exponential functions
DESCRIPTION
Exponential Functions. LSP 120 Joanna Deszcz Week 2. Linear vs. Exponential. Linear Constant rate of change (slope) y2-y1 x2-x1 Uses formula y=mx+b. Exponential For a fixed change in x there is a fixed percent change in y Percent change formula is new-old old - PowerPoint PPT PresentationTRANSCRIPT
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Linear › Constant rate of
change (slope)› y2-y1
x2-x1› Uses formula
y=mx+b
Exponential› For a fixed change
in x there is a fixed percent change in y
› Percent change formula is new-old
old› Uses formula
y=P*(1+r)x
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A fixed percent change in y indicates that a function is exponential
Formula for percent change› Difference or new - old
original old For Example: x y Percent change
0 1921 96 =(B3-B2)/B22 483 24
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x y
0 192
1 96
2 48
3 24
x y
0 0
1 1
2 4
3 9
x y
0 3
1 5
2 9
3 13
x y
0 .5
5 1.5
10 4.5
15 13.5
Which of the following are exponential? Use the percent change formula to figure it out.
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If the percent change is constant, the function is exponential
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General equation is› y = P * (1 + r)x
P is initial value or value of y when x=0 r is percent change – written as a
decimal x is input variable (usually time)
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Populations tend to growth exponentially not linearly
When an object cools (e.g., a pot of soup on the dinner table), the temperature decreases exponentially toward the room temperature
Radioactive substances decay exponentially
Bacteria populations grow exponentially
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Money in a savings account with a fixed rate of interest increases exponentially
Viruses and even rumors tend to spread exponentially through a population (at first)
Anything that doubles, triples, halves over a certain amount of time
Anything that increases or decreases by a percent
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If a quantity changes at a fixed percentage› It grows or decays exponentially
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2 ways N = P + P * r N= P * (1 + r)
› N is ending value› P is starting value and › r is percent change
By the distributive property, equations are the same
2nd version is preferred
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Same formulas except N = P - P * r N= P * (1 - r)
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Increase 50 by 10%› N= 50 + 50 * .1
= 50 + 5 = 55
OR › N = 50 * (1+.10)
= 50 * 1.10 = 55
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Sales tax in Chicago is 9.75%. You buy an item for $42.00. What is the final price of the article?
N = 42 + 42 * .0975 = 46.095 or
N = 42 * (1+.0975) = 46.095› Round value to 2 decimal places since
it’s money› Final answer: $46.01
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In 2008, the number of crimes in Chicago was 168,993. Between 2008 and 2009 the number of crimes decreased 8.7%. How many crimes were committed in 2009?
N = 168993 - 168993 * .087 = 154,290.6or
N = 168993 * (1-.087) = 154,290.6› Round to nearest whole number› 154,291 crimes
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Increase by same percent over and over If a quantity P is growing by r % each
year, › after one year there will be P*(1 + r)› after 2 years there will be P*(1 + r)2
› after 3 years there will be P*(1 + r)3 Each year the exponent increases by
one since you multiply what you already had by another (1 + r)
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Decrease by same percent over and over
If a quantity P is decaying by r % each year, › after one year there will be P*(1 - r)› after 2 years there will be P*(1 - r)2
› after 3 years there will be P*(1 - r)3
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A bacteria population is at 100 and is growing by 5% per minute.
› How many bacteria cells are present after one hour?
› How many minutes will it take for there to be 1000 cells.
Use Excel to answer each question
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Xminutes
Ypopulation0 1001 =B2*(1+.05)2 3 4 5
5% increase per minute (.05)
Multiply population by 1.05 each minute
Note: no exponent needed here since filling column gives us the population each minute
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Use the “by hand” formula with the exponent› y = P *(1+r)x
In this case:› y= 100 * (1 + .05)60
› 100 is population at start› .05 is 5% increase› 60 is number of minutes
Remember to round appropriately
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Emission of energy (or particles) from the nuclei of atoms
Atoms like to have the same # of protons and neutrons› Like to be stable
Unstable atoms are radioactive› Throw off parts to become stable
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Good› Used in medicine
to treat heart disease, cancer
› Kills bacteria like salmonella
Bad› Uncontrolled
nuclear chain reactions can cause major damage Like Chernobyl
› Can cause burns to cell mutations
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Uses Exponential functions Example:
› We date the Dead Sea Scrolls which have about 78% of the normally occurring amount of Carbon 14 in them. Carbon 14 decays at a rate of about 1.202% per 100 years.
› How old are the Dead Sea Scrolls?
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Years after death
% Carbon remaining
0 100
100 =B2-0.01202*B2
200 . .
Use the formula y=P*(1-r) (measuring decay)
Fill the table until % carbon reaches about 78%