Transcript Nth Roots and Rational Exponents

1)
12
3)
2
11) 81
59) a3/16
13) 16
1/5
63) 1 x y
15) 1/64
2/5
67) x7/6 – x11/6
17) 1216/3
19) (-11)14
23) (16)1/3 or 16
31) 21/5 or
2 5
35) 22  8 5
3
25) -8
33) 7
37)15
5
39) 1
43)
47)
d5 c
2
x9/2
1/2
5/2
51) a 49b
55) ax
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1
EXPONENTIAL FUNCTIONS
Section 5.2
Pre-Calculus AB, Revised ©2015
[email protected]
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EXPONENTIAL GROWTH VS DECAY
A.
Exponential Equation: f(x) = Cax
1.
2.
3.
B.
Exponential Equations
1.
2.
C.
Graph is above the x-axis
Domain: (–∞, ∞); Range: (0, ∞)
Exponential Growth
1.
2.
D.
C: COEFFICIENT
a: BASE
X: EXPONENT
When the base is greater than 1, a > 1
f(x) is INCREASING
Exponential Decay
1.
2.
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When the base is between zero and one, 0 < a < 1
f(x) is DECREASING
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EXAMPLE 1
Determine whether 𝒇 𝒙 =
decay.
𝟏 𝒙
represents
𝟓
exponential growth or
1
0  x
5
Exponential Decay
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YOUR TURN
Determine whether 𝒇 𝒙 = 𝟏𝟎
decay.
𝟒 𝒙
𝟑
represents exponential growth or
Exponential Growth
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TRANSLATIONS
A. Vertical Translations
1. Moving the graph up and down; the change is OUTSIDE the exponent
2. Moving UP is when the translated is added, Moving DOWN is when the graph
is subtracted
B. Horizontal Translations
1. Moving the graph up and down; the change is WITHIN the exponent
2. Moving LEFT is when the translated is added, Moving RIGHT is when the
graph is subtracted
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EXAMPLE 2
If f(x)=2x, determine the transformation to g(x) = 2x + 2
up 2
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EXAMPLE 3
If f(x)=2x, determine the transformation to g(x) = 2x + 2
left 2
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YOUR TURN
If f(x)=2x, determine the transformation to g(x) = 2x + 3 – 4
left 3, down 4
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VERTICAL STRETCHES AND COMPRESSIONS
A. Vertical Stretches and Compressions
1. Applied to the coefficient of y = Cax
2. Stretches when |c| > 1
3. Compresses when |c| < 1
B. Horizontal Stretches and Compressions
1.
2.
3.
4.
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Applied to the number in front of the exponent, d, of
y = Cadx
Take the horizontal stretch/compression and take the reciprocal
Stretches when 0 < d < 1
Compresses when d > 1
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EXAMPLE 4
If f(x) = 2x, determine the transformation to g(x) = (1/4)(2)x
1
Vertical Compression by
4
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EXAMPLE 5
If f(x) = 2x, determine the transformation to g(x) = (4)(2)x
Vertical Stretch by 4
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YOUR TURN
If f(x) = 2x, determine the transformation to g(x) = (–5)(2)x
Vertical Stretch by 5
Reflected across x  axis
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EXAMPLE 6
If f(x) = 2x, determine the transformation to g(x) = (2)0.25x
1

1/ 4
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4
Horizontal Stretch by 4
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YOUR TURN
If f(x) = 2x, determine the transformation to g(x) = (2)4x
1
1
 Horizontal Compression by
4
4
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EXAMPLE 7
If you invest in $5,000 in a stock that is increasing n value at the rate of
3% per year, then the value of your stock is given by the function f(x)
= 5000(1.03)x, where is measured in years. Assuming that the value
of your stock continues growing at this rate, how much will your
investment be worth in 4 years?
f  x   5000 1.03
f  x   5000 1.03
x
4
 $5627.54
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YOUR TURN
The number of subscribers, in millions, to basic cable TV is given this
76.7
equation, 𝒇 𝑥 =
𝑥 where x = 0 corresponds to 1970.
1+16 0.8444
Estimate the number of subscribers 1995.
f  x 
f  x 
76.7
1  16  0.8444 
76.7
1  16  0.8444 
x
25
 62.193 million
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HALF LIFE t
 1 h
N F  N0  
2
Half-life is defined as a period of time it takes for the amount of a
substance undergoing decay to decrease by half.
NF = Population Left
N0 = Initial Population
T = Elapsed Time
H = Half-Life
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EXAMPLE 8
Barium-140 has a half-life of 13 days. If there are 500 milligrams of
barium initially, how many milligrams remain after 26 days?
1
N F  N0  
2
1
N F  500  
2
t
k
26
13
NF = ??
Population Left
N0 = 500
Initial Population
T = 26
Elapsed Time
K = 13
Growth or Decay
Rate
N F  125 milligrams
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EXAMPLE 9
Barium-140 has a half-life of 13 days. If there are 500 milligrams of
barium initially, how many milligrams remain after 100 days?
N F  2.417 milligrams
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YOUR TURN
A radioactive substance has a half-life of 4 years. If there are 600
milligrams of the substance initially, how many milligrams remain after
20 years?
N F  18.75 milligrams
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ASSIGNMENT
Page 343
1-13 odd, 21, 51
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