Introduction

Two variables have a quadratic relationship if the graph of the the increase in one variable causes the other variable to gradually increase until it reaches a maximum, then the other variable begins to fall. Or, a quadratic relationship between two variables can happen if one variable’s increase causes the other variable to decrease until it reaches a minimum, then start to increase.

A quadratic relationship looks like this:

The x-axis is time and the y-axis is height. The object starts at 5 units of height when time is zero (t = 0). The object reaches its maximum at time unit 2. The maximum height appears to be 7 height units. The object then begins to fall.

This graph is also quadratic. We call this type of quadratic a negative quadratic as it has a minimum.

Example:

1.
https://www.youtube.com/watch?v=WA9POyoDaFs
The data below represents the relationship between a car’s speed at the time of an accident and the skid left by the car.

Is the relationship between miles per hour and the length of the skid quadratic?

We have not learned the tools to algebraically determine if this is a quadratic relationship, but we can make the determination visually. We should sketch out a scatter plot. To make a scatter plot, label the x-axis as the miles per hour, and the y-axis as the other variable, the length of the skid.

Next, note the highest and lowest x-value. For our data set, x = 10 is the lowest and x = 60 is highest. We make 6 vertical lines on the x-axis, one for each value of our x-variable.

Do the same for the y-axis. The lowest value is 6 and the highest value is 200. We can make however many horizontal lines we’d like. We can start at y = 0 and count by 50s, which would mean we can stop after 4 horizontal lines.

Place dots that correspond to the data points in the table. The scatter plot looks like this:

This one is tough to determine, as we cannot see the minimum to be sure. We can say that it looks quadratic. Let’s take a look at another example.

2. A business wants to charge a price for their product that maximizes their profit. If a business charges too much, customers are scared off. The following table represents the price set for different products and the total profit based on that price.

If we make a scatterplot of the data,

This data makes an upside down U, similar to the first example at the beginning of this document. That’s a quadratic shape!

Practice Problems

1. A water heater is installed in a house. The temperature for a water heater is recorded below. Is the relationship between time and temperature quadratic?

2. This data comes from fuel economy.gov and has been slightly modified to make graphing easier.

A certain vehicle’s miles per gallon is tracked based on its driving speed. Is the relationship between speed and miles per gallon a quadratic relationship?