12.5: Exploring Parabolas
Introduction
The Arc
The students were having a terrific time walking all around downtown. While they were seeing the sights and solving problems, 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 what?” Kelsey exclaimed.
“A parabola. Don’t you know what a parabola is?” Kenny asked.
Do you know what a parabola is? This lesson is all about parabolas. Pay close attention because at the end of the lesson you will need to define the term parabola.
What You Will Learn
By the end of this lesson you will be able to complete the following skills.
- Identify and compare graphs of parabolas given equations.
- Identify and compare equations of parabolas given graphs.
- Graph a parabola, given an equation.
- Write the equation of a parabola given a graph.
Teaching Time
I. Identify and Compare Graphs of Parabolas Given Equations
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.
You have studied the relationships of linear equations—their graphs were straight lines. But not all graphs will be linear. Some graphs are shaped like U's. These U shaped graphs are called parabolas. Let’s check out parabolas.
When we graphed linear equations, we made tables of value, or t-tables, using the equation that was given. For example, for the equation \begin{align*}y = x + 4\end{align*}, we use the \begin{align*}x\end{align*}-values in the t-table to find the corresponding \begin{align*}y\end{align*}-values.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
\begin{align*}-4\end{align*} | 0 |
-3 | 1 |
-2 | 2 |
-1 | 3 |
0 | 4 |
1 | 5 |
2 | 6 |
We then used the t-table to create a graph on a coordinate plane.
Now that you’ve seen so many exponents, you know that the power of \begin{align*}x\end{align*} in the equation above is
1. But what happens if we change the power to 2?
Let’s try.
Start with an equation: \begin{align*}y = x^2\end{align*}
Make a t-table just like with linear equations.
Be sure to include negative numbers, zero, and positive numbers. Then, substitute the values of \begin{align*}x\end{align*} in the equation to find the values of \begin{align*}y\end{align*}.
Start with -3.
So for the \begin{align*}x\end{align*} value of -3, the \begin{align*}y\end{align*} value is 9. Use the same process to complete your t-table.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
\begin{align*}-3\end{align*} | 9 |
-2 | 4 |
-1 | 1 |
0 | 0 |
1 | 1 |
2 | 4 |
3 | 9 |
Now you can graph the values on the coordinate plane.
Look at the difference in the graph.
This shape is called a parabola.
Equations to the \begin{align*}2^{nd}\end{align*} power are called quadratic equations and their graphs are always parabolas.
Example
Compare the graph of the equation \begin{align*}y=x^2-2\end{align*} to the one above.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
\begin{align*}-3\end{align*} | 7 |
-2 | 2 |
-1 | -1 |
0 | -2 |
1 | -1 |
2 | 2 |
3 | 7 |
Here we can see that the shape is the same but it moves down two points on the \begin{align*}y\end{align*}-axis. This is because of the constant -2.
Example
Now try the graph of \begin{align*}y=-x^2\end{align*}.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
\begin{align*}-3\end{align*} | -9 |
-2 | -4 |
-1 | -1 |
0 | 0 |
1 | -1 |
2 | -4 |
3 | -9 |
This time the graph is inverted. Instead of opening upwards, it opens downward. This is because of the coefficient -1.
As we look at parabolas we can see that the bottom point, the vertex, has moved. It is no longer on the \begin{align*}y\end{align*}-axis. As you compare graphs, you will see that the graphs change based on the values in the equation.
Notice that when you see an equation to the second power, that the graph of the equation will be a parabola.
Write this down in your notebook.
II. Identify and Compare Equations of Parabolas, Given Graphs
We’ve seen quadratic equations and that their graphs are parabolas. In the last examples, we noticed that the graphs 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.
III. Graph a Parabola, Given an Equation
You have seen a lot of graphs in this lesson. It’s your turn to graph.
First, use the given equations to complete a t-table. Be sure to include positive numbers, zero, and negative numbers in your t-table. Also, be careful with your order of operations calculate the \begin{align*}y\end{align*}-values.
Then graph your points on a coordinate plane. You can check your graphs using the knowledge that you have about the \begin{align*}a, b,\end{align*} and \begin{align*}c\end{align*} values. We know that the parabolas must be symmetrical, too.
If your graph does not fit the knowledge that you have learned about parabolas then it may indicate an error in your calculations or in graphing.
Example
Graph the equation \begin{align*}y=x^2-4\end{align*}.
Start with a t-table. Enter your \begin{align*}x\end{align*}-values.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
\begin{align*}-3\end{align*} | |
-2 | |
-1 | |
0 | |
1 | |
2 | |
3 |
Then, substitute your first value of \begin{align*}x\end{align*} in the equation.
\begin{align*}y&=(-3)^2-4\\ y&=9-4\\ y&=5\end{align*}
Do this with all of the \begin{align*}x\end{align*} values so that you can fill in the appropriate \begin{align*}y\end{align*} values.
Place your \begin{align*}y\end{align*}-value in the t-table next to its \begin{align*}x\end{align*}-value.
Your completed t-table should look like this.
Use the points from your t-table to create a graph. Your first point is (-3, 5).
Graph all of the points and connect them with a smooth parabolic shape.
Example
Graph the equation \begin{align*}y=-\frac{1}{2}x^2+5\end{align*}.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
\begin{align*}-3\end{align*} | .5 |
-2 | 3 |
-1 | 4.5 |
0 | 5 |
1 | 4.5 |
2 | 3 |
3 | .5 |
Now let’s look at how we can write an equation given a graph.
IV. Write the Equation of a Parabola, Given a Graph
Just as we have learned to make 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!
Example
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*} | -1 |
-1 | 2 |
0 | 3 |
1 | 2 |
2 | -1 |
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!
Now let’s look at the problem from the introduction.
Real-Life Example Completed
The Arc
Here is the problem from the introduction. Reread it and then define a parabola.
The students were having a terrific time walking all around downtown. While they were seeing the sights and solving problems, 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 what?” Kelsey exclaimed.
“A parabola. Don’t you know what a parabola is?” Kenny asked.
Solution to Real – Life Example
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. If the squared value is negative then the parabola opens downward.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- 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.
Time to Practice
Directions: Answer the following questions about parabolas.
- True or false. A parabola is always formed by a quadratic equation.
- True or false. If the \begin{align*}a\end{align*} value is far from zero then you have a narrow graph.
- True or false. If the \begin{align*}a\end{align*} value is close to zero then you have a narrow graph.
- True or false. The \begin{align*}c\end{align*} value helps you to predict the symmetry of a parabola.
- 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.
Directions: Match the three graphs to their equations. Describe the graphs using words. Identify their \begin{align*}a, b,\end{align*} and \begin{align*}c\end{align*} values.
- \begin{align*}y=3x^2-2\end{align*}
- \begin{align*}y=x^2+x-3\end{align*}
- \begin{align*}y=-\frac{1}{2}x^2+2\end{align*}
Graph A
Graph B
Graph C
Graph the following equations using a t-table:
- \begin{align*}y=x^2-1\end{align*}
- \begin{align*}y=-x^2+x\end{align*}
- \begin{align*}y=\frac{1}{2}x^2+1\end{align*}
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.
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