W16D3 Number e
Warm Up
1. Simplify
25x
22x
2. Simplify 46x · 43
3. What is x? 4x = 256
1. 23x
2. 46x−3
3. x = 4
Lesson 39 The number e
All examples are in W16D3 Worksheet if inquiry is preferred.
e is just like π - an irrational number represented by the letter e. e is 2.71 (approximately).
Graph y = ex
EX 1:
x
y
-1
0.4
0
1
1
2.71
2
8
5
4
3
2
1
−5 −4 −3 −2 −1
−1
1
2
3
4
5
−2
−3
−4
−5
EX 2:
Determine if each of the following are growth or decay
1. y = ex − 2
2. y = e−2x + 1
3. y =
3 3x
e −2
4
4. y = −2e5x+1
5.
1
y = − e−3x
4
6. y = 5e3x−9 − 7
Answers
1.
2.
3.
4.
5.
6.
growth
decay
growth
growth
decay
growth
EX 3:
Simplify each of the following
1.
(e4 )−2
2.
(3e2 )3
3.
27e5
6e7
4.
e3x e2x+1 e
5.
2e2 3e2x+1
√
6.
√
3
7.
Answers
1.
1
e8
54e12x
e
e2x+1
8.
9.
8e6
e2x
e
2
2.
27e6
3.
9
2e2
4.
e5x+2
5.
6e2x+3
6.
√
2e3 2
7.
√
3e4x 3 2
8.
1
e2x
9.
e4x−2
EX 3:
Identify the asymptote and y-intercept of each function
1. y = 2ex + 1
y − int(0, 3)
asym.y = 1
y = −4ex−1
−4
y − int(0,
)
e
asym.y = 0
2.
3. y = e2x+3 − 2
y − int(0, e3 − 2)
asym.y = −2
4. y = −5e−2x
−5
y − int(0,
)
e
asym.y = 0
5. y = ex+3 − 4
y − int(0, e3 − 4)
asym.y = −4
6.
3 3x 11
e −
4
4
y − int(0, −2)
11
asym.y = −
4
y=
Growth and Decay Word Probs 1. Bacteria in a culture are growing exponentially with time, as shown in the table below.
Days
Bacteria
0
2000
1
1500
2
1125
3
843.75
Write a function for the bacteria’s growth or decay.
What would the y-intercept be?
What would the asymptote be?
2. Tell whether the function represents exponential growth or exponential decay. Find the y-intercept and also the horizontal
asymptote for the function:
Movie tickets now average $9.75 a ticket, but are increasing 15% per year. How much will they cost 5 years from now?
A powerful computer is purchased for $2000,but on the average loses 20% of its value each year. How much will it be worth 4
years from now?
Dinner at your grandfather’s favorite restaurant now costs $25.25 and has been increasing steadily at 4% per year. How much
did it cost 35 years ago when he was courting your grandmother?
If a gallon of milk costs $3 now and the price is increasing 10% per year, how long before milk costs $10 a gallon?
The number of bacteria present in a colony is 180 at 12 noon and the bacteria grows at a rate of 22% per hour. How many will
be present at 8 p.m.?
6. A house purchased for $226,000 has lost 4% of its value each year for the past five years. What is it worth now?
7. A 1970 comic book has appreciated 10% per year and originally sold for $0.35. What will it be worth in 2010?8. A Honda
Accord depreciates at 15% per year. Six year ago it was purchased for$21,000. What is it worth now?
9. Inflation is at a rate of 7% per year. Today Janelle’s favorite bread costs $3.79. What would it have cost ten years ago?
10. Ryan’s motorcycle is now worth $2500. It has decreased in value 12% each year since it was purchased. If he bought it four
years ago, what did it cost new?
11. The cost of a High Definition television now averages $1200, but the cost is decreasing about 15% per year. In how many
years will the cost be under $500?
Graphing Practice
7.
f(x) = 3x − 2
8.
f(x) = 2x+1
9.
f(x) = 2−x − 5
10.
f(x) = 2(4x−1 ) − 5
11.
f(x) =
12.
f(x) = (.5)x
13.
f(x) = (−3)x
14.
f(x) = −3( 13 )x+2
15.
f(x) = 4( 23 )x − 5
1 x+2
4
+3
5
5
4
4
3
3
2
2
1
1
−5 −4 −3 −2 −1
−1
1
2
3
4
5 −5 −4 −3 −2 −1
−1
−2
−2
−3
−3
−4
−4
−5
−5
Figure 1: 1. 3x − 2
1
2
3
4
5
3
4
5
2. 2x+1
5
2
4
1
3
2
−5 −4 −3 −2 −1
−1
1
−5 −4 −3 −2 −1
−1
1
2
−2
1
2
3
4
5
−3
−2
−4
−3
−4
−5
−5
−6
Figure 2: 3. 2−x − 5
Exit Pass
1. Graph y = 2(ex ) − 3
2. What is x
6x = 216
4. 2(4x−1 ) − 5
30
6
25
5
20
4
15
3
10
2
5
1
−5 −4 −3 −2 −1
−5
1
2
3
4
5 −5 −4 −3 −2 −1
−1
−10
2
3
4
5
4
5
−2
Figure 3: 5.
1 x+2
4
6. (.5)x
+3
10
9
8
7
6
5
4
3
2
1
−1
−5 −4 −3 −2 −1
−2
−3
−4
−5
−6
−7
−8
−9
−10
1
10
9
8
7
6
5
4
3
2
1
1
2
3
4
−1
5 −5 −4 −3 −2 −1
−2
−3
−4
−5
−6
−7
−8
−9
−10
Figure 4: 7.
1 x+2
4
+3
1
2
3
8. 4( 32 )x − 5