5.2 / 5.3 - Exponential Functions, Graphs and Applications Exponential function: constant base, variable exponent. ex: f(x) = 2x g(x) = 3-x Definition: The exponential function with base ‘a’ is denoted by: f(x) = ax, where a > 0, a = 1 and x is any real number. Graphs: a-x ax (0,1) Sign of base controls y-direction ------------------------- Sign of exponent controls x-direction -a-x -ax (0,-1)

Shifting: inside/outside rules still apply: Inside ex: a(x + 1) [Graph shifts one unit to the left] Outside ex: ax - 2 [Graph shifts two units down] Reflection still controlled by negatives (previous slide) Stretch controlled by mult.& division of exponent, magnitude of base… y = 2x y = 4x

Base e ‘e’ is a constant used in many mathematical computations ‘e’ does not have an exact value because it is irrational: irrational = non-terminating / non-repeating decimal. e = 2.71828….. So approximately 2.7 As of April 2007, ‘e’ has been calculated to 100 Billion places… The first 2 million digits of e... ex is referred to as the ‘natural exponential’. In banking, ‘e’ is used to calculate continuously compounded interest…

Compound Interest Formulas ex: $9000 @ 8.5%, compounded annually for 3 years…. Vocab: Balance (A) Principal (P) Annual percentage rate (r) Number of years (t) Formulas: For ‘n’ compoundings / year: A = 9000 (1 + )(1)(3) .085 1 = $ 11,495.60 ex: $12000 @ 9%, compounded quarterly for 5 years…. A = 12000 (1 + )(4)(5) .09 4 r n A = P(1 + )nt = $ 18,726.11

Formula for Continuous Compounding A = Pert ex: $12000 @ 9%, compounded continuously for 5 years…. A = 12,000 (e)(.09)(5) = $ 18,819.75 Exponential Growth f(x) = Pax f(x) = P(1 + r)x Very similar to interest problems P = starting value a = the factor by which the quantity is changing (generally will be 1 + r where r is a percentage) x = variable for time

ex: In 1950, the world population was about 2.5 billion people and has been increasing at 1.85% per year. Write the function that gives the world population in year x, where x = 0 corresponds to 1950… f(x) = P(1 + r)x P = 2.5 a = (1 +.0185) = 1.0185 x = x f(x) = 2.5(1.0185)x ex: At the beginning of an experiment, a culture contains 1000 bacteria. Five hours later, there are 7600 bacteria. Assuming that the bacteria grow exponentially, how many bacteria will there be after 24 hours? P = 1000--->f(x) = 1000ax ---> f(5) = 1000a5 ---> 7600 = 1000a5 ---> 7.6 = a5

So… After 24 hours… = 16,900,721 General Exponential Decay ex: When tap water is filtered through a layer of charcoal and other purifying agents 30% of the chemical impurities in the water are removed. How many layers are needed to ensure that 95% of the impurities are removed from the water? f(x) = P(1 - r)x The % of impurities left after x layers… P = not given (let P = 1) (1 - r) = (1 - .3) = .7 x = layers of purifying f(x) = .7x 5% is left after x layers… find x… .05 = .7x

ENTER-- ENTER-- ENTER So x = approx. 8.4 layers… Since we can’t have a fraction of a layer, we need 9…

Radioactive Decay Vocab: Half-life: The time it takes a given quantity of a radioactive substance to decay to one half of its original mass. NOT dependent on sample size, only the half-life of the substance… f(x) = P(1 - r)x For half-life calculations, r always = .5, so… ex: An archeologist determines that a dinosaur fossil has lost 64% of its Carbon-14 content. Carbon-14 has a half-life of 5730 years. Estimate how old the fossil is.

% of Carbon-14 remaining in a given sample after x years… If P = the original amount, and 64% has been lost, we have 36% remaining… Find x…

The fossil is approx. 8445 years old.

IC 5.2 - pg 343 #’s 3-5 all, 8-10 all, 15-17 all, 20, 24-26. IC 5.2 / 5.3A IC 5.2 - pg 343 #’s 3-5 all, 8-10 all, 15-17 all, 20, 24-26. IC 5.3 - pg 353 #’s 2, 5-7 all, 10, 11, 19, 23, 26, 29, 32, 35.

IC 5.3B - pg 355 #’s 41, 42, 46, 48, 49, 50, 53.