8.1%20Exponential%20Growth

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8.1 Exponential Growth
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Objectives/Assignment

1. Graph exponential growth functions
2. Use exponential growth functions
3. Assignment 13-41 odd

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Use a calculator to graph the following
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Exponential Function

• f(x) bx where the base b is a positive number
  other than one.
• Graph f(x) 2x
• Note the end behavior
• x?8 f(x)?8
• x?-8 f(x)?0
• y0 is horizontal asymptote

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Asymptote

• A horizontal line that a graph approaches as you
  move away from the origin

The graph gets closer and closer to the line y
0 . As values Of x become larger And larger
negative Negative numbers But NEVER reaches it
2 raised to any power Will NEVER be zero!!
y 0
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Look at the activity on p. 465

• Graph y a 2x (first with a 1/3 then a 3)
• Now let a -1/5 then let a -5.compare these
  graphs with those graphed above.
• What is the effect of a on the graph of y?
• Passes thru the point (0,a) (the y intercept is
  a)
• The x-axis is the asymptote of the graph
• D is all reals (the Domain)
• R is ygt0 if agt0 and ylt0 if alt0
• (the Range)

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• Consider
• y abx
• If agt0 bgt1
• The function is an Exponential Growth Function

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Example 1

• Graph y (½) 3x
• Plot (0, ½) and (1, 3/2)
• Then, from left to right, draw a curve that
  begins just above the x-axis, passes thru the 2
  points, and moves up to the right

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D all reals R all realsgt0
y 0
Always mark asymptote!!
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Example 2

• Graph y - (3/2)x
• Plot (0, -1) and (1, -3/2)
• Connect with a curve
• Mark asymptote
• D??
• All reals
• R???
• All reals lt 0

y 0
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To graph a general Exponential Function

• y a bx-h k
• Sketch y a bx
• h ??? k ???
• Move your 2 points h units left or right and k
  units up or down
• Then sketch the graph with the 2 new points.

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Example 3 Graph y 32x-1-4

• Lightly sketch y32x
• Passes thru (0,3) (1,6)
• h1, k-4
• Move your 2 points to the right 1 and down 4
• AND your asymptote k units (4 units down in this
  case)

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D all reals R all reals gt-4
y -4
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Nowyou try one!

• Graph y 23x-2 1
• State the Domain and Range!
• D all reals
• R all reals gt1

y1
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Exponential Growth Models

• When a real life quantity increases by fixed
  percent each year (or other time period), the
  amount y of the quantity after t years can be
  modeled by
• y a(1r)t
• Where a is the initial amount and r is the
  percent increase expressed as a decimal.
• The quantity 1r is called the growth factor

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Compound Interest

• AP(1r/n)nt
• P - Initial principal
• r annual rate expressed as a decimal
• n compounded n times a year
• t number of years
• A amount in account after t years

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Compound interest example

• AP(1r/n)nt

• You deposit 1000 in an account that pays 8
  annual interest.
• Find the balance after I year if the interest is
  compounded with the given frequency.
• a) annually b) quarterly c) daily

A1000(1.08/4)4x1 1000(1.02)4 1082.43
A1000(1 .08/1)1x1 1000(1.08)1 1080
A1000(1.08/365)365x1 1000(1.000219)365
1083.28