### Nonlinear Trends in Data

Data in a scatterplot can show a relationship that is not linear. If the points in a scatterplot approximate the shape of a parabola, a quadratic function may be a better fit for the data.

A

**scatterplot**is a data display consisting of the graph of a set of ordered pairs. Lines of best fit are often used to model data shown in a scatterplot. However, the trend depicted by the data may not be linear. A**nonlinear relationship**is a relationship between two variables for which a linear model is not a good fit. A**quadratic relationship**is a nonlinear relationship for which a quadratic function is an appropriate model. If the pattern formed by the data points in a scatterplot approximates a parabola, a quadratic model is appropriate.### 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. |

**Regression**is the mathematical process for determining the equation of the graph that best models a set of data. Linear regression is used to determine the equation of a line that fits a data set. Similarly, quadratic regression is used to identify the equation of a parabola that models a data set. Like linear regression, quadratic regression is usually implemented by using technology.

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 $x$-axis at approximately $x=15$. 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.

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 $x$, and evaluate to find $y$.
If the trend continues, in 3 years, the company will make about $9.4 million in profit.

$\begin{aligned}y &= 0.1023x^2 - 0.8426x + 3.0767\\y &= 0.1023\cdot 13^2 - 0.8426\cdot 13 + 3.0767\\y &= 9.4116\end{aligned}$