Transcript Exponential Functions


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
A fixed percent change in y indicates
that a function is exponential
 Formula for percent change

› Difference
original

For Example:
or
new - old
old
x
y
0
192
1
96
2
48
3
24
Percent change
=(B3-B2)/B2
Which of the following are exponential? Use
the percent change formula to figure it out.
x
y
x
y
0
192
0
0
1
96
1
1
2
48
2
4
3
24
3
9
x
y
x
y
0
3
0
.5
1
5
5
1.5
2
9
10
4.5
3
13
15
13.5

If the percent change is constant, the
function is exponential

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)


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

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

If a quantity changes at a fixed
percentage
› It grows or decays exponentially

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


Same formulas except
N=P-P*r
 N= P * (1 - r)

Increase 50 by 10%
› N= 50 + 50 * .1
= 50 + 5
= 55
OR
› N = 50 * (1+.10)
= 50 * 1.10
= 55

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
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 = 154,290.6
or
 N = 168993 * (1-.087) = 154,290.6

› Round to nearest whole number
› 154,291 crimes
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)
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

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
X
minutes
Y
population
0
1
2
3
4
5
100
=B2*(1+.05)


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

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
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

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
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?
Years after
death
% Carbon
remaining
0
100
200
.
.

100
=B2-0.01202*B2

Use the formula
y=P*(1-r) (measuring
decay)
Fill the table until %
carbon reaches
about 78%