Chapter 6: Exponential Functions - Mr. McDonald'smrmcdonaldshomepage.weebly.com/uploads/1/3/9/2/13921997/...Chapter 6: Exponential Functions Section 6.4 149 Solving a Problem Using

Chapter 6: Exponential Functions Section 6.1

135

Chapter 6: Exponential Functions

Section 6.1: Exploring Characteristics of Exponential Functions

Terminology:

Exponential Functions:

A function of the form:

𝑦 = 𝑎(𝑏)𝑥

Where 𝑎 ≠ 0, 𝑏 > 0, 𝑏 ≠ 1

a is the coefficient, b is the base, and x is the exponent.

Exponential Functions and Their Characteristics

Graph the following exponential function and interpret its characteristics.

(a) 𝑓(𝑥) = 10𝑥

Table: Graph:

𝑥 𝑓(𝑥)

−3

−2

−1

0

1

2

3

Characteristics:

1. Number of x-intercept:___________

2. Coordinates of y-intercept:__________

3. End Behaviour:_______________

4. Domain: __________________

5. Range: ___________________

6. Number of Turning Points: ____________________

x- 4 - 2 2 4

y

- 200

200

400

600

800

1000


136

(b) 𝑔(𝑥) = (1

2)

𝑥

Table: Graph:

𝑥 𝑓(𝑥)

−3

−2

−1

0

1

2

3

Characteristics:



3. End Behaviour:_______________

4. Domain: __________________

5. Range: ___________________


(c) ℎ(𝑥) = 10(2)𝑥

Table: Graph:

𝑥 𝑓(𝑥)

−3

−2

−1

0

1

2

3

Characteristics:



3. End Behaviour:_______________

4. Domain: __________________

5. Range: ___________________


x- 4 - 2 2 4

y

- 2

2

4

6

8

10

x- 4 - 2 2 4

y

- 25

25

50

75

100

125

150

175


137

(d) ℎ(𝑥) = 8 (1

4)

𝑥

Table: Graph:

𝑥 𝑓(𝑥)

−3

−2

−1

0

1

2

3

Characteristics:



3. End Behaviour:_______________

4. Domain: __________________

5. Range: ___________________


SUMMARY:

What effect does the value of b have on the graph and characteristics of the exponential

function?

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

What effect does the value of a have on the graph and characteristics of the exponential

function?

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

x- 4 - 2 2 4

y

- 80

80

160

240

320

400

480

560


138

Section 6.2: Exploring Characteristics of Exponential Equations

Terminology:

Euler’s Number:

The symbol e is a constant known as Euler’s number. It is an irrational number

that equals 2.718… This number occurs naturally in some situations where a

quantity increases continuously, such as increasing populations.

Connecting the Characteristics of an Increasing Exponential

Function to its Equation and Graph

Predict the number of x-intercepts, the y-intercept, the end behaviour, the

domain, and the range of the following functions. Also determine whether

this function is increasing or decreasing:

(a) 𝑦 = 𝑒𝑥 1. Number of x-intercept:___________


3. End Behaviour:_______________

4. Domain: __________________

5. Range: ___________________

6. Increasing or decreasing:______________

(b) 𝑦 = 9 (2

3)

𝑥



3. End Behaviour:_______________

4. Domain: __________________

5. Range: ___________________


(c) 𝑦 = 2(5)𝑥 1. Number of x-intercept:___________


3. End Behaviour:_______________

4. Domain: __________________

5. Range: ___________________



139

(d) 𝑦 = 8 (3

4)

𝑥



3. End Behaviour:_______________

4. Domain: __________________

5. Range: ___________________


(e) 𝑦 =1

2(

5

3)

𝑥



3. End Behaviour:_______________

4. Domain: __________________

5. Range: ___________________



140

Matching an Exponential Equation with its Corresponding Graph

Which exponential function matches each graph below? Provide your reasoning

i) 𝑦 = 3(0.2)𝑥

ii) 𝑦 = 4(3)𝑥

iii) 𝑦 = 4(0.5)𝑥

iv) 𝑦 = 2(4)𝑥

x- 4 - 3 - 2 - 1 1 2 3 4

y

- 2

2

4

6

8

x- 4 - 3 - 2 - 1 1 2 3 4

y

- 2

2

4

6

8

x- 4 - 3 - 2 - 1 1 2 3 4

y

- 2

- 1

1

2

3

4

5

x- 4 - 3 - 2 - 1 1 2 3 4

y

- 2

- 1

1

2

3

4

5


141

Section 6.3: Solving Exponential Equations

Laws of Exponents

Remember from grade 10:

Law 1: Product of Powers

When we multiply two powers with the same base, we must add the values of the

exponents.

𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛

Law 2: Quotient of Powers

When we divide two powers with the same base, we must subtract the values of the

exponents.

𝑎𝑚

𝑎𝑛= 𝑎𝑚−𝑛 𝑜𝑟 𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚−𝑛

Law 3: Power of a Power

When an exponent is applied to a power, we must multiply the powers of the two

exponents.

(𝑎𝑚)𝑛 = 𝑎𝑚 ∙ 𝑛

Law 4: Power of a Product

When an exponent is applied to the product of two numbers, we use the distributive

property and apply that exponent to each of the factors.

(𝑎 ∙ 𝑏)𝑚 = 𝑎𝑚 ∙ 𝑏𝑚

Law 5: Power of a Quotient

When an exponent is applied to the quotient of two numbers, we use the distributive

property and apply that exponent to both the numerator and denominator (divisor and

dividend).

(𝑎

𝑏)

𝑚

=𝑎𝑚

𝑏𝑚


142

Law 6: Power of Zero

When a base has an exponent of zero, the value of the power is one.

𝑎0 = 1

Law 7: Power of One

When a base does not have an exponent, it actually has an exponent of one.

𝑎 = 𝑎1

Law 8: Negative Exponents

When a power has negative exponent, it can be rewritten as the reciprocal of the base

with a positive exponent.

𝑎−𝑚 =1

𝑎𝑚 𝑜𝑟 (

𝑎

𝑏)

−𝑚

= (𝑏

𝑎)

𝑚

Law 9: Rational Exponents

When a power has a rational exponent, its numerator represents the applied the

exponent and the denominator represents the index of the applied radical (root).

𝑎𝑚

𝑛⁄ = √𝑎𝑚𝑛 𝑜𝑟 𝑎

𝑚𝑛⁄ = ( √𝑎

𝑛)

𝑚

Solving Exponential Equations when Bases are Powers of One Another

It is important to note that when you are dealing with exponential equations, if there is a

single base on each side of the equation and both bases are equal, than the exponents

must also be equal. The opposite is also true.

Solve each Equation

(a) 5𝑥 = 125 (b) 32𝑥 =1

81


143

(c) 9𝑥 = √27 (d) 4𝑥+3 = 82𝑥

(e) 2(4)2𝑥 =1

32 (f) 5𝑥+2 = √125

(g) 9𝑥−1 = 3𝑥−3 (h) 4𝑥+5 = 642𝑥

Terminology:

Half-Life Exponential Function:

A function of the form

𝑦 = 𝑎 (1

2)

𝑥ℎ

Where the base is 1

2, 𝑎 ≠ 0, and ℎ ≠ 0; the value of h is called the half-life because

it corresponds to the point on the graph of the function where the function is half

of its original value, a.


144

Determining the Half-Life of an Exponential Function

Ex1. When driving underwater, the light decreases as the depth of the diver

increases. On a sunny day off the coast of Vancouver Island, a diving team recorded

100% visibility at the surface but only 25% visibility 10 m below the surface.

Determine the half-life equation and use it to determine when the visibility will be

half of that at the surface.

Ex2. On another day, the same diving team noticed that the diving conditions were

worse. They recorded 100% visibility at the surface and 3.125% visibility 8 m below

the surface. Determine the half-life equation and use it to determine when the

visibility will be half of that at the surface?


145

Solving Exponential Equations When the Bases are Not

Powers of One Another

Solve the following exponential equation. Round your answer to one decimal place.

(a) 2𝑥+1 = 5𝑥−1 (b) 3𝑥−5 = 5 𝑥

4

X=2.5 x=7.9

Done using graphs

NOTE: To do this you can either use trial and error or use your graphing calculator.

For the graphing calculator use the following steps:

1. Use the right hand side of the equation to be 𝑦1 = in the y= menu

2. Use the left hand side of the equation to be 𝑦2 = in the y= menu

3. Hit graph, then manipulate the window until you can see the intersection of the

two graphs

4. Hit 2nd and trace and select intersect.

5. Hit Enter three times and your answer will be the x value that is given.


146

Word Problems

A cup of coffee cools exponentially over time after it is brought into a car. The cooling is described by the function shown where T is the temperature of the coffee in degrees Celsius with respect to time, m,

in minutes. 𝑇(𝑚) = 75 (1

5)

𝑚

20. Determine how long it would take to reach a temperature of 3°C?

The half-life of a certain drug in the bloodstream is 6 days. If a patient is given 480 mg, algebraically determine how long it will take for the amount of drug in the patient’s body to reduce to 15 mg. Recall

that the equation for the half-life of a function is 𝐴(𝑡) = 𝐴𝑜 (1

2)

𝑡

ℎ.


147

A laboratory assistant decided to observe the reproductive properties of a new strain of bacteria. The assistant started observing a population of 300 bacteria and noted that the bacteria population doubled

every 5 minutes. This can be modeled by 𝑃(𝑡) = 300(2)𝑡

5 , determine the population after 2 hours. A university student studied and recorded the population of a bacterial culture every 30 minutes. The

exponential function that models the bacteria population during the study is 𝑃(𝑡) = 50(2)𝑡

30, use it to find the bacteria population 180 minutes after the study began.


148

Section 6.4: Exponential Regression

Creating Graphical and Algebraic Models of Given Data

Ex1. The population of Canada from 1871 to 1971 is shown in the table below. In the third

column, the values have been rounded.

(a) Using graphing technology, create a graphical model and an algebraic exponential model

for the data. State the regression equation.

(b) Assuming that the population growth continued at the same rate to 2011, estimate the

population in 2011. Round your answer to the nearest million.


149

Solving a Problem Using an Exponential Regression Model

Ex1. Sonja did an experiment to determine the cooling curve of water. She placed the same

volume of hot water in three identical cups. Then she recorded the temperature of the water in

each cup as it cooled over time. Her data for three trials is given as follows.

(a) Construct a scatter plot to display the data. Determine the equation of the exponential

regression function that models Sonja’s data.

(b) Estimate the temperature of the water 15 min after the experiment began. Round your answer

to the nearest degree.

(c) Estimate when the water reached a temperature of 30 °C. Round your answer to the nearest

minute.


150

Ex2. Emma did the same experiment, but performed only one trial. Her data is given below.

Time (min) 0 5 10 20 30 45

Temperature

(ºC) 90 78 68 52 38 26

(a) Construct a scatter plot to display Emma’s data. Determine the equation of the

exponential regression function that models her data.

(b) Compare the characteristics of the exponential regression function for Emma’s data with

the characteristics of the function for Sonja’s data. How are the graphs the same? How

are they different?

(c) At what time, to the nearest minute, did the water reach 51 °C in Emma’s experiment?

How much longer did it take than in Sonja’s experiment?


151

Section 6.5: Financial Applications Involving Exponential Functions

Terminology:

Compound Interest: The interest earned on both the original amount that was invested and any interest that has accumulated over time. The formula for compound interest is:

𝐴(𝑛) = 𝑃(1 + 𝑖)𝑛

Where 𝑨(𝒏) represents the future value, P represents the principal, i represents the interest rate per compounding period, and n represents the number of compounding periods.

Future Value: The amount that an investment will be worth after a specific period of time.

Principal: The original amount of money that is invested or borrowed.

Compounding Period: The time over which interest is calculated and paid on an investment. (NOTE: /a means per annum or per year)

Simple Interest:

The interest earned only on the original amount that was invested, any interest that has accumulated will not affect the earning of future interest. The formula for simple interest is:

𝐴 = 𝑃(1 + 𝑟𝑡) Where 𝑨represents the future value, P represents the principal, r represents the interest rate per annum, and t represents the time in years.


152

Simple Interest Ex. Jayden decides to invest $1000 of his money in a savings account that offers to pay 5%/a simple interest.

(a) Determine the equation that models this situation

(b) Determine how much his investment would be worth after 5 years.

(c) Determine how much his investment would be worth after 10 years. Ex2. Morgan invests $3200 into a savings account that offers 2.5%/a simple interest.



(c) Determine how much his investment would be worth after 10 years.

Compounded Interest Ex1. Jayden instead decides to invest $1000 of his money in a savings account that offers to pay 5%/a compounded annually.




153

(c) Determine how much his investment would be worth after 10 years.

(d) Referring to you answers to this example and the simple interest example on the previous page, which is a better choice to invest your money, simple interest or compounded interest. Explain.

Ex. Brittany invested $2500 in an account that pays 3.5% /a compounded monthly.


(b) How much would her investment be worth after 4 years (48 months)?

(c) Use graphing technology to estimate when the investment will grow to $3000. Ex2. After her account reached a value of $3000, Brittany reinvested into an account that provided 5%/a, compounded semi-annually.

(a) Determine the equation that models this situation.

(b) How much would the investment be worth after 8 years?

(c) Use graphing technology to determine when the account will be worth $7667.


154

Modelling Appreciation and Depreciation with an Exp. Function Terminology:

Appreciation: The increase in a value over time. Appreciation can be modelled using the standard exponential equation:

𝐴(𝑛) = 𝑃(1 + 𝑖)𝑛

Depreciation: The decrease in a value over time. Depreciation can be modelled using a slightly modified exponential equation:

𝐴(𝑛) = 𝑃(1 − 𝑖)𝑛 Where A(n) = future value, P = initial value, and i is the interest per annum

Ex1. Gina brought a camera for her studio two years ago. Her accountant told her that the camera will have a depreciation rate of 20% per year. When she originally purchased her camera it cost her $2000.

(a) Determine the equation that models the situation.

(b) Determine how much the camera would be worth 4 years after it was purchased.


155

Ex2. Ali inherited a set of rare silver coins from his great-grandfather. An appraiser told Ali that the coins were currently valued at $1000 and their appreciation rate will likely be 2.5%/a.

(a) Determine the equation that would model this situation.

(b) Using your equation, determine the value of the coins after 16 years. Ex3. Hector paid $ 20 000 for a new car, it is estimated to depreciate 5% annually.


(b) What would the value of the car be after 7 years? Ex4. A baseball card valued at $100 appreciates 8%/a.


(b) What would the value of the card be after 12 years?


156

Determining an Exponential Equation from a Table Time 0 2 4 6 8 Amount 1200 300 75 18.75 4.6875

X -3 0 3 6 9 Y 15 30 60 120 240

Time -10 -5 0 5 10 Volume 1 10 100 1000 10000

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Chapter 6: Exponential Functions - Mr. McDonald'smrmcdonaldshomepage.weebly.com/uploads/1/3/9/2/13921997/...Chapter 6: Exponential Functions Section 6.4 149 Solving a Problem Using