Modeling with Exponential Data

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Modeling with Exponential Data

• Finding the Constant Ratio

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Having already studied linear functions, we know
that these functions

• Have a constant growth rate (slope)
• Have an initial value (y-intercept)
• Have a slope-intercept form (ymxb)
• Have a point-slope form (ym(x-x1)y1)

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Looking at the table, we know the data is linear
because

• There is a common difference between the data
  points.
• We continue to
• ADD the same
• value to the
• previous value.

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That common difference is the slope of the line
that connects the points.
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We also have already seen that in an exponential
function

• The change is not constant.
• We do NOT add
• the same value
• to the previous
• value.

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In an exponential function, the change comes from
multiplication.

• We multiply
• the same value
• to the previous
• value.
• We have a common ratio!!!

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A growth factor for an exponential function comes
from multiplying by a common ratio.

• To find a common ratio, just divide the second
  term by the first term.
• Or divide the third term by the second term.
• Or divide the fourth term by the third term.
• And so on

4/2 2
8/4 2
16/8 2
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Lets find the common ratio for this exponential
data.

• Jack planted a mysterious bean just outside his
  kitchen window. It immediately sprouted 2.56 cm
  above the ground. Jack kept a careful log of the
  plants growth. He measured the height of the
  plant each day at 800 am and recorded these data.

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Select two consecutive terms and divide to find
the ratio.

• The catch is, the x-coordinates need to increase
  by 1. These do.

6.4/2.56 2.5
The common ratio for this data is 2.5.
16/6.4 2.5
40/16 2.5
100/40 2.5
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To write the exponential function that models
this situation

• remember that y abx.
• y (initial value)(common ratio)x
• In this problem, the initial value was 2.56 cm,
  because that is how much the mystery bean grew
  immediately.
• The common ratio we just calculated is 2.5

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So the function for this situation is y
2.56(2.5)x

• Now, how high would the plant be on the 5th day?
  on the 6th day?
• We could either continue the pattern by
  continuing to multiply by 2.5
• or we could use the function rule and substitute
  the day number in for x.

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Pick a method and find those two answers.

• On day 5, the plant was 250 cm tall.
• On day 6, the plant was 625 cm tall.

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Jacks younger brother measured the plant at 800
pm on the evening of the third day.

• He found the plant to be 63.25 cm tall. Show how
  to find this value using the equation.

800 pm of the third day is 3.5 hours after the
bean first went into the ground. We substitute
3.5 into the equation for x. y 2.56(2.5)3.5
63.24555 or 63.25 cm
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Sometimes, we just have two data points to use
instead of an entire table of data.

• Penicillin decays exponentially in the human
  body. If you receive a 300-milligram dose of
  penicillin to combat strep throat, about 180
  milligrams will still be active in your blood
  after one day.

Write the coordinates of two data points from
this situation.
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(0,300) would represent the initial amount.

• (1, 180) would represent the data for the next
  day.
• Now, use the data points to find the decay factor.

These two data points have a difference of 1
x-value. 180/300 0.6
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We can now write the function

• P 300(0.6)d
• where d is the of days since the penicillin was
  first taken and P is the amount of penicillin, in
  mg, still in the body.

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What is the domain and is it discrete or
continuous?

• The domain starts at day 0 and goes until there
  is no more penicillin left in the body. My
  calculator shows that this equation never reaches
  0 mg, but we know that to not be true. If we
  round to the nearest hundredth, after 22 days
  there should not be any trace of the penicillin.
  0 d 22
• Since we can have a partial day, the data is
  continuous.

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What is the range?

• The most medicine that we had was 300 mg and the
  least would be none, so the range is 0 P
  300.

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How much penicillin would there be 4 ½ days later?

• y 300(.6)4.5 30.1163
• There would still be 30.11 mg of penicillin in
  the body.

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Assuming that the penicillin decayed at the same
rate, how would the equation change if you had
taken a 400-milligram dose?

• The initial amount would be different so the
  equation would be y 400(0.6)x

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Try another one

• Chandler invested 1000 in a money market
  account. The next year, the value of the account
  had grown to be 1030. What function models the
  growth of the account? If he doesnt remove any
  of the money and it continues to grow at the same
  rate, how much will be in the account after 10
  years?

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(0, 1000) represents the initial investment.

• (1, 1030) represents the money in the account one
  year later.
• The growth factor is 1030/1000 1.03
• The function is M 1000(1.03)t
• After 10 years M 1000(1.03)10 1343.92

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Now it is time for your assignment.