3.6

Students are expected to:• Construct and analyse graphs and tables relating two

variables.• Solve problems using graphing technology.• Develop and apply strategies for solving problems.• Interpret solutions to equations based on context.• Explore and describe the dynamics of change depicted

in tables and graphs.• Solve problems by modeling real-world phenomena.• Model real-world phenomena with exponential

equations.• Express problems in terms of equations and vice versa.

Exponential Functions

An is a function

of the form ,

where 0, 0, and 1,

and t

expone

exponent vahe riab

ntial f

must be a .

unction

le

xy

b

b

b

a

a

First, let’s take a look at an exponential function:

2xy x y

0 1

1 2

2 4

-1 1/2

-2 1/4

Let’s examine exponential functions. They are different than any of the other types of functions we’ve studied because the independent variable is in the exponent.

xxf 2Let’s look at the graph of this function by plotting some points. x 2x

3 8 2 4 1 2 0 1

-1 1/2 -2 1/4 -3 1/8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

2

121 1 f

Recall what a negative exponent means:

BASE

xxf 2

xxf 3

Compare the graphs 2x, 3x , and 4x

Characteristics about the Graph of an Exponential Function where a > 1 xaxf

What is the domain of an exponential function?

1. Domain is all real numbers

xxf 4

What is the range of an exponential function?

2. Range is positive real numbers

What is the x intercept of these exponential functions?

3. There are no x intercepts because there is no x value that you can put in the function to make it = 0

What is the y intercept of these exponential functions?

4. The y intercept is always (0,1) because a 0 = 1

5. The graph is always increasingAre these exponential functions increasing or decreasing?

6. The x-axis (where y = 0) is a horizontal asymptote for x -

Can you see the horizontal asymptote for these functions?

All of the transformations that you learned apply to all functions, so what would the graph of look like?

xy 232 xy

up 3

xy 21up 1

Reflected over x axis 12 2 xy

down 1right 2

xy 2

Reflected about y-axis This equation could be rewritten in a different form:

x

xxy

2

1

2

12

So if the base of our exponential function is between 0 and 1 (which will be a fraction), the graph will be decreasing. It will have the same domain, range, intercepts, and asymptote.

There are many occurrences in nature that can be modeled with an exponential function. To model these we need to learn about a special base.

The Base “e” (also called the natural base)To model things in nature, we’ll need a base that turns out to be between 2 and 3. Your calculator knows this base. Ask your calculator to find e1. You do this by using the ex button (generally you’ll need to hit the 2nd or yellow button first to get it depending on the calculator). After hitting the ex,

you then enter the exponent you want (in this case 1) and push = or enter. If you have a scientific calculator that doesn’t graph you may have to enter the 1 before hitting the ex. You should get 2.718281828

Example for TI-83

xxf 2

xxf 3

xexf

This says that if we have exponential functions in equations and we can write both sides of the equation using the same base, we know the exponents are equal.

If au = av, then u = v

82 43 x The left hand side is 2 to the something. Can we re-write the right hand side as 2 to the something?

343 22 xNow we use the property above. The bases are both 2 so the exponents must be equal.

343 x We did not cancel the 2’s, We just used the property and equated the exponents.

You could solve this for x now.

Let’s try one more:

8

14 x The left hand side is 4 to

the something but the right hand side can’t be written as 4 to the something (using integer exponents)

We could however re-write both the left and right hand sides as 2 to the something.

32 22 x

32 22 xSo now that each side is written with the same base we know the exponents must be equal.

32 x

2

3x

Check:

8

14 2

3

8

1

4

1

2

3 8

1

4

12 3

Growth Decay

Exponential function – A function of the form y=abx, where b>0 and b1.

Step 1 – Make a table of values for the function.

y x 3

31

32

2 1

9

3 0

3 2

3 3

2

x y

0 1

2

3

9

27

Now that you have a data table of ordered pairs for the function, you can plot the points on a graph.

Draw in the curve that fits the plotted points.

y

x

y

x

(-2, 1/9) (0,1) (2,9)

Domain – The collection of all input values of a function. These are usually the “x” values.

Range – The collection of all output values of a function. These are usually the “y” values.

Describe the domain and range of the function y = -5x.

Domain – The domain of the function is all real numbers since the function is defined for all x-values.

Range – The range of the function is all negative real numbers.

If a quantity increases by the same proportion r in each unit of time, then the quantity displays exponential growth and can be modeled by the equation

y C r t ( )1Where

C = initial amount

r = growth rate (percent written as a decimal)

t = time where t 0

(1+r) = growth factor where 1 + r > 1

You deposit $1500 in an account that pays 2.3% interest compounded yearly,

1) What was the initial principal (P) invested?

2) What is the growth rate (r)? The growth factor?

3) Using the equation A = P(1+r)t, how much money would you have after 2 years if you didn’t deposit any more money?

3 ) A P r

A

A

t

( )

( . )

$ .

1

1500 1 0 023

1569 79

2

1) The initial principal (P) is $1500.

2) The growth rate (r) is 0.023. The growth factor is 1.023.

If a quantity decreases by the same proportion r in each unit of time, then the quantity displays exponential decay and can be modeled by the equation

y C r t ( )1

Where

C = initial amount

r = growth rate (percent written as a decimal)

t = time where t 0

(1 - r) = decay factor where 1 - r < 1

You buy a new car for $22,500. The car depreciates at the rate of 7% per year,

1) What was the initial amount invested?

2) What is the decay rate? The decay factor?

3) What will the car be worth after the first year? The second year?

1) The initial investment was $22,500.

2) The decay rate is 0.07. The decay factor is 0.93.

3 1

22 500 1 0 07

20 925

1

) ( )

, ( . )

$ ,

y C r

y

y

t

y C r

y

y

t

( )

, ( . )

$ .

1

22 500 1 0 07

19460 25

2

1) Make a table of values for the function using x-values of –2, -1, 0, 1, and 2. Graph the function. Identify the domain and range of the function. Does this function represent exponential growth or exponential decay?

2) Your business had a profit of $25,000 in 1998. If the profit increased by 12% each year, what would your expected profit be in the year 2010? Identify C, t, r, and the growth factor. Write down the equation you would use and solve.

3) Iodine-131 is a radioactive isotope used in medicine. Its half-life or decay rate of 50% is 8 days. If a patient is given 25mg of iodine-131, how much would be left after 32 days or 4 half-lives. Identify C, t, r, and the decay factor. Write down the equation you would use and solve.

yx

1

6

x yx

1

6

2 1

66

22

1 1

66

11

0 1

6

0

1 1

6

1

2 1

6

2

y

36

6

11

61

36

The domain of this function is the set of all real numbers.

The range of this function is the set of all positive real numbers.

This function represents exponential decay.

C = $25,000

T = 12

R = 0.12

Growth factor = 1.12

y C r

y

y

y

t

( )

$ , ( . )

$ , ( . )

$ , .

1

25 000 1 0 12

25 000 1 12

97 399 40

12

12

C = 25 mg

T = 4

R = 0.5

Decay factor = 0.5

y C r

y m g

y m g

y m g

t

( )

( . )

( . )

.

1

25 1 0 5

25 0 5

1 56

4

4