<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

Identification of Quadratic Models

Identify vertices and up and down parabolas

Atoms Practice
Estimated3 minsto complete
%
Progress
Practice Identification of Quadratic Models
 
 
 
MEMORY METER
This indicates how strong in your memory this concept is
Practice
Progress
Estimated3 minsto complete
%
Practice Now
Turn In
Understanding the Equations of Parabolas
License: CC BY-NC 3.0

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 U-shaped graph.Equations with the ‘\begin{align*}x\end{align*}’ variable raised to the 2nd power are called quadratic equations and their graphs are always parabolas.

Here is a quadratic equation.

\begin{align*}y=x^2 -2\end{align*}

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, you name 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\end{align*}’ and ‘\begin{align*}b\end{align*}’ are coefficients and can be either positive or negative.  The value of ‘\begin{align*}c\end{align*}’ is a constant.  All of these values affect 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

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.

2. if the graph opens upward or downward.

A positive value of ‘\begin{align*}a\end{align*}’ will give a graph that opens upwards while a negative value of ‘\begin{align*}a\end{align*}’ will give a graph that opens downwards.

What about the ‘\begin{align*}b\end{align*}’ value?

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 \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 where the graph will cross 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.

Let’s look at some graphs.

License: CC BY-NC 3.0

License: CC BY-NC 3.0

License: CC BY-NC 3.0

Looking at these graphs, and knowing what the \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} values of the quadratic equation represent, will help to determine the equation of the graph. Here is a chart to help you understand what you can determine by these graphs.

License: CC BY-NC 3.0

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 ‘\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 opens upward, the ‘\begin{align*}a\end{align*}’ value must be positive.

You also know that the ‘\begin{align*}c\end{align*}’ value tells you the \begin{align*}y\end{align*}-intercept on the graph. So, if you know the \begin{align*}y\end{align*}-intercept, then you know the ‘\begin{align*}c\end{align*}’ value.

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

Let’s look at an example.

Write the equation for the given graph.

License: CC BY-NC 3.0

First, start with what you know about the values of ‘\begin{align*}a\end{align*}’ and ‘\begin{align*}c\end{align*}’.

\begin{align*}a\end{align*}: Graph opens downward so \begin{align*}a < 0\end{align*}.

\begin{align*}c\end{align*}: The \begin{align*}y\end{align*}-intercept is (0, 3) so \begin{align*}c = 3\end{align*}.

Next, construct a table of values from the graph.

 \begin{align*}x\end{align*}  \begin{align*}y\end{align*}
-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 \begin{align*}a = -1\end{align*}.

\begin{align*}\begin{array}{rcl} y &=& ax^2 + bx + c \\ y &=& -x^2 + bx + 3 \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 + 3\end{align*}

Point: (-1, 2)

\begin{align*}\begin{array}{rcl} 2 &=& - (-1)^2 + b(-1) + 3 \\ 2 &=& -1 +-b + 3 \\ 2 &=& -b +2 \\ b &=& 0 \end{array}\end{align*}

The answer is \begin{align*}y=-x^2+3\end{align*}.

Examples

Example 1

Earlier, you were given a problem about the parabola.

License: CC BY-NC 3.0

First, start with what you know about the values of ‘\begin{align*}a\end{align*}’ and ‘\begin{align*}c\end{align*}’.

\begin{align*}a\end{align*}: Graph opens upward so \begin{align*}a > 0\end{align*}.

\begin{align*}c\end{align*}: The \begin{align*}y\end{align*}-intercept is (0, 0) so \begin{align*}c = 0\end{align*}.

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.

License: CC BY-NC 3.0

First, start with what you know about the values of ‘\begin{align*}a\end{align*}’ and ‘\begin{align*}c\end{align*}’.

\begin{align*}a\end{align*}: Graph opens upward so \begin{align*}a > 0\end{align*}.

\begin{align*}c\end{align*}: The \begin{align*}y\end{align*}-intercept is (0, -4) so \begin{align*}c = -4\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^2-4\end{align*}.

Example 3

If the \begin{align*}c\end{align*} value is 4, where is the \begin{align*}y\end{align*}-intercept of the graph?

If \begin{align*}c = 4\end{align*}, the \begin{align*}y\end{align*}–intercept or the point where the curve crosses the \begin{align*}y\end{align*}-axis is (0, 4).

Example 4

If the ‘\begin{align*}a\end{align*}’ value is −3, will the parabola open upward or downward?

If \begin{align*}a < 0\end{align*}, then the graph opens downward so when \begin{align*}a = -3\end{align*}, the graph will open downward.

Example 5

If the parabola opens upward, which value is positive \begin{align*}a, b\end{align*} or \begin{align*}c\end{align*}.

When the graph opens upward or downward, the value of ‘\begin{align*}a\end{align*}’ is affected. Therefore if the graph opens upward you know that ‘\begin{align*}a\end{align*}’ is positive.

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*} intercept is the same as the \begin{align*}c\end{align*} 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 \begin{align*}a\end{align*} value is to zero the wider the parabola.

8. True or false. The closer the \begin{align*}a\end{align*} value is to zero the narrower the parabola.

9. True or false. The \begin{align*}b\end{align*} value determines the axis of symmetry.

10. What does the \begin{align*}c\end{align*} 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 \begin{align*}a\end{align*} and \begin{align*}c\end{align*} values and a t-table to help you.

13.

License: CC BY-NC 3.0

14.

License: CC BY-NC 3.0

15.

License: CC BY-NC 3.0

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.13. 

Resources

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More

Vocabulary

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 =ax^2 + bx + c = 0, where a, b, and c are real constants and a\ne 0.

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

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0
  4. [4]^ License: CC BY-NC 3.0
  5. [5]^ License: CC BY-NC 3.0
  6. [6]^ License: CC BY-NC 3.0
  7. [7]^ License: CC BY-NC 3.0
  8. [8]^ License: CC BY-NC 3.0
  9. [9]^ License: CC BY-NC 3.0
  10. [10]^ License: CC BY-NC 3.0
  11. [11]^ License: CC BY-NC 3.0

Explore More

Sign in to explore more, including practice questions and solutions for Identification of Quadratic Models.
Please wait...
Please wait...