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 parabola?” Kelsey exclaimed.

How can Kenny explain a parabola to Kelsey?

In this concept, you will learn to understand the graphs of a parabola given equations.

### Parabola

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.

Not all graphs will be linear. Some graphs are shaped like U’s. These U-shaped graphs are called **parabolas**. A **parabola** is a graph of a quadratic function. So instead of graphing \begin{align*}y=x+4\end{align*} which forms a linear graph, parabolas are graphs of function like \begin{align*}y=x^2 +4\end{align*}. Look at the graph below that shows the difference in appearance of these two functions.

Let’s try an example where you can graph a parabola.

Graph \begin{align*}y=x^2\end{align*}.

First, start by filling in a table of values. Look at the table below. The \begin{align*}x\end{align*}-values are listed from -3 to +3. Use these values to find the corresponding \begin{align*}y\end{align*}-values. It is not necessary to use these \begin{align*}x\end{align*}-values but most of the time; they will work with the parabolas being graphed.

When filling out the table, simply put the \begin{align*}x\end{align*}-value into the equation \begin{align*}y=x^2\end{align*}. For example, when \begin{align*}x=-3, y=(-3)^2\end{align*} or \begin{align*}y=9\end{align*}.

Next, graph the values on the coordinate plane.

Look at the difference in the graph. This shape is called a parabola. Equations to the 2^{nd} power are called **quadratic equations** and their graphs are always parabolas.

Let’s look at another example.

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

First, start by filling in a table of values. Look at the table below. The \begin{align*}x\end{align*}-values are listed from -3 to +3. Use these values to find the corresponding \begin{align*}y\end{align*}-values.

When filling out the table, simply put the \begin{align*}x\end{align*}-value into the equation \begin{align*}y=x^2 -2\end{align*}. For example, when \begin{align*}x=-3, y=x^2-2, y(-3)^2 -2\end{align*} or \begin{align*}y=7\end{align*}.

Next, graph the values on the coordinate plane.

The shape is the same but it moves down two points on the y-axis. This is because of the constant -2.

Let’s try one more example.

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

First, start by filling in a table of values. Look at the table below. The \begin{align*}x\end{align*}-values are listed from -3 to +3. Use these to find the corresponding \begin{align*}y\end{align*}-values.

When filling out the table, simply put the \begin{align*}x\end{align*}-value into the equation \begin{align*}y=x^2 -4\end{align*}. For example, when \begin{align*}x=-3, y=x^2-4, y=(-3)^2 -4\end{align*} or \begin{align*}y =5\end{align*}.

Next, graph the values on the coordinate plane.

### Examples

#### Example 1

Earlier, you were given a problem about Kenny and the parabola. He wants to describe the meaning of parabola to Kelsey.

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 and therefore has a minimum point. If the squared value is negative then the parabola opens downward and therefore has a maximum point.

#### Example 2

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

First, start by filling in a table of values. Look at the table below. The \begin{align*}x\end{align*}-values are listed from -3 to +3. Use these to find the corresponding \begin{align*}y\end{align*}-values.

When filling out the table, simply put the \begin{align*}x\end{align*}-value into the equation \begin{align*}y=-x^2\end{align*}. For example, when \begin{align*}x=-3\end{align*}. or \begin{align*}y =-9\end{align*}.

Next, graph the values on the coordinate plane.

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

#### For these three examples, work with the quadratic function \begin{align*}y=-\frac{1}{2} x^2 + 5\end{align*}.

#### Example 3

First, create the t-table.

#### Example 4

Next, graph the parabola for the quadratic function \begin{align*}y= - \frac{1}{2} x^2 + 5\end{align*}.

#### Example 5

Where is the vertex of this parabola?

The vertex is the turning point of the curve. Parabolas are symmetrical, and this one could really be folded along the \begin{align*}y\end{align*}-axis at the vertex point to create two identical halves.

The vertex occurs at the point (0, 5).

### Review

Answer the following questions about parabolas.

1. True or false. A parabola is always formed by a quadratic equation.

2. True or false. A parabola can have a vertex that is positive or negative.

3. True or false. If the vertex is positive, then the parabola will be located above the \begin{align*}x\end{align*}-axis.

4. True or false. If the vertex is negative, then the vertex will be below the \begin{align*}x\end{align*}-axis.

5. True or false. All parabolas are symmetrical.

Match the three graphs to their quadratic equations.

6. \begin{align*}y=3x^2 -2\end{align*}

7. \begin{align*}y=x^2 + x-3\end{align*}

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

**Graph A**

**Graph B**

**Graph C**

Now answer these questions about the graphs.

9. What is the vertex of graph A?

10. What is the vertex of graph B?

11. What is the vertex of graph C?

12. Which graph is the narrowest graph?

Graph the following equations using a t-table:

13. \begin{align*}y=x^2 -1\end{align*}

14. \begin{align*}y=-x^2 +x\end{align*}

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

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.12.