4.1 Graph Exponential GrowthFunctions

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4.1 Graph Exponential GrowthFunctions

• p. 228
• What is an exponential function?
• What is exponential growth function?
• What is an asymptote?
• What information does the equation give you?
• How do you graph an exponential function?
• What is the equation for real life exponential
  growth? What is a, r, t?
• What is the equation for compound interest?
• What does A, P, r, n, t, stand for?

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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 an asymptote

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Asymptote

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

The graph gets closer and closer to the line y
0 . But NEVER reaches it
2 raised to any power Will NEVER be zero!!
y 0
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SOLUTION
Make a table of values.
STEP 1
STEP 2
Plot the points from the table.
Draw, from left to right, a smooth curve that
begins just above the x-axis, passes through the
plotted points, and moves up to the right.
STEP 3
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• This shows of y a 2x
• 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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• These are true of
• y abx
• If agt0 bgt1
• The function is an Exponential Growth Function
• http//my.hrw.com/math06_07/nsmedia/tools/Graph_Ca
  lculator/graphCalc.html

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D
D all reals R all realsgt0
y 0
Always mark asymptote!!
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Graph the function.
SOLUTION
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Example 1

• 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
• Find your asymptote from k
• Pick values for x. Try to make your exponent
  value 0 or 1.
• Complete your T chart (Find y).
• Sketch the graph.

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Example 2 Graph y 32x-1-4

• h 1, k -4
• asymptote y -4
• Pick x values
• x y

D all reals R all reals gt-4
1 2
-1 2
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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Write an exponential growth model giving the
number n of incidents t years after 1996. About
how many incidents were there in 2003?
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Graph the model.
Use the graph to estimate the year when there
were about 125,000 computer security incidents.
SOLUTION
STEP 1
The initial amount is a 2573 and the percent
increase is r 0.92. So, the exponential growth
model is
Write exponential growth model.
Substitute 2573 for a and 0.92 for r.
Simplify.
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The graph passes through the points (0, 2573) and
(1,4940.16). Plot a few other points. Then draw a
smooth curve through the points.
STEP 2
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STEP 3
Using the graph, you can estimate that the number
of incidents was about 125,000 during 2002 (t
6).
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Compound Interest

• Compound interest is interest paid on the initial
  investment, called the principal and on
  previously earned interest. Interest paid only
  on the principal is called simple interest.

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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

• 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
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• What is an exponential function?
• An equation with a variable as the exponent.
• What is an exponential growth function?
• f(x) abx when a gt 0 and b gt 1
• What is an asymptote?
• A line that a graph approaches as you move away
  from the origin.
• What information does the equation give you?
• y abx-h k (k - asymptote.)
• How do you graph an exponential function?
• Plot the asymptote, pick values for x to make
  exponent 0 or 1.
• What is the equation for real life exponential
  growth? What is a, r, t?
• y a(1 r)t a initial amount, r rate, t
  time
• What is the equation for compound interest?
• What does A, P, r, n, t, stand for?
• A amount in account, P principal, r rate, n
  times compounded, t of years

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4.1 Assignment

• Page 232, 4 30 even