chapter 10 sec 5 exponential functions. 2 of 16 algebra 1 chapter 10 sections 5 & 6 power of 2...

Chapter 10 Sec 5Chapter 10 Sec 5

Exponential Exponential FunctionsFunctions

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Algebra 1 Chapter 10 Sections 5 & 6

Power of 2Power of 2

Which would desire most.Which would desire most.

1.1.$1, 000, 000 in 30 days or…$1, 000, 000 in 30 days or…

2.2. 2 cents today then doubled 2 cents today then doubled for 30 days.for 30 days.

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Power of 2Power of 2221 1 = .02= .02 2211 11 = 20.48= 20.48 2221 21 = 20,971.52= 20,971.52

222 2 = .04= .04 2212 12 = 40.96= 40.96 2222 22 = 41,943.04= 41,943.04

223 3 = .08= .08 2213 13 = 81.92= 81.92 2223 23 = 83,886.08= 83,886.08

2244 = .16 = .16 221414 = 163.84 = 163.84 222424 = 167,772.16 = 167,772.16

2255 = .32 = .32 221515 = 327.68 = 327.68 222525 = 335,544.32 = 335,544.32

226 6 = .64= .64 2216 16 = 655.36= 655.36 2226 26 = 671,088.64= 671,088.64

2277 = 1.28 = 1.28 221717 = 1,310.72 = 1,310.72 222727 = 1,342,177.28 = 1,342,177.28

2288 = 2.56 = 2.56 221818 = 2,621.44 = 2,621.44 222828 = 2,684,354.56 = 2,684,354.56

2299 = 5.12 = 5.12 221919 = 5,242.88 = 5,242.88 222929 = 5,368,709.12 = 5,368,709.12

221010 = 10.24 = 10.24 222020 = 10,485.76 = 10,485.76 223030 = 10,737,418.24 = 10,737,418.24

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Exponential FunctionExponential Function

From the previous example we can see, to From the previous example we can see, to find the amount of money accumulated, find the amount of money accumulated, y, y, over over xx amount of days can be written amount of days can be written as: as: y = y = 22xx. .

This type of function, in which the This type of function, in which the variable is the exponent, is called anvariable is the exponent, is called an exponential functionexponential function..

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Graph an exponential Function with Graph an exponential Function with a a >1>1

a. Graph y = a. Graph y = 44xx. State the y-intercept. State the y-intercept

xx 44xx yy

-2 4-2 1/16

-1 4-1 1/4

0 40 1

1 41 4

2 42 16

3 43 64

b. b. Use graph to find approximate value ofUse graph to find approximate value of 441.81.8

441.81.8~ 12.12573252~ 12.12573252

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Graph an exponential Function with 0 < Graph an exponential Function with 0 < a < 1a < 1

a. Graph . State the y-intercepta. Graph . State the y-intercept

xx (1/2)(1/2)XX yy

-3 (1/2)-3 8

-2 (1/2) -2 4

-1 (1/2) -1 2

0 (1/2)0 1

1 (1/2)1 1/2

2 (1/2)2 1/4

b. b. Use graph to find approx value ofUse graph to find approx value of (1/2)(1/2)-2.5-2.5

(1/2)(1/2)-2.5 -2.5 ~ 5.656854249~ 5.656854249

y 12

x

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Identify Exponential BehaviorIdentify Exponential BehaviorDetermine whether each set of data displays.Determine whether each set of data displays.

Method 2. Graph the data.Method 2. Graph the data.Method 1. Look for a pattern.Method 1. Look for a pattern.The domain values are at regular The domain values are at regular intervals of 10. See if there is a intervals of 10. See if there is a common factor among the range..common factor among the range..8080 40 20 10 5 2.540 20 10 5 2.5

Since the domain values are at Since the domain values are at regular intervals and the range have regular intervals and the range have a common factor. The data is a common factor. The data is probably exponential, involving probably exponential, involving (1/2)(1/2)xx

xx

yy

0 10 20 30 40 50

80 40 20 10 5 2.5

1

2

1

2

1

2

1

2

1

2

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Identify Exponential BehaviorIdentify Exponential BehaviorDetermine whether each set of data displays.Determine whether each set of data displays.

Method 2. Graph the data.Method 2. Graph the data.Method 1. Look for a pattern.Method 1. Look for a pattern.The domain values are at regular The domain values are at regular intervals of 10. The range values intervals of 10. The range values have a common difference 6.have a common difference 6.

xx

yy

0 10 20 30 40 50

15 21 27 33 39 45

Chapter 10 Sec 6Chapter 10 Sec 6

Exponential Exponential Growth/DecayGrowth/Decay

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General EquationGeneral Equation

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Exponential GrowthExponential Growth

In 1971, there were 294,105 females participating in high In 1971, there were 294,105 females participating in high school sports. Since then, that number has increased an school sports. Since then, that number has increased an average of 8.5% per year.average of 8.5% per year.

a. Write an equation to represent the number of females a. Write an equation to represent the number of females participating in high school sports since 1971.participating in high school sports since 1971.

y = y = CC(1 + (1 + rr))tt

y = y = 294,105294,105(1 + (1 + 0.0850.085))tt

y = y = 294,105(1.085)294,105(1.085)tt

b. How many female students participated in 2001?b. How many female students participated in 2001? y = y = 294,105(1.085)294,105(1.085)tt tt = 2001 - 1971 or 30 = 2001 - 1971 or 30 y = y = 294,105(1.085)294,105(1.085)3030

y ~ 3,399,340y ~ 3,399,340

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Compound interestCompound interest

The equation The equation , where , where AA is the amount of the is the amount of the investment, investment, PP is the principal (initial amount invested), is the principal (initial amount invested), rr is the annual is the annual rate of interest expressed as a decimal, rate of interest expressed as a decimal, nn is the number of times the is the number of times the interest is compounded each year, and interest is compounded each year, and tt is the number of years the is the number of years the money is invested.money is invested.Example: Use info on right. If money was invested Example: Use info on right. If money was invested at 6% per year compounded semiannually (2 times a at 6% per year compounded semiannually (2 times a year), How much money would there be in 2026?year), How much money would there be in 2026?

P =P = 24, 24, rr = 6% or 0.06, = 6% or 0.06, n n = 2= 2

tt = 2026 - 1626 = 400 = 2026 - 1626 = 400

A P 1 r

n

nt

A P 1 r

n

nt

A 24 1 0.06

2

2 400

24 1.03 800

000,000,000,447$1047.4 11 A

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General EquationGeneral Equation

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Exponential DecayExponential Decay

In 1950, the use of coal by residential and commercial users was 114.6 million tons. Because of cleaner fuels the use of coal has decreased by 6.6% per year

a. Write an equation to represent the use of coal since 1950.y = C(1 - r)t

y = 114.6(1 - 0.066)t

y = 114.6(0.934)t

b. Estimate the amount of coal that will be used in 2015. y = 114.6(0.934)t t = 2015 - 1950 or 65 y = 114.6(0.934)65

y ~ 1.35 million tons of coal.

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DepreciateDepreciate

Sometimes items decrease in value or depreciate. Cars, office equipment depreciate as they get older. You can use the exponential decay formula to determine the value of an item at a given time.

A farmer buys a tractor for $50,000. If the tractor depreciates 10% per year, find the value of the tractor in 7 years.

y = C(1 - r)t

y = 50000(1 - 0.10)7

y = 50000(0.90)7 use a calculator… y ~ 23,914.85

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Daily AssignmentDaily Assignment

• Chapter 10 Sections 5 & 6Chapter 10 Sections 5 & 6• Study Guide (SG)Study Guide (SG)

• Pg 139 – 142 AllPg 139 – 142 All

chapter 10 sec 5 exponential functions. 2 of 16 algebra 1 chapter 10 sections 5 & 6 power of 2...

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