12.13: Understanding the Equations of Parabolas
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.
Here is a quadratic equation.
Equations to the
The graph of a parabola can change position, direction, and width based on the coefficients of
Standard form of a quadratic equation:
Once again, the
How about the
This reflection line is called the axis of symmetry which we will learn about in the next lessons.
The
Finally, the
Write down information about the
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
Graph A  Graph B  Graph C 

opens upward  opens downward  opens upward 
fairly wide  neither narrow nor wide  fairly narrow 
symmetrical on 
symmetry is right of the 
symmetrical on 



equation is 
equation is 
equation is 



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
We also know that the
If we have a graph, then we can also work backwards—we can fill in a ttable using the points we see on the graph. Then, by looking for a pattern in the ttable, 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
The





\begin{align*} 1\end{align*}  \begin{align*}{\color{white}} 2\end{align*} 
\begin{align*}{\color{white}} 0\end{align*}  \begin{align*}{\color{white}} 3\end{align*} 
\begin{align*}{\color{white}} 1\end{align*}  \begin{align*}{\color{white}} 2\end{align*} 
\begin{align*}{\color{white}} 2\end{align*}  \begin{align*} 1\end{align*} 
Standard form: \begin{align*}y=ax^2+bx+c\end{align*}
\begin{align*}c = 3\end{align*} so \begin{align*}y=ax^2+bx+3\end{align*}
The graph goes downward and we see the same pattern as with \begin{align*}y = x^2\end{align*}.
The equation may be \begin{align*}y=x^2+3\end{align*}.
Test point (1, 2) \begin{align*}\rightarrow\end{align*} does \begin{align*}2=(1)^2+3\end{align*}?
Yes! Our equation is correct!
Answer each question about parabolas.
Example A
If the \begin{align*}c\end{align*} value is \begin{align*}4\end{align*}, where is the yintercept of the graph?
Solution: \begin{align*}4\end{align*}
Example B
If the \begin{align*}a\end{align*} value is \begin{align*}3\end{align*}, will the parabola open upward or downward?
Solution: Downward
Example C
If the parabola opens upward, which value is positive \begin{align*}a, b\end{align*} or \begin{align*}c\end{align*}.
Solution: \begin{align*}a\end{align*}
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 \begin{align*}a\end{align*} value will be positive.
The parabola is symmetrical on the yaxis. The yintercept is zero.
The equation is \begin{align*}y=x^2\end{align*}.
If we substitute an ordered pair into the equation, we can check our work.
\begin{align*}4 = 2^2\end{align*}
\begin{align*}4 = 4\end{align*}
Our equation works!
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.
Figure out the equation for the following parabola.
Solution
First, notice that the yintercept is 4.
You can also see that the graph is symmetrical with the yaxis. This means that the first part of our equation is \begin{align*}x^2\end{align*}.
Now we can write the entire equation.
\begin{align*}y=x^24\end{align*}
This is our answer.
Video Review
Determining the Equation of a Parabola from a Graph
Practice
Directions: Answer the following questions about parabolas.
 True or false. All parabolas are symmetrical.
 True or false. The \begin{align*}y\end{align*} intercept is the same as the \begin{align*}c\end{align*} value.
 A parabola with a positive squared value opens _____________.
 A parabola with a negative squared value opens _____________.
 What is the vertex of the parabola?
 True or false. A parabola always forms a U shape.
 True or false. The closer the \begin{align*}a\end{align*} value is to zero the wider the parabola.
 True or false. The closer the \begin{align*}a\end{align*} value is to zero the narrower the parabola.
 True or false. The \begin{align*}b\end{align*} value determines the axis of symmetry.
 What does the \begin{align*}c\end{align*} value indicate?
 True or false. A linear equation will have a graph that is a parabola.
 True or false. A quadratic equation and a linear equation will have a similar graph.
Write the equations of the following graphs. Use the \begin{align*}a\end{align*} and \begin{align*}c\end{align*} values and a ttable to help you.
 .
 .
 .
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Term  Definition 

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 and identify the equation of a parabola.