1. Unit 3: Exponential and Logarithmic Functions
Section 3.1
Graphs of Exponential Functions including transformations and base ‘e’
Notes:
Exponential Function- function containing a variable exponent
f(x) = 𝑏 𝑥 where b is a positive constant other than 1
(b > 0 and b ≠ 1) think about why b ≠ 1
Using a half sheet of graph paper, sketch the graph of f(x) = 2 𝑥
Then find the following:
lim 𝑥→−∞ 2 𝑥
lim 𝑥→∞ 2 𝑥 =0
= ∞

2. Sketch the graphs of the following functions on the same set of axes as you graphed f(x) = 2 𝑥 .
g(x) = 3 𝑥 h(x) = 4 𝑥 j(x) = 𝑥 k(x) = 𝑥 l(x) = 𝑥
Then come up with as many properties of exponential functions that you can (there are a total of 7)
Domain is (-∞, ∞) Range is (0, ∞)
Horizontal asymptote at y = 0
y-intercept = 1
If b > 1, graph is increasing
If 0 < b < 1, graph is decreasing
When b > 1, as b increases, the graph gets steeper
When 0 < b < 1, as b decreases, the graph gets steeper
It is a 1:1 function

3. Using the transformation rules we learned in Chapter 1, describe the transformations that are occurring in the examples below. Be sure to list them in the correct order.
f(x) = −2 (−𝑥+4)
g(x) = 3· 2 (𝑥−3) − 6
h(x) = 1 2 ( −2 (𝑥+1) ) + 5
left 4, reflect over the x and y axis
right 3, vertically stretched by a factor of 3, down 6
left 1, vertically shrunk by a factor of ½, reflect over the x-axis, up 5

4. Natural Base ‘e’
The natural base ‘e’ is an irrational number symbolized by the letter e.
It was discovered by a Swiss mathematician Leonard Euler who proved that:
lim 𝑛→∞ 𝑛 𝑛 e
where e ≈ …
e is called the natural base
What does the graph of f(x) = 𝑒 𝑥 look like? Sketch the graph on the same grid as all the others.

5. Example
In 1969, the world population was 3.6 billion, with a growth rate of 2% per year. The function
f(x) = 3.6 𝑒 0.02𝑥
describes world population, f(x), in billions, x years after Find the world population for 2020.
In 2000, the world population was approximately 6 billion but the growth rate had slowed to 1.3%. Find the world population in 2050.
f(x) = 6 𝑒 0.013𝑥 𝑤ℎ𝑒𝑟𝑒 𝑥=# 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠 𝑠𝑖𝑛𝑐𝑒 2000
9.98 billion
11.49 billion

6. Compound Interest
Interest computed on original investment as well as any already accumulated interest.
𝐴= 𝑃(1+ 𝑟 𝑛 ) 𝑛𝑡
Where:
A = final balance
P = principal (original investment)
r = annual percentage rate in decimal form
n = number of compounding periods per year
t = number of years
Be sure to know the following:
Annually: n = 1
Semi-annually: n = 2
Quarterly: n = 4

7. Some banks use continuous compounding where the number of compounding periods increases infinitely.
as n →∞, (1+ 1 𝑛 ) 𝑛 →𝑒
Therefore:
A = Pert is the formula for continuous compounding
Example:
Suppose $10000 is invested at an annual rate of 8%. Find the balance in the account after 5 years when interest is compounded:
a) Quarterly b) continuously
𝐴= 10000( ) 4·5
A = 10000e.08·5
A = $14,918.25
A = $14,859.47

8. Homework:
Pg /19-24 all, 29, 39, 51
Day 2:
Pg. 397/ 53,55,57,61,63,67,73