Upload
pearl-day
View
218
Download
4
Tags:
Embed Size (px)
Citation preview
1.Warm-Up 4/21
D
36+
36โ
26
46
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.
3-1 Exponential Functions
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.
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 (๐ฅ )=3๐ฅ
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:
(โโ ,โ)(0 ,โ)
(0 ,1)๐ฆ=0
lim๐ฅโโโ
๐ (๐ฅ)=0 lim๐ฅโโ
๐ (๐ฅ)=โand
Increasing :(โโ ,โ)
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 (๐ฅ )=2โ๐ฅ
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:
(โโ ,โ)(0 ,โ)
(0 ,1)๐ฆ=0
lim๐ฅโโโ
๐ (๐ฅ)=โ lim๐ฅโโ
๐ (๐ฅ)=0and
Decreasing :(โโ, โ)
Example 2a: Use the graph of to describe the transformation of the function. Then sketch both functions.
๐ (๐ฅ )=2๐ฅ+1
x y
โ 4 .125
โ 3 .25
โ 2 .5
โ 1 1
0 2
1 4
2 8
๐ (๐ฅ )=2๐ฅ
x y
โ 3 .125
โ 2 .25
โ 1 .5
0 1
1 2
2 4
3 8
๐ (๐ฅ )= ๐ (๐ฅ+1)The graph of is the graph of translated 1 unit to the left.
Example 2b: Use the graph of to describe the transformation of the function. Then sketch both functions.
h (๐ฅ )=2โ๐ฅ
x y
โ 3 8
โ 2 4
โ 1 2
0 1
1 .5
2 .25
3 .125
๐ (๐ฅ )=2๐ฅ
x y
โ 3 .125
โ 2 .25
โ 1 .5
0 1
1 2
2 4
3 8
h (๐ฅ )= ๐ (โ ๐ฅ)The graph of is the graph of reflected in the y-axis.
Example 2c: Use the graph of to describe the transformation of the function. Then sketch both functions.
๐ (๐ฅ )=โ3(2)๐ฅ
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 (๐ฅ )=2๐ฅ
x y
โ 3 .125
โ 2 .25
โ 1 .5
0 1
1 2
2 4
3 8
๐ (๐ฅ )=โ3 ๐ (๐ฅ)The graph of is the graph of reflected in the x-axis and expanded vertically by a factor of 3.
Example 3a: Use the graph of to describe the transformation of the function. Then sketch both functions.
๐ (๐ฅ )=๐4๐ฅ
x y
โ 3 6.1x10โ6
โ 2 3.4x10โ4
โ 1 .01832
0 1
1 54.598
2 2981
3 162755
๐ (๐ฅ )=๐๐ฅ
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 (๐ฅ )= ๐ (4 ๐ฅ )The graph of is the graph of compressed horizontally by a factor of 4.
Example 3b: Use the graph of to describe the transformation of the function. Then sketch both functions.
๐ (๐ฅ )=๐โ๐ฅ+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 (๐ฅ )=๐๐ฅ
x y
โ 3 .04979
โ 2 .13534
โ 1 .36788
0 1
1 2.7183
2 7.3891
3 20.086
๐ (๐ฅ )= ๐ (โ๐ฅ )+3The graph of is the graph of reflected in the y-axis andtranslated 3 units up.
Example 3c: Use the graph of to describe the transformation of the function. Then sketch both functions.
๐ (๐ฅ )=12๐๐ฅ
x y
โ 3 .02489
โ 2 .06767
โ 1 .18394
0 .5
1 1.3591
2 3.6945
3 10.043
๐ (๐ฅ )=๐๐ฅ
x y
โ 3 .04979
โ 2 .13534
โ 1 .36788
0 1
1 2.7183
2 7.3891
3 20.086
๐ (๐ฅ )=12๐ (๐ฅ )
The graph of is the graph of compressed vertically by a factor of .
Example 4: Krysti invest $300 in an account with 6% interest rate, making no other deposits or withdrawals. what will Krystiโs account balance be after 20 years if the interest is compounded:
a. semiannually?
b. Monthly?
c. Daily?
๐ด=๐ (1+ ๐๐ )
๐๐ก
P = 300, r = 0.06, t = 20
n = 2๐ด=300(1+ 0.06
๐ )๐(20)
๐ด=300(1+ 0.062 )
2 (20 )
๐ดโ978.61
n = 12๐ด=300(1+ 0.06
12 )12(20)
๐ดโ993.06
n = 365๐ด=300(1+ 0.06
365 )365(20)
๐ดโ995.94
Example 5: Suppose Krysti invest $300 in an account with 6% interest rate, making no other deposits or withdrawals. What will Krystiโs account balance be after 20 years if the interest is compounded continuously?
๐ด=๐ ๐๐ ๐กP = 300, r = 0.06, t = 20
๐ด=300๐.06 (20)
๐ดโ996.04
a. 1.42% annually
b. 1.42% continuously
Example 6: Mexico has a population of approximately 110 million. If Mexicoโs population continues to grow at the described rate, predict the population of Mexico in 10 and 20 years.
๐=๐ 0๐๐๐ก
N0 = 110,000,000
r = 0.0142
,t = 10 and t = 20
๐=๐ 0 (1+๐ )๐ก
๐=110,000,000 (1+0.0142 )10 ๐=110,000,000 (1+0.0142 )20
๐ โ126,656,869 ๐ โ145,836,022 k = 0.0142
๐=110,000,000๐0.0142(10)
๐ โ126,783,431๐=110,000,000๐0.0142(20)
๐ โ146,127,622
โโ1math!
3-1 Assignment: TX p166, 4-32 EOE Test Corrections Due Friday 4/25Chapter 3 test Thursday 5/1