Lesson 13 Introducing Exponential Equations

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Lesson 13Introducing Exponential Equations

• Integrated Math 10 - Santowski

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Comparing Linear Exponential Models

• Data set 1 Can this data be modeled with a
  linear relation? Why/why not? How do you know?
• Data set 2 - Can this data be modeled with a
  linear relation? Why/why not? How do you know?

x 0 1 2 3 4 5 7
y 5 10 15 20 25 30 40
x 0 1 2 3 4 5 7
y 16 24 36 54 81 121.5 273.375
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HINT

• Any obvious number patterns in the data??

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Comparing Linear Exponential Models

• Data set 3 Can this data be modeled with a
  linear relation? Why/why not? How do you know?
• Data set 4 - Can this data be modeled with a
  linear relation? Why/why not? How do you know?

x 0 1 2 3 4 5 6
y 50 75 112.5 168.75 253.13 379.69 569.53
x 0 1 2 3 4 5 6
y 5 7.75 10.5 13.25 16 18.75 21.5
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Comparing Linear Exponential Models

• Data set 5 Can this financial data be modeled
  with a linear relation? Why/why not? How do you
  know? What equation summarizes the data
  relationship and what do the s in the eqn
  represent?
• Data set 6 - Can this financial data be modeled
  with a linear relation? Why/why not? How do you
  know? What equation summarizes the data
  relationship and what do the s in the eqn
  represent?

x 0 1 2 3 4 5 7
y 5000 6000 7000 8000 9000 10000 12000
x 0 1 2 3 4 5 7
y 5000 6000 7200 8640 10368 12441.6 17915.90
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Summary

• What NEW pattern/relationships have you seen in
  the data sets?
• How can we write equations for our data that
  reflect our new patterns/relationships?

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Summary

• How can we write equations for our data that
  reflect our new patterns/relationships?
• If you have correctly answered this question, go
  to slides 12 - 16

• How can we write equations for our data that
  reflect our new patterns/relationships?
• If you have NOT correctly answered this question,
  go to slides 9 - 11

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Additional data sets (if needed)
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Exploring Exponential Equations

• Data set A This data can be modeled with a
  exponential relation. How do you know? What
  equation summarizes the data relationship?
• Data set B - This data can be modeled with a
  exponential relation. How do you know? What
  equation summarizes the data relationship?

x 0 1 2 3 4 5 6
y 1 2 4 8 16 32 64
x 0 1 2 3 4 5 6
y 729 243 81 27 9 3 1
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Exploring Exponential Equations

• Data set C This data can be modeled with a
  exponential relation. How do you know? What
  equation summarizes the data relationship?
• Data set D - This data can be modeled with a
  exponential relation. How do you know? What
  equation summarizes the data relationship?

x 0 1 2 3 4 5 6
y 0.3 0.6 1.2 2.4 4.8 9.6 19.2
x 0 1 2 3 4 5 6
y 100 125 156.25 195.31 244.14 305.18 381.47
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Exploring Exponential Equations

• Data set E This data can be modeled with a
  exponential relation. How do you know? What
  equation summarizes the data relationship?
• Data set F - This data can be modeled with a
  exponential relation. How do you know? What
  equation summarizes the data relationship?

x 0 1 2 3 4 5 6
y 50 30 18 10.8 6.48 3.89 2.33
x 0 1 2 3 4 5 6
y 360 120 40 13.33 4.44 1.48 0.494
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Application of Exponential Models
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Modeling Example 1

• Investment data Mr. S has invested some money
  for Andrews post-secondary education (not too
  hopeful for an athletic scholarship for my
  son!!!!)
• (a) How can you analyze the numeric data (no
  graphs) to conclude that the data is exponential
  ? i.e how do you know the data is exponential
  rather than linear?
• (b) Graph the data on a scatter plot
• (c) Write an equation to model the data. Define
  your variables carefully.

Time (years) 0 1 2 3 4 5 6 7 8
Value of investment (000s ) 8 8.480 8.989 9.528 10.000 10.706 11.348 12.029 12.751
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Modeling Example 2

• The following data table shows the relationship
  between the time (in hours after a rain storm in
  Manila) and the number of bacteria (/mL of
  water) in water samples from the Pasig River
• (a) How can you analyze the numeric data (no
  graphs) to conclude that the data is exponential
  ? i.e how do you know the data is exponential
  rather than linear?
• (b) Graph the data on a scatter plot
• (c) Write an equation to model the data. Define
  your variables carefully.

Time (hrs) 0 1 2 3 4 5 6 7 8
of Bacteria 100 196 395 806 1570 3154 6215 12600 25300
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Modeling Example 3

• The value of Mr. Ss car is depreciating over
  time. I bought the car new in 2002 and the value
  of my car (in thousands) over the years has been
  tabulated below
• (a) How can you analyze the numeric data (no
  graphs) to conclude that the data is exponential
  ? i.e how do you know the data is exponential
  rather than linear?
• (b) Graph the data on a scatter plot
• (c) Write an equation to model the data. Define
  your variables carefully.

Year 2002 2003 2004 2005 2006 2007 2008 2009 2010
Value 40 36 32.4 29.2 26.2 23.6 21.3 19.1 17.2
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Modeling Example 4

• The following data table shows the historic world
  population since 1950
• (a) How can you analyze the numeric data (no
  graphs) to conclude that the data is exponential
  ? i.e how do you know the data is exponential
  rather than linear?
• (b) Graph the data on a scatter plot
• (c) Write an equation to model the data. Define
  your variables carefully.

Year 1950 1960 1970 1980 1990 1995 2000 2005 2010
Pop (in millions) 2.56 3.04 3.71 4.45 5.29 5.780 6.09 6.47 6.85
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SUMMARY

• Write a SINGLE equation that summarizes the data
  relationships you have investigated this lesson.

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SUMMARY ? Exponential Growth Equations

• In general, the algebraic model for exponential
  growth is y c(a)x where a is referred to as the
  growth rate (provided that a gt 1) and c is the
  initial amount present and x is the number of
  increases given the growth rate conditions.
• All equations in this section are also written in
  the form y c(1 r)x where c is a constant, r
  is a positive rate of change and 1 r gt 1, and x
  is the number of increases given the growth rate
  conditions.

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SUMMARY ?Exponential Decay Equations

• In general, the algebraic model for exponential
  decay is y c(a)x where a is referred to as the
  decay rate (and a is lt 1) and c is the initial
  amount present.
• All equations in this section are in the form y
  c(1 r)x or y cax, where c is a constant, r is
  a rate of change (this time negative as we have a
  decrease so 1 r lt 1), and x is the number of
  increases given the rate conditions