Lesson 7

Lesson 7: Modelling Data Using Exponential and Logarithmic Regressions

Focus

Imagine your favourite warm drink, perhaps tea or hot chocolate. The drink is nice and hot when it’s first made; however, the drink will cool off as it sits. Clearly the temperature is decreasing, but what would a graph of temperature versus time look like? Would it be a straight line? A curve? One way to determine what this graph looks like is to collect data and perform a regression.

In this lesson you will learn about exponential and logarithmic regression, and you will use those equations to model real-world problems.

Lesson Outcomes

At the end of this lesson, you will be able to

• use technology to determine exponential and logarithmic regression equations
• use a regression equation to answer questions in a real-world context

Lesson Question

You will investigate the following question: How can an exponential or logarithmic model be used to solve problems?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 7 Assignment (Download the Lesson 7 Assignment and save it in your course folder now.)
• course folder submissions from Try This activities
• additions to Glossary Terms
• work under Project Connection

Launch

Do you have the background knowledge and skills you need to complete this lesson successfully? Launch will help you find out.

Before beginning this lesson, you should be able to

• use technology to determine the equation of a line of best fit
• use technology to determine the equation of a curve of best fit (quadratic and polynomial)
• use a regression equation to answer questions in a real-world context

Are You Ready?

Complete these questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

1. Logos-to-Go is a company that travels around Alberta supplying teams with hoodies featuring unique logos. The company needs to buy the plain hoodies in bulk to help save money. However, Logos-to-Go does not want to be left with too much stock at the end of each season. The price per hoodie is related to the size of the bulk order. Seven of Logos-to-Go past orders are listed in the following table:
   
   

   Number of Hoodies	Cost per Hoodie ($)
   400	22
   600	19
   250	27
   720	18
   330	25
   820	16
   500	21

   1. Use technology to create a scatter plot and determine an equation for the linear regression function that models this data. Answer
   2. What do the slope and y-intercept in the regression equation represent in this question? Answer
   3. Use the linear regression function model to extrapolate and find the size of order needed to achieve a cost of $15 per hoodie. Answer
2. Chuck is a pitcher for his local slow-pitch team. The height of the ball after being pitched is listed in the following table:
   
   

   Time (s)	0	0.33	0.67	1.0	1.3
   Height (m)	1	2.8	3.5	3.1	1.6

   1. Use technology to create a scatter plot and determine an equation for the quadratic regression function that models this data. Answer
   2. Using your model, what is the maximum height that the ball will reach? Answer
   3. State appropriate values for domain and range in this situation. Answer

If you answered the Are You Ready? questions without difficulty, move to Discover.

If you found the Are You Ready? questions difficult, complete Refresher.

Refresher

Review Try This 2 of Math 30-2: Module 4: Lesson 2 to review creating a graph and equation based on a linear set of data and extracting information from the graph and equation.

Open “Linear Regressions” to see how to make predictions using a linear model.

Review Try This 1 of Math 30-2: Module 4: Lesson 3 to review creating a graph and equation based on a set of quadratic data and extracting information from the graph and equation.

Go to “Quadratic Regressions” to see how to make predictions using a quadratic model.

Go back to the Are you Ready? section and try the questions again. If you are still having difficulty, contact your teacher.

Discover

Try This 1

Each morning a teacher reads the newspaper. He does this with a nice fresh cup of coffee. Many mornings, however, he becomes so focused on his newspaper that the coffee becomes cold and unpleasant to drink.

The following data shows the temperature of the cup of coffee as it sits cooling.

1. Use your graphing calculator to plot the data.
2. Is the temperature increasing or decreasing?
3. What is the shape of your scatter plot? What kind of model do you expect to use with this data?
4. Use your calculator to find a regression equation.
5. Graph the regression equation on the same graph as the scatter plot.
6. What is an appropriate domain and range for this situation?

Explore

In Try This 1, you may have discovered that the model that would best fit the cooling cup of coffee is the exponential regression equation y = 82.63(0.956)x. This is the form of exponential equation you saw in Lesson 1 (y = abx, b > 0, b ≠ 1). The data points lie close to this function; therefore, it is a good model for the data.

Adapted from  Stockbyte/Thinkstock

So far in this lesson, you have looked at exponential regressions. Next, you will focus on logarithmic regressions.

Try This 2

The Iverson family ordered some trees from the provincial shelterbelt program to plant on their acreage.

The following data shows the average growth rate of 20 weeping willow trees after planting. At the time of planting, all the trees were 2 ft in height and were 1 y old.

1. Use your graphing calculator to plot the data.
2. Is the average height increasing or decreasing?
3. What is the shape of your scatter plot? What kind of model do you expect to use with this data?
4. What is an appropriate domain and range in this situation?

Save your responses in your course folder.

If you identified that the weeping willow tree growth looked like a logarithmic function, you are correct. The growth is also an increasing function, as the trees are getting taller as time passes.

Most graphing calculators provide the logarithmic regression equation in the form y = a + b ln x. The abbreviation ln is referring to loge and is usually called a natural logarithm. The number e is an irrational number, like π, and is approximately 2.718. Natural logarithms are important in advanced mathematics, so most regression programs use it instead of log10.

Determine a logarithmic regression equation for the data in Try This 2.

If you are unsure how to calculate a logarithmic regression, see your calculator manual or contact your teacher.

The logarithmic regression for the tree data is y = −0.068 + 8.136 (ln x). Make sure that you are able to do this on your calculator. Graph this function on the same grid as the scatter plot.

The screen on your graphing calculator should look similar to the following:

Self-Check 2

Using your logarithmic regression model from the weeping willow tree example, answer the following questions.

1. Interpolate: What was the average height of the trees at 2.5 y of age (to the nearest tenth of a foot)? Answer
2. Extrapolate: What is the predicted average height of the trees at 20 y of age (to the nearest tenth of a foot)? Is this prediction realistic? Answer
3. According to your model, if the average height of the trees is 17 ft, what is the age of the trees to the nearest tenth of a year? Answer

A logarithmic regression equation

• can be of the form y = a + b ln x
• has end behaviour that extends from quadrant IV to quadrant I (increasing) or quadrant I to quadrant IV (decreasing)

In Try This 3, you reviewed the characteristics of exponential and logarithmic equations and their graphs. These characteristics are summarized in the following table. You used these characteristics to match equations to their graphs.

Self-Check 3

Connect

Lesson 7 Assignment

Lesson 7 Summary

In this lesson you looked at data that modelled either exponential or logarithmic behaviour. You were introduced to both exponential and logarithmic regression. The general formula for exponential functions, y = a(b)x, was applied to the exponential data, and the formula y = a + b ln x was used when investigating and using logarithmic regression.