Exploring Exponential Functions



Exploring Exponential Functions

Warm-up: Evaluate each expression for the given value of x.

1. 2x for x = 3 2. 4x+1 for x = 1 3. 23x + 4 for x = -1

4. 3x3x-2 for x=2 5. (½)x for x = 0 6. 2x for x = -2

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March Madness Math

Exponential Function

y = abx

x is real

a ≠ 0

b > 0

How many grains of rice would be on…

Square 0:

Square 1:

Square 2:

Square 3:

Square 4:

Square 5:

Square 6:

Write a NOW-NEXT equation to model the number of grains of rice in a given square:

NEXT= starting at:

Write an explicit equation to model the number of grains of rice in a given square:

y =

How many grains of rice would be on…

Square 10:

Square 20:

Square 30:

Square 40:

Square 50:

Square 60:

Square 63:

y = abx y = abx

Notice: The value of y approaches 0 as it goes in one direction. y=0 is the __________.

List the equations beneath:

Exponential Growth: Exponential Decay:

U.S. Population

⊗ In 2000, the annual rate of increase in the U.S. population was about 1.24%.

⊗ In 2000 the population was about 281 million people.

1. Find the growth factor for the U.S. population. (Don’t forget how you convert from a %.)

2. Suppose the rate of increase continues to be 1.24%.

Write functions to model U.S. population growth.

NEXT = y=

Starting @

3. Predict the U.S. population in 2015.

4. If you were going to graph this, what would the y-intercept be?

What would the negative x-values mean?

Determine whether each function represents exponential growth or decay.

Identify the growth/decay factor.

1. y = 50(0.24)x 2. y= 0.5(9)x 3. y = 2(1/3)x 4. y = 10x

Determine whether each function represents exponential growth or decay.

Identify the growth/decay factor.

1. y = 100(0.12)x 2. y 0.2(5)x 3. y = 16(1/2)x 4. y = 4.7x

Kalamazoo Teen Wins Big Lottery Prize - $20,000

A Kalamazoo teenager has just won $20,000 from a Michigan lottery ticket that she got as a birthday gift from her uncle. In a new lottery payoff scheme, the teen (whose name has been withheld) has two payoff choices: One option is to receive $1,000 payments each year for the next 20 years. In the other plan, the lottery will invest $10,000 in a special savings account that will earn 8% interest compounded annually for 10 years. At the end of that time she can withdraw the balance of the account.

1. Which of the two payoff methods would you choose?

2. Which method do you think would give the greatest total payoff? (Just guess.)

3. About how much money do you think would be in the special savings account at the end of 10 years? (Just guess.)

4. Now let’s figure it out:

Method 1:

$1000 x ____ years = ____________________ total

Method 2:

∝ after 1 year: ∝ growth rate =

∝ after 2 years:

∝ after 3 years:

Write equations to model Method 2.

∝ NEXT = ∝ y =

Starting at

5. Let’s try some similar situations.

Write equations to model:

← Initial investment of $15,000 earning 4% interest

∝ Initial investment of $5,000 earning 12% interest

What will the balances be after 10 years?

Example 1 - Graph y = 4(2)x

Make a table of values.

What does a do to the graph?

Example 2 - Graph y = 24 (1/2)x

Will the graph ever go below 0?

What is the asymptote?

(Circle one.) All Some No exponential functions have asymptotes.

Warm-up:

1. Which function represents exponential growth?

A. y = 35x1.35 B. y = 35(0.35)x

C. y = 35(1.35)x D. y = 35 + (0.35)x

2. What is the equation of the asymptote of y = 15(1/3)x?

A. y = 1 B. y = 0 C. y = x D. y = 1/3

Car Depreciation

Initial value =

Value after one year =

What percent of the initial value

does the car have after one year?

percent lost = difference in value

initial value

percent remaining = 1 - percent lost

Decay rate =

Write functions to model the value of the car.

NEXT = y =

Predict the value of the car after six years.

The table gives the 1998 gross domestic product and the real growth rate for several countries.

a. Explain how a negative growth rate affects the equation for an exponential model.

b. Write a function for each country to model the GDP.

Suppose the given real growth rates continue.

Predict the gross domestic product for each country in 2005.

|Country |1998 Gross |1998 Real |

| |Domestic |Growth Rate |

| |Product | |

| |(billions) | |

|Armenia |$9.2 |6% |

|Canada |$688.3 |3% |

|Oman |$18.6 |-8.5% |

|Paraguay |$19.8 |-0.5% |

Analyze the Graph

Which function represents the graph?

A. y = (1/3)2x

B. y = 2(1/3)x

C. y = -2(1/3)x

Exponential Regression

▪ If you have ____________________________________ above the x-axis you can find an exponential function that ________________________ goes through these points.

▪ If you have _______________________________________ above the x-axis you can find a ______________________ exponential function that goes close to these points.

Example: (2, 2), (3, 4)

Try graphing these two and just free-hand an exponential curve through them.

Finding an exact exponential equation through two points:

1. Use the general form: y = abx

2. Temporarily plug in one of points for x & y.

3. Solve for a. (You will have b in your equation.)

4. Go back to general form: y = abx

Temporarily plug in other point for x & y AND formula you just got in for a.

5. Solve for b.

6. Solve for a.

7. Permanently plug in a and b into general form.

Example: (2, 2), (3, 4)

Example: Write an exponential equation that goes through (-1, 8 1/3) and (2, 1.8).

The table shows the number of degrees above room temperature for a cup of coffee after x minutes of cooling.

1. Graph the data on the calculator.

2. Find an exponential regression

equation to fit the data.

3. Graph your regression equation.

Half-Life Formula

[pic]

_______________________: time required for half of the nuclei in a substance to decay

__________________ _________________: time required for the body to eliminate half of the radiation acquired from exposure

y = 9(3)x+1 y = -9(3)x -4

y = 9(3)x-3 -1

Warm-up: What is the asymptote of y= -4([pic])x-3?

True or False? You will make more $ in an account compounded monthly than in one compounded annually.

You will make more $ in an account compounded more or less frequently?

How often would an account be compounded that made the maximum amount of $?

Continuously Compounded Interest Formula

A=Pert

Suppose you invest $1050 at an annual interest rate of 5.5% compounded continuously. How much will you have in the account after five years?

Suppose you invest $1300 at an annual interest rate of 4.3% compounded continuously. How much will you have in the account after three years?

Logarithms

The logarithm to the base b of y: If y = bx, then ____________

Example 1 - Write in logarithmic form.

A. 25=32 B. 361/2=6

C. (1/2)3=1/8 D. 100=1

Example 2 - Write in exponential form.

A. log1/2 32 = -5 B. log27 3 = 1/3

Example 3 - Evaluate.

A. log8 64 B. log16 4

A common logarithm is a log with base ________.

____________ = log y No base? - understood _________.

Example 4 - Evaluate.

A. log 1000= B. log 1/10 =

An extra example from homework worksheet:

55. For each pH given, find the concentration of hydrogen ions [H+].

Use the formula pH=-log[H+]. 7.3

Logarithm Properties and Formulas

Change of Base Formula

For any positive numbers, M, b, and c, with b ≠ 1 and c ≠ 1,

logb M =

Example 1 - Convert to a ratio of common logs.

A. log3 15 = B. log11 121 =

C. log12 12 = D. log3 1/9 =

Example 2 - Evaluate using a calculator.

A. log 14 B. log 1

C. log3 6561 D. log1/3 (1/27)

Product Property

For any positive numbers, M, N, and b, b≠1,

logb MN =

Quotient Property

For any positive numbers, M, N, and b, b≠1,

logbM/N =

Power Property

For any positive numbers, M, N, and b, b≠1,

logbMx =

Example: log √5x =

Example 1 - Rewrite as a single logarithm.

A. log3 20 - log3 4 =

B. log2 y + log2 x =

C. 3log 2 + log 4 - log 16 =

Example 2 - Expand each logarithm.

A. log5 x/y =

B. log 3r4 =

C. log2 7b =

D. log (y/3)2 =

Example 3 - Identify properties used.

A. log2 8 - log2 4 = log2 2

B. logb x3y = 3logb x + logby

C. 3logb 4 - 3logb 2 = logb 8

D. log5 2 + log5 6 = log5 12

Solving Exponential Equations

(For these problems approximate to __________ decimal places unless told otherwise.)

Example 1 - Solve 73x = 20

Example 2 - Solve 7 - 52x - 1 = 4

Example 3 – Solve 5 – 3x = -40

Example 4 – Solve 3x+ 4 = 101

Warm-up: Solve 14x+1 = 36

The Number e

π is an I _ _ _ _ _ _ _ _ _ number. Π ≈

e is also an _______________ number. E ≈

Evaluate:

e4= e-3= e1/2=

Natural Logarithms

Remember with common logs, no base is an understood ______.

Natural log has base ____, written ____.

ln x =

Example 1 - Solve 7e2x + 2.5=20

Example 2 -

A spacecraft can attain a stable orbit 300 km above Earth if it reaches a velocity of 7.7 km/s. The formula for a rocket’s maximum velocity v in kilometers per second is v = -0.0098t + c ln R. The booster rocket fires for t seconds and the velocity of the exhaust is c km/s. The ratio of the mass of the rocket with the fuel to its mass without fuel is R. Suppose a rocket used to propel a spacecraft has a mass ratio of 25, an exhaust velocity of 2.8 km/s, and a firing time of 100s. Can the spacecraft attain a stable orbit 300 km above Earth?

Translating Exponential and Logarithmic Functions

y = a(b)x-h +k

y=logb(x-h)+k

What do you think happens if there is a negative in front?

Examples:

y = 8(.5)x y = 8(.5)x+2 +3

What is the asymptote? What is the asymptote?

What are the intercepts? What are the intercepts?

What is the "end behavior"? What is the "end behavior"?

y = -3(2)x +1

What is the asymptote?

What are the intercepts?

What is the "end behavior"?

[pic]

[pic]

Steps to Graphing Logarithms:

1. Asymptote: y=logb(x-h)+k

2. Pick numbers to plug in for x.

Graph y=log2(x+3)+1

Solving Logarithmic Equations

Example 5 – Solve log (3x + 1) = 5

Example 6 – Solve 1 + log (7 – 2x) = 0

Example 7 – 2log x – log 3 = 2

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How does the number of teams left in each round compare to the number of teams in the previous round?

NEXT=

starting at:

y =

Compound Interest Formula

[pic]

Or [pic]

To "free" a variable from an exponent, get the term with the exponent __________________.

Then ______________________!

To "free" a variable from a log, get a single logarithm __________________.

Then ____________________!

Graph

y = log (x-1)-2

Graph

y = -ln (x+1)

Hints: Asymptote is always a ____________ line.

Be careful!

What if there are no ( )’s?

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