EXPONENTIAL FUNCTION where a bmi01000971.schoolwires.net/cms/lib05/MI01000971...Exponential Growth functions are also used in real-life scenarios involving compound interest. Compound

7.1 Graph Exponential growth functions No graphing calculators!!!!

EXPONENTIAL FUNCTION A function of the form xbay where a 0 and the base b is a positive number other

than one.

xy 31 a = ______

b = _______

HA = ________________

Horizontal Asymptote

Horizontal Asymptote: A horizontal line that the graph approaches but does not touch or cross*

Domain: _____________________________

Range: ______________________________

x y Ordered Pairs

1 ,

0 ,

1 ,

2 ,

EXPONENTIAL GROWTH A function of the form xbay where a > 0 and the base b > 1

b: b is a called the growth factor

When b is a number greater than 1, the graph will

be an exponential growth function. In an exponential

growth function, we see the y values growing over

time by the same multiple, called the growth factor.

a: a is the y-intercept of the graph, written (0, a)

b must be greater than 1, and a must be greater than

0 in order for the graph to be exponential growth

When a is a number greater than 1, the y values will

be more stretched out, causing the graph to stretch.

When a is a number between 0 and 1, the y values

will be closer together, causing the graph to shrink.

If you were looking at the table and did not know the equation, how could you determine the growth factor?

xy 42

1 a = _______

b = _______

HA = ________________

Domain: _____________________________________ Range: _____________________________________

x y Ordered Pairs

1 ,

0 ,

1 ,

2 ,

Exponential Growth with Shifts (translations): kbay hx

a = y – intercept before the shifts take place

b = growth factor

h = the value that is being added or subtracted from x. It affects

all of the x values of the problem by shifting them LEFT or

RIGHT.

If h is being subtracted in the problem, you will add it to

each of your x values (RIGHT).

If h is being added in the problem, subtract it from each of

your x values (LEFT).

k = whatever number is being subtracted or added to the

OUTSIDE of the function. K affects al the y values of the

graph by shifting them UP or DOWN.

If k is positive, add it to each y value (UP).

If k is negative, subtract it from each y value

(DOWN).

Example: 232 1 xy

a = _______

b = _______

Step 1: Identify a, b, h, k.

Step 2: Find the x and y values for xy 32 (no shifts)

Step 3: Apply h to each of the x values.

Step 4: Apply k to each one of the y values.

Step 5: Determine and graph the HA.

Step 6: Create ordered pairs and graph.

Step 7: Determine the Domain and Range.

h = _______

ONLY affects your x values Describe shift: _____________

k = _______

ONLY affects your y values Describe shift: _____________

HA = ________________

Domain: __________________________

Range: ___________________________

x Apply h

if necessary y Apply k

if necessary

1

,

0

,

1

,

2

,

Example: 142

1 2 xy

a = _______

b = _______


Step 2: Find the x and y values for xy 4

2

1 (no shifts)






h = _______

ONLY affects your x values Describe shift: _____________

k = _______

ONLY affects your y values Describe shift: _____________

HA = ________________

Domain: __________________________

Range: ___________________________

x Apply h


if necessary

1

,

0

,

1

,

2

,

EXPONENTIAL GROWTH MODELS

When a real-life quantity increases by a fixed percent each year (or other time period), the amount of the quantity (y)

after t years can be modeled by the equation:

tray 1

where a is the initial amount of the quantity and r is the percent of increase, written as a decimal.

Example: In the last 12 years, an initial population of 38 buffalo in a state park grew by about 7% per year.

a. Write an exponential growth model giving the number y of buffalo after t years.

b. About how many buffalo were in the park after 7 years?

a.

t

t

y

ray

07.138

1

b. 707.138 y

NOTE: For the story problems, formulas will be provided on the quiz and test, but will NOT be labeled. You must know what each letter represents in the formula, as well as when to use the formula!

Exponential Growth functions are also used in real-life scenarios involving compound interest. Compound interest is paid on

an initial investment, called the principal, AND on previously earned interest. So, we are trying to figure out how much money

this principal and its interest are worth after interest is calculated on the initial principal, then added to the principal and

recalculated, n times per year, over t number of years.

Consider an initial principal (P) deposited in an account that pays interest at an annual rate (r), expressed as a decimal,

compounded n times per year. The amount (A) in the account after t years is given by the equation:

nt

n

rPA

1

Example: You deposit $2900 in an account that pays 3.5% annual interest. Find the balance after 3 year if the interest is

compounded monthly.

nt

n

rPA

1

7.2 Graph Exponential decay functions No graphing calculators!!

EXPONENTIAL DECAY A function of the form xbay where a > 0 and the base b is between 0 and 1

b: b is a called the decay factor

When b is a number between 0 and 1, the graph will

be an exponential decay function. In an exponential

decay function, we see the y values decreasing over

time by the same multiple, called the decay factor.

a: a is the y-intercept of the graph, written (0, a)

When a is a number greater than 1, the y values will

be more stretched out, causing the graph to stretch.

When a is a number between 0 and 1, the y values

will be closer together, causing the graph to shrink.

x

y

2

13 a = _______

b = _______

DECAY FACTOR

HA = ________________

Domain: _____________________________________ Range: _____________________________________

x y Ordered Pairs

2 ,

1 ,

0 ,

1 ,

Exponential DECAY with Shifts (translations): kbay hx

a = y – intercept before the shifts take place

b = decay factor

h = the value that is being added or subtracted from x. It

affects all of the x values of the problem by shifting them LEFT

or RIGHT.

If h is being subtracted in the problem, you will add it to

each of your x values (RIGHT).

If h is being added in the problem, subtract it from each of

your x values (LEFT).

k = whatever number is being subtracted or added to the

OUTSIDE of the function. K affects al the y values of the

graph by shifting them UP or DOWN.

If k is positive, add it to each y value (UP).

If k is negative, subtract it from each y value (DOWN).

Example: 22

13

1

x

y

a = _______

b = _______


Step 2: Find the x and y values for

x

y

2

13 (no shifts)






h = _______

ONLY affects your x values

k = _______

ONLY affects your y values

HA = ________________

Domain: __________________________

Range: ___________________________

x Apply h


if necessary

2

,

1

,

0

,

1

,

Example: 33

12

1

x

y

a = _______

b = _______


Step 2: Find the x and y values for

x

y

3

12 (no shifts)






h = _______

ONLY affects your x values

k = _______

ONLY affects your y values

HA = ________________

Domain: __________________________

Range: ___________________________

x y

2 ,

1

,

0

,

1

,

EXPONENTIAL DECAY MODELS

When a real-life quantity decreases by a fixed percent each year (or other time period), the amount y of the quantity after t

years cab be modeled by the equation:

tray 1

where a is the initial amount of the quantity and r is the is the percent of decrease, written as a decimal.

Example: A new television costs $1200. The value of the television decreases by 21% each year.

a. Write an exponential decay model giving the television's value y (in dollars) after t years.

b. Estimate the value after 2 years.

a. tray 1

b.

7.3 USE FUNCTIONS INVOLVING e

xey 3 a = _______

b = _______

r = _______

Growth or Decay

HA = _______________

Domain: _____________________________________ Range: _____________________________________

x y Ordered Pairs

2 ,

1 ,

0 ,

1 ,

Natural base e Functions

A function on the form rxeay is called a natural base exponential function.

If a > 0 and r > 0, the function is an exponential GROWTH function.

If a > 0 and r < 0, the function is an exponential DECAY function.

e

e is an irrational (non-repeating, not-terminating) constant. Because it is non-repeating and non-terminating, we

have a button in our calculator that represents e , so that we do not have to round it every time we want to use it.

e is often referred to as “the natural base”. One reason is because of where we see it represented in nature. In

nature, many populations grow exponentially, in a manner that is best modeled by e . In general, e helps us

model quantities from nature to money that grow and decay continuously. We will see e used as a base.

EXAMPLE: The population of bacteria in a certain culture at time t is given by , where t is measured in hours.

From the equation, we can see that 100 bacteria are initially present. After 10 hours, where t = 10, there are

bacteria present. After 100 hours, roughly bacteria are present.

Find e on your calculator and use it to evaluate the following values. Round answers to two decimal places.

(2nd

LN, close exponent in parenthesis)

2e = 7.39

2

5

e = 0.08 5e = 9.36

Instead of a number, use e as a base to graph exponential growth and decay functions.

Example: 12 2 xey

a = _______

b = _______ r = _______

Step 1: Identify a, b, r, h, k.

Step 2: Find the x and y values for xey 22

(no shifts)






h = _______

Describe shift: ____________________

k = _______

Describe shift:

_______________________

Growth or Decay

HA = ________________

Domain: __________________________

Range: ___________________________

x y

2

,

1

,

0

,

1

,

Example:

2135.0

xey

a = _______

b = _______ r = _______

Step 1: Identify a, b, r, h, k.

Step 2: Find the x and y values for xey 35.0

(no shifts)






h = _______

Describe shift: ____________________

k = _______

Describe shift:

_______________________

Growth or Decay

HA = ________________

Domain: __________________________

Range: ___________________________

x y

2

,

1

,

0

,

1

,

Use your exponent rules to simplify expressions with e.

e6 e

3

9e

29e

95 xx ee

(4e3x

)2

xe616

5

23

8

4

e

e

69e

CONTINUOUSLY COMPOUNDED INTEREST

When interest is compounded continuously, the amount A in an account after t years is

given by the formula A Pert where P is the principal and t is the annual interest rate expressed as a decimal.

NOTE: In our other interest model, the interest was compounded a certain amount of times.

You deposit $4800 in an account that pays 6.5% annual interest compounded continuously. What is the balance after 3 years?

4 2

6 18

e e

7.4 Evaluate Logarithms and Graph Logarithmic Functions

EXPONENTIAL FORM

QUESTION: What is the result of raising a base of 2 to the third power?

ANSWER: After 222 , we see the answer is 8.

823 ybx

Base Exponent Result Base Exponent Result

LOGARITHMIC FORM By definition a logarithmic situation and an exponential situation are inverses of one

another, so the question needs to change…

Before: QUESTION: What is the result of raising a base of 2

to the third power?

823

Base Exponent Result

After: QUESTION: Base 2 raised to what power gives you 8?

823

328

38log2

Pronounced: “Log base 2 of 8 is 3”

The answer to the logarithm is the exponent needed to

take a base of 2 to 8. The answer to a logarithm is always

an exponent, and that’s how the question is different from

exponential form.

Note: the base must be a positive number

Rewrite the equation in exponential form. (DO THE LOG LOOP)

93

29log

2

3

The LOG LOOP’s job is to take us from

unfamiliar log form to the very familiar

exponential form.

2

15log 25 2

100

1log10

Evaluate the logarithm. (TO FIND ITS VALUE, DO THE LOOP, THEN FIND THE EXPONENT NEEDED)

1log 5.

6log36

64

1log8

CALCULATOR

A common logarithm is one with a base of 10. It is denoted as 10log or simply log . Your calculator is

designed to find the value of common logarithms, so when you press the log button, your calculator is

assuming that you are typing in something that has a base of 10.

Use a calculator to evaluate the logarithm. Round your answer to three decimal places.

903.

8log8log 10

assametheis

213.6log

Base e

e gets its own special logarithm. Anytime we take the elog , we can replace this notation with ln.

ln stands for the “logarithm of natural base e”. To avoid saying that entire phrase we also call ln the

“natural log” for short. This also makes taking the log of e extremely easy to put in the calculator.

lnlog e


204.1

3.0ln3.0log

assametheise

7log e

INVERSE

We have already studied and manipulated several types of inverses. Squaring and square rooting, cubing

and cube rooting, and we know that that a form and its inverse can be changed from one to the other by

switching the place of the x and y and resolving for y. We also know that inverses reflect over the line

xy on the coordinate plane. (For a refresher graph 2xy and xy on your calculator.)

Because logarithmic form and exponential form are inverses of one another, we can convert from one to its

inverse (the other) by following the same method.

Find the inverse of the function. (switch the place of the x and y, then resolve for y)

xy 6log Have : LOG FORM Want: It’s inverse: EXP. FORM

First invert: yx 6log

Now let’s change it into the correct form. How can we turn a

log into an exponential??? THE LOOP!!

3ln xy Have : ___________ Want: ____________

7.4 DAY #2!

xy 6log2

1 Have : ___________ Want: ____________

xy 6 Have : ___________ Want: ____________

INVERSE

Think about inverses: 2x or 3 3x or 1

6

6

1 . What do inverses do when one is applied to the other?

Consider the problems below. In each problem we see the log form and exponential form of the same base.

Simplify.

8

6 6log f.

When you have a log base 6, right next to an exponential base 6, they cancel each other. The answer is 8!

81log9

nothing cancels….hmm… x10log77

When you have an exponential base 7, right next to a log base 7, they cancel each other.

The answer is ______.

)13(log44x

64log4

GRAPHING

1. Plot some convenient points by thinking about the log loop. “3 raised to ___ = ___”

2. Apply any necessary shifts.

3. Determine the vertical asymptote, by setting the x value and its contained ( ) = 0

4. Draw a curve through the points.

5. State the domain and range.

x

xy

y

3

log 3

VA = ________ Domain: _____________________________ Range: _____________________________

x y Ordered Pairs

,

,

,

3log 2 xy

h = _______

VA = ________ Domain: _____________________________ Range: _____________________________

x y Ordered Pairs

,

,

,

41log2

xy

h = _______ k = _______

VA = ________ Domain: _____________________________ Range: _____________________________

x y Ordered Pairs

,

,

,

7.5 day #1 Apply Properties of logarithms

NOTE: In the following properties, the base cannot be negative, nor can it = 1. Also, the numbers you are taking the log of must be positive.

PRODUCT

PROPERTY

8log3log 1010 x = x24log10

21loglog1010

yy =

QUOTIENT

PROPERTY

5log20log 1010 x = x4log10

xx 2log6log10

2

10 =

POWER

PROPERTY

3log2 7 = 9log3log 7

2

7

25log2

110 = 2

1

10 25log =

xln6 =

What we are doing is learning ways to CONDENSE logarithmic expressions. In other words, we will be given an expression and asked to simplify it as much as possible. These properties give us the rules for doing so.

We are adding two logs

of the same base.

To create our answer, we start by writing down a log with the same base as the logs in the problem. We will then multiply what we

were taking the logs of

We are subtracting two logs of the same base.

To create our answer, we start by writing down a log with the same base as the logs in the problem. We will then divide what we were taking the log of in each log of the problem (first divided by second)

We are multiplying a log by a number.

To create our answer, we start by writing down a log with the same base as the log in the problem. We will then take the number being multiplied and bring it to the exponent position of the number or expression that we’re taking the log of. We then simplify if possible.

Condense the following logarithmic expressions. 1) Always start with the numbers that are multiplying by the logs. Move those numbers to the exponent position

and simplify before moving on. 2) Move from left to right and use the properties to condense the logs

1. 4log100log 44

2. 4ln12ln

3.

4

1log2

4. yx ln4ln6

5. yx log4log5

6. yx 444 log4log72log5

7. xln2

1ln240ln

8. 26loglog3

14log 555 x

9. yx ln4ln32ln6

10.

50log

2log25log

4log5log

4log5.05log2

4log5.04log20log2

3

33

5.0

3

2

3

33

333

CHANGE OF

BASE

THEOREM (CALCULATOR)

Your calculator is designed to find the value of common logarithms, so when you press the log button,

your calculator is assuming that you are typing in something that has a base of 10.

For all other logarithms that do not have a base of 10, we have to apply the change of base theorem and

put them into base 10 so they can be entered into the calculator.

Example:

5log

17log

5log

17log17log

10

10

5


6log/256log:

6log

256log256log

10

10

6

calculatorin

8.45log 3.2

7.5 Day #2

EXPANSION

We can use the properties of logs to condense logarithms, and we can also use them to expand a

logarithm. In other words, given the answer to a log problem, you will recreate the original log problem that

created that answer.

Expand the expression. (THINK: WHAT ARE THE NUMBERS AND VARIABLES IN THE ANSWER DOING? WHAT ADDITION OR

SUBTRACTION PROBLEM WOULD HAVE CREATED THAT? )

x4log 3

How could we have gotten 4 connected to x? Was it an addition

problem between logs or a subtraction problem?

5

12ln

43log x

y

x

3ln

31

2ln yx

x7log5

46ln x

5

log2

8

yx

You will know you are done with the problem when each log has only one variable or number behind it, and that number or variable does not have an exponent!!!

Expand the expression using 7log 2 and/or 3log 2 . Then find its value using the following information:

Use: 807.27log585.13log 22 and

21log 2

392.4

585.1807.2

3log7log 22

49log 2

3

7log 2

7.6 Solve exponential and logarithmic equations (Note: all rounding in this section will be done to three decimal places.)

SOLVING EXPONENTIAL EQUATIONS

What you’ll see: equations in which the variable is stuck up in the exponent

What you want: to get the variable out of the exponent so it can be solved for

Method: 1. Equating exponents

2. Taking the logarithm of both sides

3. Taking the natural logarithm of both sides

Method 1: EQUATING EXPONENTS

When it can be used: This method can be used when the bases on each side of the equal sign are the same or when the bases are not the same, but can be rewritten to become the same. Supporting Property:

72 88 xx

11664 xx

Method 2: TAKING THE LOGARITHM OF BOTH SIDES

Steps: 1. Isolate the base that contains the exponent. 2. Take that logarithm of both sides. 3. Solve what remains. Round to the hundredths place. 4. Check for extraneous solutions.

1164 x

1610

19310

32

32

x

x

Method 3: TAKING THE NATURAL LOGARITHM (LN) OF BOTH SIDES

Steps: 1. Isolate the base that contains the exponent. 2. Take that natural log of both sides 3. Solve what remains. Round to the hundredths place. 4. Check for extraneous solutions.

4

123

3

3

x

x

e

e

684

3 3 xe

7.6 Day #2

SOLVING LOGARITHMIC EQUATIONS

What you’ll see: equations with logarithms on one or both sides

What you want: to simplify the logarithms as much as possible and solve for the variable

Method: 1. Equating logarithms

2. Converting from logarithmic to exponential form

Method 1: EQUATING LOGARITHMS

When it can be used: After each side of the equation has been simplified (all product, quotient, and power properties applied), this method can be used when each side is left with a single log. Supporting Property:

Steps: 1. Starting on one side at a time, use the product, quotient, and power properties to do all addition, subtraction, mult. 2. Check to make sure that each side contains a single log. 3. By property of equality, cancel the logs. 4. Solve what remains. Round to the hundredths place. 5. Check for extraneous solutions.

xx 5log2log77

93log72log55

xx

5.23ln5.23ln

5.4

276

8719

87ln19ln

x

x

xx

xx

Method 2: CONVERTING FROM LOGARITHMIC TO EXPONENTIAL FORM (THE LOOP)

When it can be used: After each side of the equation has been simplified (all product, quotient, and power properties applied), this method can be used when there is only one log on one side.

Steps: 1. Starting on one side at a time, use the product, quotient, and power properties to do all addition, subtraction, mult. of logs. If the equation only contains one log, make sure that the log is completely alone. 2. Check to make sure that only one side contains a log. 3. Use the log loop to rewrite the log as an exponent. 4. Solve what remains (you may have to factor). 5. Check for extraneous solutions.

64log2

x

213ln4 x

32log

32loglog

2

2

22

xx

xx

210loglog44

xx

7.7 Write and Apply Exponential and Power Functions

Write an exponential function y = abx whose graph passes through (1, 10) and (4, 80).

Step 1: Substitute the coordinates of the two given points into y = abx.

Step 2: Solve for a in the first equation to obtain a, and then substitute this expression for a into the second equation.

Step 3: Because b = ________ , it follows that a = ________________ So, y = ________________ .

Determine whether an exponential model is appropriate for the data. If so, find the exponential model (equation/function) for the data.

Savings: The table shows the amount (y) in a savings account (x) years after the account was opened.

x 0 1 2 3 4 5 6 7

y 210 255 310 377 459 557 677 822

To determine whether this information would be best modeled by an exponential equation, we will make a table of the

points (x, ln y). Once we create the table, we will make a scatterplot of these points in the calculator. If the modeled points

lie along a linear pattern, then we’ll know that an exponential model is best for the original data.

x 0 1 2 3 4 5 6 7

y

ln 210=

5.35

Enter the original data into your calculator and use the Exponential Regression feature to get the exponential model.

y = ______________________

Write a power function y = axb whose graph passes through (2, 4) and (6, 10).

Step 1: Substitute the coordinates of the two given points into y = axb

Step 2: Solve for a in the first equation to obtain a, and then substitute this expression for a into the second equation.

Step 3: Because b = ________ , it follows that a = ________________ So, y = ________________ .

Determine whether a power model is appropriate for the data. If so, find the power model (equation/function) for the data.

Birds: The table shows the typical wingspan (x), in feet, and the typical weights (y), in pounds, for several types of birds. .

cuckoo crow curlew goose vulture

Wingspan, x 1.90 2.92 6.41 5.35 8.40

Weight, y .23 1.04 1.69 6.76 16.03

To determine whether this information would be best modeled by a power function, we will make a table of the

points (ln x, ln y). Once we create the table, we will make a scatterplot of these points in the calculator. If the modeled points

lie along a linear pattern, then we’ll know that a power model is best for the original data.

x

y

Enter the original data into your calculator and use the Power Regression feature to get the power model.

y = ______________________

TI-83 and TI-84

Exponential Regression

1. Press STAT. 2. Arrow over to CALC. 3. Choose #0 ExpReg. 4. You should see ExpReg on the screen. 5. Press ENTER. The screen will show you the coefficients and constant of the

exponential equation.

Power Regression

1. Press STAT. 2. Arrow over to CALC. 3. Choose A PwrReg. 4. You should see PwrReg on the screen. 5. Press ENTER. The screen will show you the coefficients and constant of the

power equation.

Documents

EXPONENTIAL FUNCTION where a bmi01000971.schoolwires.net/cms/lib05/MI01000971...Exponential Growth functions are also used in real-life scenarios involving compound interest. Compound