2.  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

3.  A fixed percent change in y indicates that a function is exponential  Formula for percent change › Difference or new - old original old  For Example: xyPercent change 0192 196=(B3-B2)/B2 248 324

4. xy 0192 196 248 324 xy 00 11 24 39 xy 03 15 29 313 xy 0.5 51.5 104.5 1513.5 Which of the following are exponential? Use the percent change formula to figure it out.

5.  If the percent change is constant, the function is exponential

6.  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)

7.  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

8.  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

9.  If a quantity changes at a fixed percentage › It grows or decays exponentially

10.  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  2 nd version is preferred

11.  Same formulas except  N = P - P * r  N= P * (1 - r)

12.  Increase 50 by 10% › N= 50 + 50 *.1 = 50 + 5 = 55 OR › N = 50 * (1+.10) = 50 * 1.10 = 55

13.  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

14.  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

15.  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)

16.  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

17.  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

18. X minutes Y population 0100 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

19.  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

21.  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

22.  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

23.  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?

24. Years after death % Carbon remaining 0100 =B2-0.01202*B2 200..  Use the formula y=P*(1-r) (measuring decay)  Fill the table until % carbon reaches about 78%