line of symmetry is a line that can be drawn to divide a figure into equal halves. |

Parabola |
A parabola is the characteristic shape of a quadratic function graph, resembling a "U". |

Quadratic Equation |
A quadratic equation is an equation that can be written in the form , where , , and are real constants and . |

Vertex |
The vertex of a parabola is the highest or lowest point on the graph of a parabola. The vertex is the maximum point of a parabola that opens downward and the minimum point of a parabola that opens upward. |

### Image Attributions

Here you'll understand the graphs of a parabola given equations.