Warmup ‐‐ On Student Notes for Day 3
Identify growth/decay factor, growth/decay rate and initial value
(A through H)
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Standard 6.3
Writing Exponential Equations
Standard 6.3
Proficient: Write exponential growth and decay models from context and make predictions.
Expert: Write exponential equations given a table.
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Identify growth/decay factor, growth/decay rate and initial value
(A through H)
A.
B.
C.
D.
E.
F.
G.
H.
growth/decay:
factor:
rate:
initial value:
growth/decay:
factor:
rate:
initial value:
growth/decay:
factor:
rate:
initial value:
growth/decay:
factor:
rate:
initial value:
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Review: Calculate growth factor from % increase:
I.
50% J. 5% K.
0.5% L. 0.05%
Calculate decay factor from % decrease:
M.
50% N. 5% O.
0.5% P. 0.05%
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Context Problems
Example 1A. You start a walking program. You start by walking 2 miles. Then each week you increase your distance 5% per week. Write an exponential growth function to model the situation. How far will you walk in the tenth week?
General equation:
Final equation:
Initial value:
Prediction:
Growth rate:
Growth factor:
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Example 1B. An initial population of 20 mice triples each year for 8 years. Write an equation that models this population. What is the mouse population after 5 years?
General equation:
Final equation:
Initial value:
Prediction:
Growth rate:
Growth factor:
6
Example 2A: You bought a car for $16,000.You expect the car to lose value, or depreciate, at a rate of 12% per year. Write an exponential decay model to represent this situation. How much will the car be worth in 3 years?
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Example 2B: Membership in an after‐school athletic club declined at a rate of 5% per year for the period 2000‐2005. There were 54 members in 2000. Identify the initial amount, the decay rate, and the decay factor, then write an equation that describes this situation. In what year did the club have 45 members?
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Example 1C: A musician is saving money to buy a new snare drum. The musician puts $100 in a savings account that pays 3% annual interest compounded yearly. Write a function that models the amount of money in the account over time. The musician wants a drum that costs $149.95. Will there be enough in the account after 3 years? Explain.
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Example 2C: The half‐life of a medication is the time it takes for the medication to reduce to half of its original amount in a patient’s bloodstream. A certain antibiotic has a half‐life of about 8 hours. A patient takes 500 milligrams of the medication. Write a function that models the amount of the medication in the patient’s bloodstream over time. How much of the 500 milligram dose will be in the patient’s bloodstream after 24 hours?
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Example 1D: A tree’s cross‐sectional area taken at a height of 4.5 feet from the ground is called its basal area and is measured in square inches. Tree growth can be measured by the growth of the tree’s basal area. The initial basal area and annual growth rate for two particular trees are shown in the table below. Write a function that models the basal area A of each tree over time. In about how many years will the trees be the same height? 11
Expert Extras
Exponential Equations from a Table of Data (Expert)
Step 1) Find the growth/decay factor by calculating the ratio of outputs (a.k.a. y2/y1)
Step 2) Find the initial value from the table of values if the point with x = 0 is given, since y = a when x = 0. If this value is not given, use one of the points to solve for a.
Step 3) Write growth/decay equation using calculated values for a and b.
Example 3A.
Growth/Decay Factor
Initial Value
Equation
Example 3B.
Growth/Decay Factor
Initial Value
Equation
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3C. 3D . 3E.
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