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(a)
(b)
(c)
(d)
Find the accumulated value of a CD of $20,000 for 3 yearsat an interest rate of 3.1% if the money is compounded continuously?
50,690218,760,38421,94921,860
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The exponential equation 13.49 .967 1 predicts the number of O-rings
that are expected to fail at the temperature x F on the space shuttles. The O-rings were used to seal the connections between d
x
o
f x
ifferent sections of the shuttleengines. Use a calculator to find the number expected to fail at the temperature of 40 degrees.
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.8
The population of a small polynesian island can be modeled by
the equation f(x)=8e . Round off your answers to the nearest integer.a. How many people originally went to this small island? (x=0)b. H
x
ow many people will be living on this island 10 years laterdue to general population growth with no new immigration.
Section 5.2/5.4Exponential and Logarithmic Functionsp. 339: 28-36 (even), 37p. 358: 12-20 (even), 31-38
Definition of the Exponential Function The exponential function f with
base b is defined by f(x) =bx or y = bx Where “b” is a positive
constant other than 1 and greater than 0.
YES f(x) = 2x
f(x) = 10x
f(x) = 3x+1 f(x) = (1/2)x-1
NO f(x) = x2
f(x) = 1x f(x) = (-1)x f(x) = xx
Exponential Growth vs. Decay: y = bx
Growth b>1 Domain: (-∞,∞) Range: (0,∞) y-intercept is (0,1) As x increases, for b >
1, f(x) also increases without bound
The x-axis (y = 0) is the asymptote
Decay 0<b<1 Domain: (-∞,∞) Range: (0,∞) y-intercept is (0,1) As x increases, for
0<b<1, f(x) decreases, approaching zero
The x-axis (y = 0) is the asymptote
Example 1Graph the following two equations: f(x)= , f(x)= 44
Draw the asymptotes.
xx
Consider the function y = 2x. How would the graph change given the transformations below?
f(x) = 2x-1
f(x) = 2x + 4
f(x) = 2x - 2
f(x) = 2x + 3
Transformations of Exponential Functions
Transformation
Equations Description
Vertical Translation
G(x) = bx + cG(x) = bx – c
• Shifts the graph of f(x) = bx UP c units
• Shifts the graph of f(x) = bx DOWN c units
Horizontal Translation
G(x) = bx+c
G(x) = bx-c• Shifts the graph of f(x) = bx LEFT
c units• Shifts the graph of f(x) = bx RIGHT
c unitsReflection G(x) = -bx
G(x) = b-x• Reflects the graph f(x) bx about
the x-axis• Reflects the graph f(x) bx about
the y-axisVertical Shrinking or Stretching
G(x) = cbx • If c > 1, vertical stretch• If 0< c < 1, vertical shrink
Horizontal stretching or shrinking
G(x) = bcx • If c > 1, horizontal shrink• If 0< c < 1, horizontal stretch
Example Use the graph of f(x)=4 to obtain the graph of g(x)=4 3.What is the domain and range of each function?
x x
Example 2Use the graph of f(x)=4 to obtain the graph of g(x)=4Find the domain and range for the g(x) function.
x x
Example Use the graph of f(x)=4 to obtain the graph of g(x)=2 4Find the domain and range for the g(x) function.
x x
Logarithmic functions
Definition: Let b > 0 and b not equal to 1. Then y is the logarithm of x to the base be written: y = logbx if and only if by = x
In other words, a logarithmic graph is the inverse of an exponential.
Graphing logarithms To graph y = logb x Rewrite as an
exponential equation: by = x
Make an x/y table, filling in y first.
Graph points. y = log3x
x Y
-1
0
1
2
x
y
Other logarithms
Common Logarithm: (base 10)log x = y 10y = x
Natural Logarithm: (base e)logex = y ln e = y ey = x
Properties of y = logbx y = log bx OR
x = by
Domain
Range
Asymptotes (line that graph
approaches, but does not touch)
Point on all graphs
Determining the Domains of Logarithmic Functions.FYI: the range never changes!
Remember the “argument” must be positive (> 0)
f(x) = log2 (x – 1)
f(x) = (log3 x) – 1
f(x) = log4 |x|
f(x) = log5 ( x2 – 4)
Transformations of Logarithmic Functions
Transformation
Equations Description
Vertical Translation
G(x) = logbx + c
G(x) = logbx - c
• Shifts the graph of f(x)= logbx UP c units
• Shifts the graph of f(x)= logbx DOWN c units
Horizontal Translation
G(x) =logb (x + c)
G(x) = logb (x - c)
• Shifts the graph of f(x)= logbx LEFT c units, VERTICAL ASYMPTOTE (x = -c)
• Shifts the graph of f(x)= logbx RIGHT c units, VERTICAL ASYMPTOTE (x = c)
Reflection G(x) = -logbx
G(x) = logb (-x)
• Reflects the graph f(x)= logbx about the x-axis
• Reflects the graph f(x)= logbx about the y-axis
Vertical Shrinking or Stretching
G(x) =c logbx • If c > 1, vertical stretch• If 0< c < 1, vertical shrink
Horizontal stretching or
shrinkingG(x) = logb(cx)
• If c > 1, horizontal shrink• If 0< c < 1, horizontal stretch
Example3
3
Use transformations to graph g(x)=2+log ( 3). Start with log .
xx
x
y
Example
3Use transformations to graph g(x)=-log x
x
y
Example
3Use transformations to graph g(x)=log ( )x
x
y
Closure
4Find the domain of f(x) = log ( 5)x
0
0
Use the formula R=log to solve the problem. If an
earthquake is 100 times as intense as a zero-level quake(I=100 I ), what is its magnitude on the Richter Scale?
II
2
Find the domain of each function.a. f(x)= ln (x-3)
b. h(x)=ln x