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# 12.13: Understanding the Equations of Parabolas

Difficulty Level: Basic Created by: CK-12
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Have you ever seen a parabola? Can you write the equation of a parabola? Take a look at this dilemma.

This parabola was created because of an equation.

Use this Concept to learn how to identify the equation of a parabola from a graph.

### Guidance

Quadratic equations have graphs that are parabolas.

A parabola is a U shaped graph. Quadratic equations have graphs that are parabolas.

$y=x^2-2$

Equations to the $2^{nd}$ power are called quadratic equations and their graphs are always parabolas.

The graph of a parabola can change position, direction, and width based on the coefficients of $x^2$ and $x$ as well as the constant. Because those pieces of the equation are so important, we have named them in what is called the standard form .

Standard form of a quadratic equation: $y=ax^2+bx+c$ (where $a$ cannot be zero) – notice that $a, b$ and $c$ will be coefficients and can be either positive or negative. This will also effect the parabola that is graphed.

Once again, the $a$ value can predict two things—1) how wide the graph will be and 2) if the graph opens upward or downward. Generally speaking, the further the $a$ value is from zero, the narrower the graph; the closer the $a$ value is to zero, the wider the graph. Also, a positive value of $a$ will give us a graph that opens upwards while a negative value of $a$ will open downward.

How about the $b$ value? You may have noticed that all of the parabolas are symmetrical—they are the same on both sides, as if they were reflected on a mirror that were right down the middle of the graph.

This reflection line is called the axis of symmetry which we will learn about in the next lessons.

The $b$ value helps us to predict the axis of symmetry.

Finally, the $c$ value, determines the $y$ - intercept of the graph—it tells us where the graph will hit the $y$ -axis. When the $c$ value was 3, the graph crossed the $y$ -axis at 3.

Write down information about the $a, b$ and $c$ values of the quadratic equation.

Let’s look at some graphs.

When you look at these graphs, we can determine certain things by looking at each and by knowing what the $a,b$ and $c$ values of the quadratic equation help us to determine. Here is a chart to help you understand what we can determine by these graphs.

Graph A Graph B Graph C
opens upward opens downward opens upward
fairly wide neither narrow nor wide fairly narrow
symmetrical on $y$ -axis symmetry is right of the $y$ -axis symmetrical on $y$ -axis
$y$ -intercept is 3 $y$ -intercept is 2 $y$ -intercept is -3
equation is $y=\frac{1}{2}x^2+3$ equation is $y=-x^2+x+2$ equation is $y=2x^2-3$
$a=\frac{1}{2},b=0,c=3$ $a=-1,b=1,c=2$ $a=2,b=0,c=-3$

Now you can see how the graphs of each equation provides us with information.

You have learned to write linear equations based on linear graphs, we can also find a quadratic equation by using the parabola.

We know that the $a$ value tells if the graph goes upward or downward. So, if the graph goes downward, the $a$ value must be negative. If the graph goes upward, the $a$ value must be positive.

We also know that the $c$ value tells us the $y$ -intercept on the graph. So, if we know the $y$ -intercept, then we know the $c$ value.

If we have a graph, then we can also work backwards—we can fill in a t-table using the points we see on the graph. Then, by looking for a pattern in the t-table, we can derive the equation.

It is a bit like being a detective when you have to figure out the equation!

Take a look at this one.

Write the equation for the given graph.

Graph opens downward so $a < 0$ .

The $y$ -intercept is 3 so $c = 3$ .

$x$ $y$
$-2$ $-1$
$-1$ ${\color{white}-} 2$
${\color{white}-} 0$ ${\color{white}-} 3$
${\color{white}-} 1$ ${\color{white}-} 2$
${\color{white}-} 2$ $-1$

Standard form: $y=ax^2+bx+c$

$c = 3$ so $y=ax^2+bx+3$

The graph goes downward and we see the same pattern as with $y = -x^2$ .

The equation may be $y=-x^2+3$ .

Test point (1, 2) $\rightarrow$ does $2=(-1)^2+3$ ?

Yes! Our equation is correct!

#### Example A

If the $c$ value is $4$ , where is the y-intercept of the graph?

Solution: $4$

#### Example B

If the $a$ value is $-3$ , will the parabola open upward or downward?

Solution: Downward

#### Example C

If the parabola opens upward, which value is positive $a, b$ or $c$ .

Solution: $a$

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

Here is the parabola once again.

The parabola opens upward, so we know the $a$ value will be positive.

The parabola is symmetrical on the y-axis. The y-intercept is zero.

The equation is $y=x^2$ .

If we substitute an ordered pair into the equation, we can check our work.

$4 = -2^2$

$4 = 4$

Our equation works!

### Vocabulary

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

Figure out the equation for the following parabola.

Solution

First, notice that the y-intercept is -4.

You can also see that the graph is symmetrical with the y-axis. This means that the first part of our equation is $x^2$ .

Now we can write the entire equation.

$y=x^2-4$

### Practice

1. True or false. All parabolas are symmetrical.
2. True or false. The $y$ intercept is the same as the $c$ value.
3. A parabola with a positive squared value opens _____________.
4. A parabola with a negative squared value opens _____________.
5. What is the vertex of the parabola?
6. True or false. A parabola always forms a U shape.
7. True or false. The closer the $a$ value is to zero the wider the parabola.
8. True or false. The closer the $a$ value is to zero the narrower the parabola.
9. True or false. The $b$ value determines the axis of symmetry.
10. What does the $c$ value indicate?
11. True or false. A linear equation will have a graph that is a parabola.
12. True or false. A quadratic equation and a linear equation will have a similar graph.

Write the equations of the following graphs. Use the $a$ and $c$ values and a t-table to help you.

1. .
1. .
1. .

Basic

Mar 19, 2013

Aug 21, 2014