This parabola was created from an equation. What was the equation?
In this concept, you will learn to understand the equation of a parabola.
Parabolas
A parabola is a Ushaped graph.Equations with the ‘
Here is a quadratic equation.
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 ‘
1. how wide the graph will be
Generally speaking, the further the ‘
2. if the graph opens upward or downward.
A positive value of ‘
What about the ‘
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. The ‘
Finally, the
Let’s look at some graphs.
Looking at these graphs, and knowing what the
Now you can see how the graphs of each equation provide you with information.
You have learned to write linear equations based on linear graphs, you can also find a quadratic equation by using the parabola.
You know that the ‘
You also know that the ‘
If you have a graph, then you can also work backwards. In other words, you can fill in a ttable using the points you see on the graph. Then, by looking for a pattern in the ttable, you can derive the equation.
Let’s look at an example.
Write the equation for the given graph.
First, start with what you know about the values of ‘
Next, construct a table of values from the graph.


2  1 
1  2 
0  3 
1  2 
2  1 
Then, put what you know into the standard form of the quadratic equation. Since the graph goes over one and down 1, you know that
Then, test a point on the graph to find the value of
Point: (1, 2)
The answer is
Examples
Example 1
Earlier, you were given a problem about the parabola.
First, start with what you know about the values of ‘
Next, construct a table of values from the graph.
\begin{align*}x\end{align*} 
\begin{align*}y\end{align*} 
2  4 
1  1 
0  0 
1  1 
2  4 
Then, put what you know into the standard form of the quadratic equation. Since the graph goes over one and up 1, you know that \begin{align*}a = 1\end{align*}
\begin{align*}\begin{array}{rcl} y &=& ax^2 + bx + c \\ y &=& x^2 + bx \end{array}\end{align*}
Then, test a point on the graph to find the value of \begin{align*}b\end{align*}
\begin{align*}y = x^2 + bx\end{align*}
Point: (1, 1)
\begin{align*}\begin{array}{rcl} 1 &=& (1)^2 + b(1) \\ 1 &=& 1+ b \\ b &=& 0 \end{array} \end{align*}
The answer is \begin{align*}y=x^2\end{align*}
Example 2
Figure out the equation for the following parabola.
First, start with what you know about the values of ‘\begin{align*}a\end{align*}
\begin{align*}a\end{align*}
\begin{align*}c\end{align*}
Next, construct a table of values from the graph.
\begin{align*}x\end{align*} 
\begin{align*}y\end{align*} 
2  0 
1  3 
0  4 
1  3 
2  0 
Then, put what you know into the standard form of the quadratic equation. Since the graph goes over one and up 1, you know that \begin{align*}a = 1\end{align*}
\begin{align*}\begin{array}{rcl} y &=& ax^2 + bx + c \\ y &=& x^2 + bx  4 \end{array}\end{align*}
Then, test a point on the graph to find the value of \begin{align*}b\end{align*}
\begin{align*}y= x^2 + bx  4\end{align*}
Point: (1, 3)
\begin{align*}\begin{array}{rcl} 3 &=& (1)^2 + b (1) 4 \\ 3 &=& 1 + b 4 \\ 3 &=& b 3 \\ b &=& 0 \end{array}\end{align*}
The answer is \begin{align*}y=x^24\end{align*}
Example 3
If the \begin{align*}c\end{align*}
If \begin{align*}c = 4\end{align*}
Example 4
If the ‘\begin{align*}a\end{align*}
If \begin{align*}a < 0\end{align*}
Example 5
If the parabola opens upward, which value is positive \begin{align*}a, b\end{align*}
When the graph opens upward or downward, the value of ‘\begin{align*}a\end{align*}
Review
Answer the following questions about parabolas.
1. True or false. All parabolas are symmetrical.
2. True or false. The \begin{align*}y\end{align*}
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 \begin{align*}a\end{align*}
8. True or false. The closer the \begin{align*}a\end{align*}
9. True or false. The \begin{align*}b\end{align*}
10. What does the \begin{align*}c\end{align*}
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 \begin{align*}c\end{align*}
13.
14.
15.
Review (Answers)
To see the Review answers, open this PDF file and look for section 12.13.