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.
\begin{align*}y=x^2-2\end{align*}
Equations to the \begin{align*}2^{nd}\end{align*} 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 \begin{align*}x^2\end{align*} and \begin{align*}x\end{align*} 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: \begin{align*}y=ax^2+bx+c\end{align*} (where \begin{align*}a\end{align*} cannot be zero) – notice that \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} will be coefficients and can be either positive or negative. This will also effect the parabola that is graphed.
Once again, the \begin{align*}a\end{align*} 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 \begin{align*}a\end{align*} value is from zero, the narrower the graph; the closer the \begin{align*}a\end{align*} value is to zero, the wider the graph. Also, a positive value of \begin{align*}a\end{align*} will give us a graph that opens upwards while a negative value of \begin{align*}a\end{align*} will open downward.
How about the \begin{align*}b\end{align*} 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 \begin{align*}b\end{align*} value helps us to predict the axis of symmetry.
Finally, the \begin{align*}c\end{align*} value, determines the \begin{align*}y\end{align*}-intercept of the graph—it tells us where the graph will hit the \begin{align*}y\end{align*}-axis. When the \begin{align*}c\end{align*} value was 3, the graph crossed the \begin{align*}y\end{align*}-axis at 3.
Write down information about the \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} 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 \begin{align*}a,b\end{align*} and \begin{align*}c\end{align*} 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 \begin{align*}y\end{align*}-axis | symmetry is right of the \begin{align*}y\end{align*}-axis | symmetrical on \begin{align*}y\end{align*}-axis |
\begin{align*}y\end{align*}-intercept is 3 | \begin{align*}y\end{align*}-intercept is 2 | \begin{align*}y\end{align*}-intercept is -3 |
equation is \begin{align*}y=\frac{1}{2}x^2+3\end{align*} | equation is \begin{align*}y=-x^2+x+2\end{align*} | equation is \begin{align*}y=2x^2-3\end{align*} |
\begin{align*}a=\frac{1}{2},b=0,c=3\end{align*} | \begin{align*}a=-1,b=1,c=2\end{align*} | \begin{align*}a=2,b=0,c=-3\end{align*} |
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 \begin{align*}a\end{align*} value tells if the graph goes upward or downward. So, if the graph goes downward, the \begin{align*}a\end{align*} value must be negative. If the graph goes upward, the \begin{align*}a\end{align*} value must be positive.
We also know that the \begin{align*}c\end{align*} value tells us the \begin{align*}y\end{align*}-intercept on the graph. So, if we know the \begin{align*}y\end{align*}-intercept, then we know the \begin{align*}c\end{align*} 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 \begin{align*}a < 0\end{align*}.
The \begin{align*}y\end{align*}-intercept is 3 so \begin{align*}c = 3\end{align*}.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
\begin{align*} -2\end{align*} | \begin{align*} -1\end{align*} |
\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 y-intercept 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 y-axis. The y-intercept 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 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 \begin{align*}x^2\end{align*}.
Now we can write the entire equation.
\begin{align*}y=x^2-4\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 t-table to help you.
- .
- .
- .
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.
Concept Nodes:
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.