Lesson 1

Lesson 1: Exponential Functions

Focus

How can you describe the growth and decay of certain elements? One way is to say that an amount has doubled, tripled, halved, and so on.

An example of a quantity that keeps decreasing by the same amount over a specific time is the mass of a radioactive isotope. A radioactive isotope is a chemical element that is unstable and breaks down or decays into other elements. The time it takes for the element to decay to half its original amount is called the element’s half-life.

In Canada, about one out of every three patients entering a hospital will undergo some form of nuclear-medicine procedure.

In this lesson you will examine quantities that are increasing or decreasing by the same amount over time. What would the graph of the mass of a medical isotope look like over time? Is there a type of function that can be used to describe this relationship? If so, how can you use these types of functions in other applications?

Lesson Outcomes

At the end of this lesson you will be able to

• sketch a graph of an exponential function
• apply transformations to the graph of an exponential function
• identify characteristics of the graph of an exponential function
• solve problems involving exponential growth or decay

Lesson Question

You will investigate the following question:

• How can exponential functions be graphed and analyzed?

Assessment

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

• completion of the Lesson 1 Assignment (Download the Lesson 1 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Glossary Terms and Formula Sheet
• work under Project Connection

Self-Check activities are for your own use. You can compare your answers to suggested answers to see if you are on track. If you have difficulty with concepts or calculations, contact your teacher.

Remember that the questions and activities you will encounter provide you with the practice and feedback you need to successfully complete this course. You should complete all questions and place your responses in your course folder. Your teacher may wish to view your work to check on your progress and to see if you need help.

Time

Each lesson in Mathematics 30-1 Learn EveryWare is designed to be completed in approximately two hours. You may find that you require more or less time to complete individual lessons. It is important that you progress at your own pace, based on your individual learning requirements.

This time estimation does not include time required to complete Going Beyond activities or the Module Project.

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

• identify powers
• evaluate expressions
• apply transformations

Are You Ready?

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

1. Identify the coefficient, base, exponent, and power of the following expressions.
   1. 2x5 Answer
   2. (−3x)−4 Answer
   3. −x3 Answer
2. Evaluate the following expressions.
   1. −23 Answer
   2. (−3)3 Answer
   3. Answer
   4. 4−3 Answer
   5. x0 Answer
3. Given the graph of y = f(x), sketch the graph of Describe the transformations of the graph.
   
   
    
   
   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 the definition of power and how to evaluate powers in Power.

View “Negative Exponent Intuition” to review negative exponents.

View Graphing Transformations to review graphing using transformations. You might also review the transformations summary sheet you made in Module 1.

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

Discover

A value can increase very quickly when doubled over a period of time. This idea is illustrated in Try This 1.

Try This 1

Logan asked his father for a weekly allowance. Logan suggested he get one penny the first week and that the allowance double each of the following weeks. Logan’s father thought he should investigate the idea a little further before deciding. He made the following chart:

1. Complete the rest of the chart. Sketch a graph of allowance (cents) as a function of the week. Describe the shape of the graph.
2. How could you determine the amount of allowance for any specified week?
3. After six weeks Logan would still be receiving less than a dollar per week. Determine the allowance, in dollars, Logan would receive in week 12. (This would be about three months.)

Save your responses in your course folder.

Share 1

With a partner or group discuss the following question based on the graph you created in Try This 1:

Should Logan’s father agree to Logan’s idea for his allowance? Why or why not?

If required, save a record of your discussion in your course folder.

Explore

In Try This 1 you worked with an example of exponential growth. You may have determined that after about three months, in week 12, Logan’s allowance would have been 4096¢, or $40.96.

The relationship that describes the growth of Logan’s allowance in cents, A, as a function of week, w, is A = 2w. You will notice that the variable w is an exponent; therefore, A = 2w is considered an exponential function.

An exponential function is a function in the form y = cx, where c is a constant, c > 0, c ≠ 1, and x is a variable.

Different letters can be used to represent the base of an exponential function. A common form is y = bx. In your textbook and in this course, the letter c is used to represent the base, y = cx. This is to avoid confusion with the transformation parameters a, b, h, and k that were introduced in the first module on transformations.

In Try This 2 you will explore the graphs of exponential functions.

Try This 2

Open Exponential Functions.

You may have noticed from Try This 2 that the graph increases when c > 1. This graph can be used to describe exponential growth.

Source: Pre-Calculus 12.
Whitby, ON: McGraw-Hill Ryerson, 2011.
Reproduced with permission.

When 0 < c < 1, the graph decreases. This graph can be used to describe exponential decay.

Source: Pre-Calculus 12.
Whitby, ON: McGraw-Hill Ryerson, 2011.
Reproduced with permission.

The characteristics of all exponential functions of the form y = cx, c > 0, c ≠ 1 are as follows:

• The domain is {x|x ∈ R}.
• The range is {y|y > 0, y ∈ R}.
• There is no x-intercept.
• The y-intercept is 1.
• The horizontal asymptote is y = 0.

Self-Check 1

In Try This 2 you examined the graphs of different exponential functions. You determined characteristics of each function from the function’s graph. In Try This 3 you will look at determining the equation of an exponential function when given the function’s graph.

Try This 3

The medical isotope Iodine-131 is produced at Chalk River Laboratories in Ontario. The isotope is used in nuclear medicine. A radioactive sample of Iodine-131 has a half-life of 8 d. This means that after eight days, half of the original amount of the sample has decayed. The original sample is 1.0 g. The graph of the amount of Iodine, A, remaining over time, t, in 8-d intervals is shown.

1. What are the domain and range of this graph?
2. Based on the shape of the graph, is c > 1 or 0 < c < 1?
3. Place the three data points in a table like the one that follows. As the x-value increases by 1, by what factor does the y-value change? This factor describes the pattern of decay.
   
   
    

   x	y

4. Use your answer from question 3 as the c-value in an exponential function that would model the relationship between the amount of Iodine-131, A, and time, t, in 8-d intervals.
5. To verify your function from question 4, substitute the point (2, 0.25) into your function. Is your function correct?
6. Estimate how many days it would take for there to be 0.0625 g of Iodine-131 remaining. Explain how you determined your answer.

Save your responses in your course folder.

In Try This 3 you may have determined that the function that modelled the amount of Iodine-131, A, and time, t, in 8-d intervals was . The function A = 2−t could also be used, since .

The base is one-half because the amount of Iodine-131 is decreasing by half over each 8-d interval. The domain of the function is {t|t ≥ 0, t ∈ R}, and the range is {A|0 < A ≤ 1, A ∈ R}. You may have determined that it would take 4 eight-day intervals, or 32 d, for 0.0625 g of Iodine-131 to be left.

View Determining an Exponential Function to see an example of how to determine the equation of an exponential growth function from its graph.

Self-Check 2

Remember that when you sketch a graph, you must apply the transformations in the order of stretches and reflections before translations.

In Try This 4 you may have found that the parameters a, b, h, and k change the graph of an exponential function in the form f(x) = a(c)b(x–h) + k in the ways summarized in the following chart. Note: You may find it helpful to look at Multiple Transformations from Try This 4 as you review this chart.

Parameter	Transformation	Example
a	vertical stretch by a factor of |a| about the x-axis when a < 0, reflection in the x-axis mapping notation: (x, y) → (x, ay)
b	horizontal stretch by a factor of  about the y-axis when b < 0, reflection in the y-axis mapping notation:
h	horizontal translation of h units left or right mapping notation: (x, y) → (x + h, y)
k	vertical translation of k units up or down mapping notation: (x, y) → (x, y + k)

In Try This 5 you learned how to determine the equation of a function that models a given situation. You may have written the function as , where

• c = 1.6
  This value describes the rate of growth. The increase is 60%, so 1 + 0.60 = 1.6.
  
•  
  This value describes a horizontal stretch by a factor of 4. The increase occurs every 4 d.
• a = 300
  This value describes a vertical stretch by a factor of 300. The initial number of bacteria is 300.

Connect

Lesson 1 Assignment

Lesson 1 Summary

In this lesson you explored exponential functions and how they can be used to describe growth or decay of quantities.

The characteristics of all exponential functions of the form y = cx, c > 0, c ≠ 1 are as follows:

• If c > 1, the function is increasing and is an exponential growth function.
• If 0 < c < 1, the function is decreasing and is an exponential decay function.
• The domain is {x|x ∈ R}.
• The range is {y|y > 0, y ∈ R}.
• There is no x-intercept.
• The y-intercept is 1.
• The horizontal asymptote is at y = 0.

When an exponential function is in the form y = a(c)b(x–h) + k, the parameters a, b, h and k correspond to the following transformations:

Exponential functions can be used to model real-life applications of exponential growth or decay. In the next lesson you will explore how to solve exponential equations.