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 or new - old

original old For Example: x y Percent change

0 192

1 96 =(B3-B2)/B2

2 48

3 24

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.

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 = 154,290.6or

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

Xminutes

Ypopulation

0 100

1 =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

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

100 =B2-0.01202*B2

200  

.  

.  

Use the formula y=P*(1-r) (measuring decay)

Fill the table until % carbon reaches about 78%