Quadratic Functions and Modeling

Quadratic Modeling

Using a Parabola to Model Data

Scatterplot Parabola of Best Fit
The scatterplot shows a trend that is modeled by a parabola, which can be sketched by hand or generated using technology to approximate the data. A parabola can be used to approximate the trend. Note that a curve of best fit may not touch all or even any of the points on the scatterplot.

Step-By-Step Example
Applying a Quadratic Model

The table shows data collected in an experiment measuring the height of an object at certain points in time.

Time (s) Height (m)
0 1
2 159
4 269
5 275
7 342
9 330
10 246
12 220
14 86

Estimate the time at which the object landed on the ground.

Step 1
Make a scatterplot of the data set. The scatterplot shows that a quadratic model is probably a good fit for the data.
Step 2
Sketch a parabola that approximates the pattern shown by the data set. The parabola shown models the height of the object over time.
Solution
The parabola intersects the xx-axis at approximately x=15x=15. So the object hit the ground after approximately 15 seconds.

Quadratic Functions of Best Fit

Technology can be used to generate a quadratic function that best fits the quadratic relationship between two variables.
There are many computer programs and calculators with the capability to find the equation of the parabola that best fits a set of data. Spreadsheets are commonly used to create scatterplots, fitted curves, and equations of the curves of best fit. A quadratic model generated by technology can be used to make predictions.
Step-By-Step Example
Modeling Data with a Parabola

A corporation uses the data in the table to track its profits for 10 years and wants to do an analysis for a presentation.

Year Profit (millions of $)
1 2.5
2 1.9
3 1.2
4 1.1
5 1.4
6 1.7
7 2.4
8 3.1
9 3.9
10 4.6

Use a spreadsheet to create a scatterplot of the data table, and estimate the corporation's profit in 3 more years.

Step 1
Enter the data into a spreadsheet.
Step 2

Select the simple scatterplot from the chart options.

In most spreadsheets, the data should be highlighted before selecting the chart. If a chart with no points appears, the data set needs to be highlighted.

Create the graph, and label the axes.
Step 3
Using the available trend line or regression function, fit the parabola to the data, and calculate the quadratic equation.
Solution
Use the equation to estimate the company's profit in 3 more years. Substitute 13 into the equation for xx, and evaluate to find yy.
y=0.1023x20.8426x+3.0767y=0.10231320.842613+3.0767y=9.4116\begin{aligned}y &= 0.1023x^2 - 0.8426x + 3.0767\\y &= 0.1023\cdot 13^2 - 0.8426\cdot 13 + 3.0767\\y &= 9.4116\end{aligned}
If the trend continues, in 3 years, the company will make about $9.4 million in profit.