4.1 Polynomial Functions and their Graphs
•
End behavior:
◦
Look at leading coefficient/exponent and check sign
I◦ f polynomial is factored, check sign of each factor and multiply
•
Graphing a polynomial :
◦
Factor
◦
Find x- and y-intercepts
◦
Find end behavior
◦
Either use test points between the intercepts or memorize the shape around zeros
depending on the multiplicity:
― If multiplicity is 1, then it crosses the x-axis in a straight line
― If multiplicity is even, then it turns back around
― If multiplicity is odd > 1, then it "squiggles" through the x-axis
4.2 Dividing Polynomials
•
Long Division: Make sure to fill in missing powers. Write answers in the form
•
Synthetic Division : Only works for division by (x - c). Again make sure to fill
in 0's
for missing powers Will not be covered, but you may use it if you remember it.
•
Factor Theorem: c is a zero of P ↔ (x - c) is a factor of P(x)
4.3 No longer covered in Math 22
4.4 No longer covered in Math 22
4.5 Rational Functions
•
Horizontal asymptotes: n is the degree of the numerator, m is the degree of the
de-
nominator
◦
n > m: no horizontal asymptote
◦ n = m: horizontal asymptote is
◦
n < m: horizontal asymptote is y = 0
•
Vertical asymptotes: zeros of the denominator (that do not cancel with the
numerator)
•
Graphing rational functions:
◦
Factor numerator and denominator
◦
Find x- and y-intercepts
◦
Find horizontal and vertical asymptotes
◦
Either use test points between intercepts/vertical asymptotes or use the shape
around vertical asymptotes/intercepts to determine the shape of the graph
•
Slant asymptote: only exists if the degree of the numerator is one greater than
the
degree of the denominator: use long/synthetic division
5.1 Exponential Functions
•
f(x) = ax, memorize the graph:
◦
Horizontal asymptote y = 0
◦
no vertical asymptote
◦
Domain = (-∞,∞)
◦
Range = (0,∞)
•
Compound interest formula: 
•
Continuously compounded interest: 
5.2 Logarithmic Functions
•
Definition of logarithm: 
•
Properties:
• 
memorize the graph:
◦
Vertical asymptote: x = 0
◦
no horizontal asymptote
◦ Domain = (0,∞)
◦
Range = (-∞,∞)
•
Finding the domain of logarithmic function: logarithms only defined for positive
num -
bers
• Common log: 
•
Natural log: 
5.3 Laws of Logarithms
• 
• 
• 
•
no laws for 
•
Change of base :
where c can be any positive number
5.4 Exponential and Logarithmic Equations
• Solving exponential equations:
◦
Isolate the exponential term on one side
◦
Take logarithm of both sides:
― If there is only one exponential term, use that base for the log
― If there is an exponential term on both sides, use either the common or
natural
log
◦
Pull the exponent to the front and solve the equation
•
Solving logarithmic equations:
◦
If there are multiple logarithmic terms, combine them into one using logarithmic
laws
◦
Isolate the logarithmic term on one side
◦
Raise the base of the logarithm to the left and the right side of the equation
◦
Use the property 
to get rid of the log
◦
Solve the equation
•
Two special cases of exponential equations:
◦
Combination of exponential and polynomial terms: try to factor
◦
Sum of multiply exponential terms: try to use substitution
5.5 Modeling with Exponential and Logarithmic Functions
•
Exponential growth model: 
•
To solve any problem you usually have to find n0 and r
•
Formulas and logarithmic scales
9.1 Sequences and Summation Notation
•
A few commonly used terms:
◦ 
for sequences alternating in sign
◦
2n for even numbers
◦
2n - 1 for odd numbers
•
Recursively defined sequences, Fibonacci Numbers
•
Partial sums
•
Sigma notation:
•
Properties:
9.2 Arithmetic Sequences
• Arithmetic sequence: a, a + d, a + 2d,…
• 
•
Gauss:
•
Partial sums of an arithmetic sequence: 
9.3 Geometric Sequences
•
Geometric sequence: a, ar, ar2,…
• 
•
Partial sums of an arithmetic sequence: 
•
Sum of an infinite geometric series:
