Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Algebra 1

8.12.3 Making Predictions Using a Quadratic Model

Algebra 18.12.3 Making Predictions Using a Quadratic Model
Search for key terms or text.

Activity

A helium balloon is released and floats up before coming back down to the ground.

The following data set shows the height in feet, y y , of the balloon corresponding to the number of minutes passed, x x .

x 1 x 1 (minutes) y 1 y 1 (height in feet)
1 99.7
1.5 130
3 149
4.5 172
7 108
9 84
1.

Use the Desmos below. Click the + symbol in the top left and select a table. Enter the data set into the table.

Access multimedia content

Use the instruction line, y 1 ~ a x 1 2 + b x 1 + c y 1 ~ a x 1 2 + b x 1 + c , to find the curve of best fit.

What is the equation of the curve of best fit? Round coefficients to the nearest hundredth.

2.

How high will the balloon be after 4 minutes? (Hint: Graph the line x = 4 x = 4 to help identify where it intersects with the curve of best fit.)

3.

How high will the balloon be after 10 minutes?

4.

At what time will the balloon first be at a height of 95 feet? (Graph the line y = 95 y = 95 .)

5.

At what time will the balloon return to a height of 95 feet?

6.

When will the balloon reach the ground, or have a height of 0 feet?

Video: Using Technology to Find The Quadratic Regression

Watch the following video to learn more about finding quadratic regression.

Access multimedia content

Self Check

A baseball is thrown into the air. The following data set shows the height in feet, y , of the baseball corresponding to the number of seconds passed, x .

Approximately how long will it be before the baseball hits the ground?

  1. 7.18 seconds
  2. 6.75 seconds
  3. 5.74 seconds
  4. 2.73 seconds

Additional Resources

Making Predictions Using a Quadratic Model

A filtration system is being tested in a new aquarium. A pollutant is continuously added to the aquarium, and the filtration system is turned on until it has removed all of the pollutant from the water.

The following data set shows the milliliters (mL) of pollutant present, y y , corresponding to the number of minutes that have passed, x x , since the experiment started.

Access multimedia content

Let’s enter the instruction line, y 1 ~ a x 1 2 + b x 1 + c y 1 ~ a x 1 2 + b x 1 + c , to find the curve of best fit.

A quadratic regression that closely approximates the 6 plotted points r-squared is approximately 0.99 the output summary shows that A, B, and C are approximately negative 0.29, 37.74, and 95.95, respectivelya vertical line is drawn through x equals 12, with the point (12, 507,492) labeled.

The curve of best fit is determined to be y = − 0.29 x 2 + 37.74 x + 95.95 y = − 0.29 x 2 + 37.74 x + 95.95 .

We can use this curve of best fit to answer questions and make predictions about the amount of pollutant that will be in the water (mL) and the time it takes to completely clean the tank of pollutant.

How many mL of pollutant will be in the water after 12 minutes?

  • To answer this question, we want to know the y y -value on the curve of best fit when the x x -value is 12.
  • We can graph the line x = 12 x = 12 and see where it intersects with the curve of best fit.
A quadratic regression that closely approximates the 6 plotted points r-squared is approximately 0.99 the output summary shows that A, B, and C are approximately negative 0.29, 37.74, and 95.95, respectivelya vertical line is drawn through x equals 12, with the point (12, 507,492) labeled.
  • By moving the cursor over the intersection point, we see that the y y -value is approximately 507.49.
  • This means that after 12 minutes, there will be approximately 507.49 mL of pollutant in the water.

At what time(s) will there be 1000 mL of pollutant in the water?

  • To answer this question, we want to know the x x -value on the curve of best fit when the y y -value is 1000.
  • We can graph the line y = 1000 y = 1000 and see where it intersects with the curve of best fit.
A quadratic regression that closely approximates the 6 plotted points r-squared is approximately 0.99 the output summary shows that A, B, and C are approximately negative 0.29, 37.74, and 95.95, respectivelya vertical line is drawn through x equals 12, with the point (12, 507,492) labeled.
  • The line y = 1000 y = 1000 intersects with the curve of best fit at two points. This means there are two times when there is 1000 mL of pollutant. We see that the x x -values are approximately 31.51 and 99.91.
  • In other words, there will be 1000 mL of pollutant at approximately 31.51 minutes and again at approximately 99.91 minutes after the experiment starts.

How long after the experiment has started will the filtration system have removed all of the pollutant from the aquarium?

  • To answer this question, we want to know the x x -value of the x x -intercept of the curve of best fit.
  • If we graph y = 0 y = 0 , we can click on the intersection point to find its coordinates. The x x -intercept of the curve of best fit is ( 133.92 , 0 ) ( 133.92 , 0 ) .
  • So, approximately 133.92 minutes after the experiment starts, there will be no pollutant left in the water.

Try it

Try It: Making Predictions Using a Quadratic Model

Use the same data set from the example in the section above.

Access multimedia content

The data set shows the milliliters (mL) of pollutant present, y y , corresponding to the number of minutes that have passed, x x , since the experiment started.

1. How many mL of pollutant will be in the water after 47 minutes?

2. At what time(s) will there be 1215 mL of pollutant in the water?

PreviousNext
Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

Citation information

© May 21, 2025 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.