# Math 41 Study Guide

**4.1 Polynomial Functions and their Graphs
**

• End behavior:

o Look at leading coefficient /exponent and check sign

o If polynomial is factored, check sign of each factor and multiply

• Graphing a polynomial:

o Factor

o Find x- and y-intercepts

o Find end behavior

o 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

• Synthetic Division: Only works for division by (x − c). Again make sure to fill in 0’s

for missing powers

• Remainder Theorem: to find P(c) carry out a synthetic division for c, the remainder

is P(c)

• Factor Theorem : c is a zero of P (x − c) is a factor of P(x)

**4.3 Real Zeros of Polynomials
**

• Rational Zeros Theorem: The possible rational zeros of a polynomial are of the form

where p is a factor of the constant coefficient and q is a factor of the leading

coefficient

• How to find all zeros of a polynomial:

o Try previous factoring methods like substitution or grouping , if this does not work

then:

o List all possible rational zeros using the Rational Zeros Theorem

o Test the possible zeros

o If you find a zero, factor it out

o Repeat from the top until your polynomial is quadratic, then factor/complete the

square/ quadratic formula

**4.4 Complex Zeros and the Fundamental Theorem of Algebra
**

• Fundamental Theorem of Algebra: every polynomial of degree n has precisely n zeros

(zeros of multiplicity k are counted k times )

• Conjugate Zeros Theorem: If a complex number is a zero of polynomial with real

coefficient, then its conjugate is also a zero

**4.5 Rational Functions
**

• Horizontal asymptotes: n is the degree of the numerator, m is the degree of the denominator

o n > m: no horizontal asymptote

o n = m: horizontal asymptote is

o n < m: horizontal asymptote is y = 0

• Vertical asymptotes: zeros of the denominator (that do not cancel with the numerator)

• Graphing rational functions:

o Factor numerator and denominator

o Find x- and y-intercepts

o Find horizontal and vertical asymptotes

o 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) = a

^{x}, memorize the graph:

o Horizontal asymptote y = 0

o no vertical asymptote

o Domain = (−∞,∞)

o Range = (0,∞)

• Compound interest formula:

• Continuously compounded interest: A(t) = e

^{rt}

**5.2 Logarithmic Functions
**

• Definition of logarithm:

• Properties:

• , memorize the graph:

o Vertical asymptote: x = 0

o no horizontal asymptote

o Domain = (0,∞)

o Range = (−∞,∞)

• Finding the domain of logarithmic function: logarithms only defined for positive numbers

• 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:

o Isolate the exponential term on one side

o 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

o Pull the exponent to the front and solve the equation

• Solving logarithmic equations:

o If there are multiple logarithmic terms, combine them into one using logarithmic

laws

o Isolate the logarithmic term on one side

o Raise the base of the logarithm to the left and the right side of the equation

o Use the property to get rid of the log

o Solve the equation

• Two special cases of exponential equations:

o Combination of exponential and polynomial terms: try to factor

o Sum of multiple 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 and r

• Formulas and logarithmic scales

**6.1 Angle Measure
**

• Relationship between Degrees and Radians:

o convert from degrees to radians by multiplying by

o convert from radians to degrees by multiplying by

• Coterminal angles: Angle between 0° and 360° degrees (or 0 and 2π )

• Length of a circular arc: s = rθ ( θ in rad)

• Area of a circular sector: ( θ in rad)

• Linear Speed and Angular Speed: and

• Relationship between linear and angular speed: v = rω

**6.2 Trigonometry of Right Triangles
**

• Trigonometric Ratios:

• Values of the trig ratios for angles 30° , 45° and 60°

• Solving right triangles

Prev | Next |