1. 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)

2. 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

3. 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 = ….. 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…

4. Compound Interest Formulas
ex: 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)(3)
.085 1
= $ 11,495.60
ex: 9%, compounded quarterly for 5 years….
A = ( )(4)(5)
.09 4 r n
A = P( )nt
= $ 18,726.11

5. Formula for Continuous Compounding
A = Pert
ex: 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

6. 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 = ( ) =
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 = >f(x) = 1000ax
---> f(5) = 1000a5
---> 7600 = 1000a5
---> 7.6 = a5

7. 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

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

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.

10. % 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…

11. The fossil is approx. 8445 years old.

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

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