 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: xyPercent change =(B3-B2)/B

xy xy xy xy 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  2 nd version is preferred

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

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

 Sales tax in Chicago is 9.75%. You buy an item for $ What is the final price of the article?  N = *.0975 = or  N = 42 * ( ) = › 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 = *.087 = 154,290.6 or  N = * (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 =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 0100 =B *B  Use the formula y=P*(1-r) (measuring decay)  Fill the table until % carbon reaches about 78%