1. warm-up 4/21 d. rigor: you will learn how to evaluate, analyze, graph and solve exponential...

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1. Warm-Up 4/21 D 3 6 + 3 6 โˆ’ 2 6 4 6

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Page 1: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

1.Warm-Up 4/21

D

36+

36โˆ’

26

46

Page 2: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population
Page 3: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

Rigor:You will learn how to evaluate, analyze, graph

and solve exponential functions.

Relevance:You will be able to solve population problems and

solve half-life chemistry problems using exponential functions.

Page 4: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

3-1 Exponential Functions

Page 5: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

Algebraic Functions are functions are solved using algebraic operations.Transcendental Functions are functions that can not be expressed in terms of algebraic operations. They transcend Algebra.

Exponential and Logarithmic Functions are transcendental functions.

Page 6: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

Example 1a: Sketch and analyze the graph of the function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing, or decreasing.๐‘“ (๐‘ฅ )=3๐‘ฅ

x y

โ€“ 3 0.04

โ€“ 2 0.11

โ€“ 1 0.33

0 1

1 3

2 9

3 27

Domain:Range:y-intercept:Asymptotes:End Behavior:

Increasing/Decreasing:

(โˆ’โˆž ,โˆž)(0 ,โˆž)

(0 ,1)๐‘ฆ=0

lim๐‘ฅโ†’โˆ’โˆž

๐‘“ (๐‘ฅ)=0 lim๐‘ฅโ†’โˆž

๐‘“ (๐‘ฅ)=โˆžand

Increasing :(โˆ’โˆž ,โˆž)

Page 7: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

Example 1b: Sketch and analyze the graph of the function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing, or decreasing.๐‘“ (๐‘ฅ )=2โˆ’๐‘ฅ

x y

โ€“ 3 8

โ€“ 2 4

โ€“ 1 2

0 1

1 .5

2 .25

3 .125

Domain:Range:y-intercept:Asymptotes:End Behavior:

Increasing/Decreasing:

(โˆ’โˆž ,โˆž)(0 ,โˆž)

(0 ,1)๐‘ฆ=0

lim๐‘ฅโ†’โˆ’โˆž

๐‘“ (๐‘ฅ)=โˆž lim๐‘ฅโ†’โˆž

๐‘“ (๐‘ฅ)=0and

Decreasing :(โˆ’โˆž, โˆž)

Page 8: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population
Page 9: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

Example 2a: Use the graph of to describe the transformation of the function. Then sketch both functions.

๐‘” (๐‘ฅ )=2๐‘ฅ+1

x y

โ€“ 4 .125

โ€“ 3 .25

โ€“ 2 .5

โ€“ 1 1

0 2

1 4

2 8

๐‘“ (๐‘ฅ )=2๐‘ฅ

x y

โ€“ 3 .125

โ€“ 2 .25

โ€“ 1 .5

0 1

1 2

2 4

3 8

๐‘” (๐‘ฅ )= ๐‘“ (๐‘ฅ+1)The graph of is the graph of translated 1 unit to the left.

Page 10: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

Example 2b: Use the graph of to describe the transformation of the function. Then sketch both functions.

h (๐‘ฅ )=2โˆ’๐‘ฅ

x y

โ€“ 3 8

โ€“ 2 4

โ€“ 1 2

0 1

1 .5

2 .25

3 .125

๐‘“ (๐‘ฅ )=2๐‘ฅ

x y

โ€“ 3 .125

โ€“ 2 .25

โ€“ 1 .5

0 1

1 2

2 4

3 8

h (๐‘ฅ )= ๐‘“ (โˆ’ ๐‘ฅ)The graph of is the graph of reflected in the y-axis.

Page 11: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

Example 2c: Use the graph of to describe the transformation of the function. Then sketch both functions.

๐‘— (๐‘ฅ )=โˆ’3(2)๐‘ฅ

x y

โ€“ 3 โ€“ .375

โ€“ 2 โ€“ .75

โ€“ 1 โ€“ 1.5

0 โ€“ 3

1 โ€“ 6

2 โ€“ 12

3 โ€“ 24

๐‘“ (๐‘ฅ )=2๐‘ฅ

x y

โ€“ 3 .125

โ€“ 2 .25

โ€“ 1 .5

0 1

1 2

2 4

3 8

๐‘— (๐‘ฅ )=โˆ’3 ๐‘“ (๐‘ฅ)The graph of is the graph of reflected in the x-axis and expanded vertically by a factor of 3.

Page 12: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

Example 3a: Use the graph of to describe the transformation of the function. Then sketch both functions.

๐‘Ž (๐‘ฅ )=๐‘’4๐‘ฅ

x y

โ€“ 3 6.1x10โ€“6

โ€“ 2 3.4x10โ€“4

โ€“ 1 .01832

0 1

1 54.598

2 2981

3 162755

๐‘“ (๐‘ฅ )=๐‘’๐‘ฅ

x y

โ€“ 4 .01832

โ€“ 3 .04979

โ€“ 2 .13534

โ€“ 1 .36788

0 1

1 2.7183

2 7.3891

3 20.086

4 54.598

๐‘Ž (๐‘ฅ )= ๐‘“ (4 ๐‘ฅ )The graph of is the graph of compressed horizontally by a factor of 4.

Page 13: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

Example 3b: Use the graph of to describe the transformation of the function. Then sketch both functions.

๐‘ (๐‘ฅ )=๐‘’โˆ’๐‘ฅ+3

x y

โ€“ 3 23.086

โ€“ 2 10.389

โ€“ 1 5.7183

0 4

1 3.3679

2 3.1353

3 3.0498

๐‘“ (๐‘ฅ )=๐‘’๐‘ฅ

x y

โ€“ 3 .04979

โ€“ 2 .13534

โ€“ 1 .36788

0 1

1 2.7183

2 7.3891

3 20.086

๐‘ (๐‘ฅ )= ๐‘“ (โˆ’๐‘ฅ )+3The graph of is the graph of reflected in the y-axis andtranslated 3 units up.

Page 14: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

Example 3c: Use the graph of to describe the transformation of the function. Then sketch both functions.

๐‘ (๐‘ฅ )=12๐‘’๐‘ฅ

x y

โ€“ 3 .02489

โ€“ 2 .06767

โ€“ 1 .18394

0 .5

1 1.3591

2 3.6945

3 10.043

๐‘“ (๐‘ฅ )=๐‘’๐‘ฅ

x y

โ€“ 3 .04979

โ€“ 2 .13534

โ€“ 1 .36788

0 1

1 2.7183

2 7.3891

3 20.086

๐‘ (๐‘ฅ )=12๐‘“ (๐‘ฅ )

The graph of is the graph of compressed vertically by a factor of .

Page 15: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population
Page 16: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

Example 4: Krysti invest $300 in an account with 6% interest rate, making no other deposits or withdrawals. what will Krystiโ€™s account balance be after 20 years if the interest is compounded:

a. semiannually?

b. Monthly?

c. Daily?

๐ด=๐‘ƒ (1+ ๐‘Ÿ๐‘› )

๐‘›๐‘ก

P = 300, r = 0.06, t = 20

n = 2๐ด=300(1+ 0.06

๐‘› )๐‘›(20)

๐ด=300(1+ 0.062 )

2 (20 )

๐ดโ‰ˆ978.61

n = 12๐ด=300(1+ 0.06

12 )12(20)

๐ดโ‰ˆ993.06

n = 365๐ด=300(1+ 0.06

365 )365(20)

๐ดโ‰ˆ995.94

Page 17: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

Example 5: Suppose Krysti invest $300 in an account with 6% interest rate, making no other deposits or withdrawals. What will Krystiโ€™s account balance be after 20 years if the interest is compounded continuously?

๐ด=๐‘ƒ ๐‘’๐‘Ÿ ๐‘กP = 300, r = 0.06, t = 20

๐ด=300๐‘’.06 (20)

๐ดโ‰ˆ996.04

Page 18: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

a. 1.42% annually

b. 1.42% continuously

Example 6: Mexico has a population of approximately 110 million. If Mexicoโ€™s population continues to grow at the described rate, predict the population of Mexico in 10 and 20 years.

๐‘=๐‘ 0๐‘’๐‘˜๐‘ก

N0 = 110,000,000

r = 0.0142

,t = 10 and t = 20

๐‘=๐‘ 0 (1+๐‘Ÿ )๐‘ก

๐‘=110,000,000 (1+0.0142 )10 ๐‘=110,000,000 (1+0.0142 )20

๐‘ โ‰ˆ126,656,869 ๐‘ โ‰ˆ145,836,022 k = 0.0142

๐‘=110,000,000๐‘’0.0142(10)

๐‘ โ‰ˆ126,783,431๐‘=110,000,000๐‘’0.0142(20)

๐‘ โ‰ˆ146,127,622

Page 19: 1. Warm-Up 4/21 D. Rigor: You will learn how to evaluate, analyze, graph and solve exponential functions. Relevance: You will be able to solve population

โˆšโˆ’1math!

3-1 Assignment: TX p166, 4-32 EOE Test Corrections Due Friday 4/25Chapter 3 test Thursday 5/1