Aim: What is an exponential function?

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Aim What is an exponential function?

• Do Now
• Solve/evaluate 64 23n 1

Rewrite in exponential form
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Properties of Exponents
am an amn
(am)n amn
(ab)m ambm
a-n 1/an, a? 0
a0 1
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Types of exponents
a0 1
beware! -23/2 is not the same as (-2)3/2
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Exponential Function - Outbreak
Disease Control - Outbreak
y 2x
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Exponential Function
y a bx
. . have variables as exponents, and where a ?
0, base b gt 0, and also b ? 1.
The domain (x) is the set of real numbers (-?,
?) The range (y) is the set of positive real
numbers (0, ?)
The x-axis is a horizontal asymptote ax ? 0 as
x ? -?
If b gt 1, the graph is increasing and continuous
What happens as b increases in value?
y 1 2x y 2x
As b increases in value from 1, the slope of the
graph gets steeper
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The b Affect
y a bx
(0,1)
If b gt 1, the graph is increasing - Growth
the graph is decreasing - Decay
If 0 lt b lt 1
If b is a positive number other than 1, the
graphs of y bx and y (1/b)x are reflections
through the y-axis of each other
What if b 1?
horizontal line y 1
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The b Affect b lt 0
y a bx
What if b lt 0?
Let b (-2)?
y 1 (-2)x
if b lt 0, no longer the exponential function
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The a Affect
y a bx
a 4
a 2
a 1
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The a Affect
y a bx
Graph f(x) -2x -1 2x
Graph f(x) -(1/2)x -1 (1/2)x
Graph f(x) -6x -1 (6)x
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The a affect
Graph the exponential functions
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Model Problem
Write an exponential function y abx for a graph
that includes (2, 2) and (3, 4)
general form
substitute (2,2) for x y
solve for a
substitute for a
substitute (3,4) for x y
b 2
simplify solve for b
a 1/2
substitute solve for a
substitute for a and b
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Whered e Come From?
Graph
y ? 2.7183
y ? 2.7183 is asymptotic to f(x).
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Natural Exponential Function
f(x) ex
f(x) (2.71828. . .)x
7.389056099 0.3678794412 1.616074402
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Transforming Functions

• If k and h are positive numbers and
• f(x) is a function, then
• f(x) k shifts f(x) up k units
• f(x) k shifts f(x) down k units
• f(x h) shifts f(x) left h units
• f(x h) shifts f(x) right h units

f(x) (x h)2 k - parabolic
f(x) x h k - absolute value
ex. f(x) (x 4)2 4 is the image of g(x)
x2 after a shift of 4 units to the right and
four units up or a translation of T4,4.
NOTE k is not the y-intercept. what is the
y-intercept for f(x) (x 4)2 4?
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Transforming the Exponential Function
Graph the exponential functions y 4x, y 4x
2, y 4x 3
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Transforming the Exponential Function
Graph the exponential functions y 2x3, y
2x-2, y 2x1 2
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Transforming the Exponential Function
Write the equation and graph the exponential
function f(x) 2x after a reflection in both the
x- y-axes, and a vertical translation of 4.
Graph y 2x
Graph y (1/2)x
Reflect thru x-axis y -(1/2)x
Shift up 4 units y -(1/2)x 4
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Transforming the Exponential Function
Write the equation and graph the exponential
function after a reflection in the x-axis,
vertically stretched by a factor of 2, a vertical
translation of 2 and a horizontal of 2.
Graph y 2x
Reflect thru x-axis y -2x
Shift right 2 units y -(1/2)x
2
Shift up 2 units y -(1/2)x 2
2
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Transforming the Exponential Function
The graph of y (1/4)x is translated 5 units
upward, 8 units to the right and then reflected
in the x-axis. Whats the equation?
f(x) -f(x)
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Transforming the Exponential Function
The graph of y (1/4)x is translated 5 units
upward, 8 units to the right and then reflected
in the x-axis. Whats the equation?
f(x) -f(x)