Math 118: Compendium
This document contains all of the definitions, facts, and theorems you need to memorize for weekly assessments.
Quick metric for success: During the quiz, if I were to walk up to you and ask you any definition from this list, you will be able to recall it by heart.
Week 1
1.0: The Beginning
Term, Factor
Terms are entities separated by subtraction and addition.
Factors are entities separated by multiplication.
Terms are entities separated by subtraction and addition.
Factors are entities separated by multiplication.
Global context, local context
Global context refers to the context of the entire expression.
Local context refers to a context smaller than the entire expression.
Global context refers to the context of the entire expression.
Local context refers to a context smaller than the entire expression.
1.1: Real Numbers
Properties of Fractions
Suppose $a,b,c,d$ are real numbers.
- $\dfrac{a}{b}\cdot \dfrac{c}{d} = \dfrac{ac}{bd}$
- $\dfrac{a}{b}\div \dfrac{c}{d} = \dfrac{a}{b}\cdot \dfrac{d}{c}$
- $\dfrac{a}{c} \pm \dfrac{b}{c} = \dfrac{a \pm b}{c}$
- $\dfrac{a}{b} + \dfrac{c}{d}\qquad $ Find the LCD, then use Fraction Law 3.
- $\dfrac{ac}{bc} = \dfrac{a}{b}$
Properties of Real Numbers
Let $a, b, c$ be real numbers.
- Commutative Properties
- $a + b = b + a$
- $ab = ba$
- Associative Properties
- $(a + b) + c = a + (b + c)$
- $(ab)c = a(bc)$
Tip: in English, the commutative/associative properties only interact within the same context level (e.g. only in L3)!
- Distributive Property
- $a(b + c) = ab + ac$
- $(b+c)a = ab + ac$
Tip: in English, the distributive law describes how terms and factors interact.
This is the only way they can interact; anything else is false.
Properties of Negatives
Suppose $a,b$ are real numbers.
- $(-1)a = -a$
- $-(-a) = a$
- $(-a)(-b) = ab$
- $-(a+b) = -a-b$
If $a$ is a real number, then the absolute value of $a$ is \[\lvert a \rvert = \begin{cases} a & a \geq 0 \\ -a & a < 0 \end{cases}\]
1.2: Exponents and Radicals
The $n$th power of $a$ is \[a^n = a\cdot a \cdot \cdots a\]
Zero and negative exponents are defined as
\[a^0 = 1 \qquad \qquad a^{-n} = \dfrac{1}{a^n}\]
Laws of Exponents
- $a^ma^n = a^{m+n}$
- $\dfrac{a^m}{a^n} = a^{m-n}$
- $(a^m)^n = a^{mn}$
- $(ab)^n = a^nb^n$
- $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$
- $\left(\dfrac{a}{b}\right)^{-n} = \dfrac{b^n}{a^n}$
- $\dfrac{a^{-n}}{b^{-m}} = \dfrac{b^m}{a^n}$
The principal nth root of $a$ is defined as \[\sqrt[n]{a} = b \qquad \text{means} \qquad b^n = a\] If $n$ is even, we must have $a \geq 0, b \geq 0$.
For any rational exponent $\dfrac{m}{n}$ in lowest terms, we define \[a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}\]
Week 2
1.3: Expanding and Factoring
Like terms
Like terms are terms with the same factors except for the numerical factor (called the coefficient).
Like terms are terms with the same factors except for the numerical factor (called the coefficient).
- $(A + B)(A - B) = A^2 - B^2$
- $(A + B)^2 = A^2 + 2AB + B^2$
- $(A - B)^2 = A^2 - 2AB + B^2$
Memorize and be able to use the four methods of factoring.
- $A^2 - B^2 = (A-B)(A+B)$
- $A^2 + 2AB + B^2 = (A+B)^2$
- $A^2 - 2AB + B^2 = (A-B)^2$
| Number of Terms | Factoring Methods to Try (in order) |
|---|---|
| 2 terms | GCF, $A^2 - B^2$ |
| 3 terms | GCF, $ax^2 + bx + c, A^2 + 2AB + B^2, A^2 - 2AB + B^2$ |
| 4 terms | GCF, grouping |
| $\geq$ 5 terms | GCF |
Week 3
1.4: Dealing with Rational Expressions
We wrote this down already, but it's worth really memorizing it so you know how to manipulate fractions.
Properties of Fractions
-
$\dfrac{a}{b}\cdot \dfrac{c}{d} = \dfrac{ac}{bd}$
Meaning Multiplying fractions requires multiplying the entire global context of the numerators and the entire global context of the denominators. -
$\dfrac{a}{b}\div \dfrac{c}{d} = \dfrac{a}{b}\cdot \dfrac{d}{c}$
Meaning Dividing fractions requires taking the reciprocal of the right fraction, then multiplying. -
$\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a + b}{c}$
Meaning Adding fractions with the same denominator requires adding the entire global context of the numerators together. - $\dfrac{a}{b} + \dfrac{c}{d}\qquad $ Find the LCD, then use Fraction Law 3.
-
$\dfrac{ac}{bc} = \dfrac{a}{b}$
Meaning Cancelling an entity requires the entity to be a global factor.
1.5: Equations
An equation is a statement where two mathematical expressions are equal.
Properties of Equality
- Given $A = B$, we also know $A + C = B + C$.
- Given $A = B$, we also know $A\cdot C = B\cdot C$ provided $C \neq 0$.
Solving an equation means finding all values of the variables which makes the equation true.
A linear equation is an equation of the form \[ax + b = 0\]
with $a, b \in \mathbb{R}$ and $a \neq 0$.
The zero-product property says \[A\cdot B = 0 \qquad \text{if and only if}\qquad A = 0 \quad \text{or} \quad B = 0\]
The solutions of the equation \[x^2 = c\] are $x = \sqrt{c}$ and $x = - \sqrt{c}$.
The solutions of the quadratic equation $ax^2 + bx + c = 0$ where $a \neq 0$ are \[x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Week 4
1.6: Complex Numbers
complex number
A complex number has the form \[a + bi\] where $a, b$ are real numbers and $i^2 = -1$. Also, $i = \sqrt{-1}$.
$a$ is called the real part, while $b$ is called the imaginary part.
For the quadratic equation $ax^2 + bx + c = 0$, the quantity $b^2 - 4ac$ is called the discriminant.
- If $b^2 - 4ac > 0$, there are two distinct (different) real solutions.
- If $b^2 - 4ac = 0$, there is one real solution.
- If $b^2 - 4ac < 0$, there are two distinct complex solutions.
1.8: Inequalities
Properties of Inequalities
- If $A \leq B$, then $A \pm C \leq B \pm C$.
English Adding the same number on both sides does not flip the inequality. - If $C > 0$, then given $A \leq B$, we know $A\cdot C \leq B\cdot C$.
English Multiplying a positive number on both sides does not flip the inequality. - If $C < 0$, then:
- Given $A \leq B$, we know $A\cdot C \geq B \cdot C$
- Given $A < B$, we know $A\cdot C > B \cdot C$
English Multiplying a negative number on both sides flips the inequality.
1.9: Coordinate Plane, Graphs of Equations
The coordinate plane describes where a point is in two dimensional space.
A graph of an equation in $x$ and $y$ is all points $(x, y)$ which make the equation true.
The $x$-intercept of a graph is the $x$-coordinate where the graph intersects the $x$-axis.
The $y$-intercept of a graph is the $y$-coordinate where the graph intersects the $y$-axis.
The $y$-intercept of a graph is the $y$-coordinate where the graph intersects the $y$-axis.
1.10: Lines
The slope of a line that runs through two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is \[m = \text{slope} = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}\]
The slope of a line that runs through two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is \[m = \text{slope} = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}\]
The point-slope form of a line is an equation of a line that passes through $(x_1, y_1)$ and has slope $m$: \[y - y_1 = m(x - x_1)\]
- The vertical line through $(a, b)$ is $x = a$.
- The horizontal line through $(a, b)$ is $y = b$.
2.1: Functions
A function $f$ is a rule that assigns to each element $x$ in a set $A$ to exactly one element, called $f(x)$ in a set $B$.
domain
The domain of a function is the set of all possible inputs, when evaluated, gives you a real number.
The domain of a function is the set of all possible inputs, when evaluated, gives you a real number.
Week 5
2.2: Graphs of Functions
The graph of a function with domain $A$ is the set of all possible coordinates $(x, f(x))$ which make the equation true.
Vertical Line Test
A curve in the plane is the graph of a function if no vertical intersects the curve more than once.
A curve in the plane is the graph of a function if no vertical intersects the curve more than once.
2.3: Using the Graph
Domain from a Graph
Imagine the points drop straight down by gravity:
Solving $f(x) = g(x)$ and $f(x) < g(x)$
The solutions to the equation $f(x) = g(x)$ are the $x$-values where the graphs intersect.
The solutions to the inequality $f(x) < g(x)$ are the $x$-values where the graph of $f(x)$ is under $g(x)$.
The solutions to the equation $f(x) = g(x)$ are the $x$-values where the graphs intersect.
The solutions to the inequality $f(x) < g(x)$ are the $x$-values where the graph of $f(x)$ is under $g(x)$.
increasing, decreasing function
$f$ is increasing on an interval $I$ if $f(x_1) < f(x_2)$ whenever $x_1 < x_2$ in $I$.
$f$ is decreasing on an interval $I$ if $f(x_1) > f(x_2)$ whenever $x_1 < x_2$ in $I$.
$f$ is increasing on an interval $I$ if $f(x_1) < f(x_2)$ whenever $x_1 < x_2$ in $I$.
$f$ is decreasing on an interval $I$ if $f(x_1) > f(x_2)$ whenever $x_1 < x_2$ in $I$.
local maximum, local minimum
The number $f(a)$ is a local maximum value of $f$ if $f(a) \geq f(x)$ when $x$ is near $a$.
The number $f(a)$ is a local minimum value of $f$ if $f(a) \leq f(x)$ when $x$ is near $a$.
The number $f(a)$ is a local maximum value of $f$ if $f(a) \geq f(x)$ when $x$ is near $a$.
The number $f(a)$ is a local minimum value of $f$ if $f(a) \leq f(x)$ when $x$ is near $a$.
2.4: Average Rate of Change of a Function
average rate of change
The average rate of change (ARoC) of $y = f(x)$ on the interval $(a, b)$ is \[\text{ARoC} = \dfrac{\text{change in } y}{\text{change in } x} = \dfrac{f(b) - f(a)}{b - a}\] The ARoC is the slope of the line through $(a, f(a))$ and $(b, f(b))$.
The average rate of change (ARoC) of $y = f(x)$ on the interval $(a, b)$ is \[\text{ARoC} = \dfrac{\text{change in } y}{\text{change in } x} = \dfrac{f(b) - f(a)}{b - a}\] The ARoC is the slope of the line through $(a, f(a))$ and $(b, f(b))$.
2.6: Transformations of Functions
Parent functions are functions that have different shapes, based on their formula. Memorize the following.
$\large f(x) = x$
$\large f(x) = x^2$
$\large f(x) = \sqrt{x}$
$\large f(x) = \lvert x \rvert$
$\large f(x) = x$
$\large f(x) = x^2$
$\large f(x) = \sqrt{x}$
$\large f(x) = \lvert x \rvert$
$\large f(x) = \sqrt{x}$
$\large f(x) = \lvert x \rvert$
Vertical Shifts of Graphs
To graph $y = f(x) + c$, shift $f(x)$ upwards $c$ units.
To graph $y = f(x) - c$, shift $f(x)$ downwards $c$ units.
Vertical Shifts of Graphs
To graph $y = f(x) + c$, shift $f(x)$ upwards $c$ units.
To graph $y = f(x) - c$, shift $f(x)$ downwards $c$ units.
To graph $y = f(x) - c$, shift $f(x)$ downwards $c$ units.
Horizontal Shifts of Graphs
To graph $y = f(x - c)$, shift $f(x)$ right $c$ units.
To graph $y = f(x + c)$, shift $f(x)$ left $c$ units.
Horizontal Shifts of Graphs
To graph $y = f(x - c)$, shift $f(x)$ right $c$ units.
To graph $y = f(x + c)$, shift $f(x)$ left $c$ units.
To graph $y = f(x + c)$, shift $f(x)$ left $c$ units.
Reflecting Graphs
To graph $y = -f(x)$, reflect $f(x)$ around the $x$-axis.
To graph $y = f(-x)$, reflect $f(x)$ around the $y$-axis.
Reflecting Graphs
To graph $y = -f(x)$, reflect $f(x)$ around the $x$-axis.
To graph $y = f(-x)$, reflect $f(x)$ around the $y$-axis.
To graph $y = f(-x)$, reflect $f(x)$ around the $y$-axis.
Vertical Stretching and Shrinking
To graph $y = c\cdot f(x)$:
To graph $y = c\cdot f(x)$:
If $c > 1$, stretch $f(x)$ vertically by a factor of $c$.
If $0 < c < 1$, shrink $f(x)$ vertically by a factor of $c$.
Vertical Stretching and Shrinking
To graph $y = c\cdot f(x)$:
To graph $y = c\cdot f(x)$:
If $c > 1$, stretch $f(x)$ vertically by a factor of $c$.
If $0 < c < 1$, shrink $f(x)$ vertically by a factor of $c$.
If $0 < c < 1$, shrink $f(x)$ vertically by a factor of $c$.
Horizontal Stretching and Shrinking
To graph $y = f(c\cdot x)$:
To graph $y = f(c\cdot x)$:
If $c > 1$, shrink $f(x)$ horizontally by a factor of $\frac{1}{c}$.
If $0 < c < 1$, stretch $f(x)$ horizontally by a factor of $\frac{1}{c}$.
Horizontal Stretching and Shrinking
To graph $y = f(c\cdot x)$:
To graph $y = f(c\cdot x)$:
If $c > 1$, shrink $f(x)$ horizontally by a factor of $\frac{1}{c}$.
If $0 < c < 1$, stretch $f(x)$ horizontally by a factor of $\frac{1}{c}$.
If $0 < c < 1$, stretch $f(x)$ horizontally by a factor of $\frac{1}{c}$.
If there is a horizontal shift + another horizontal transformation, make sure the coefficient of $x$ is $1$ before identifying transformations.
Week 6
2.7: New Functions From Old
$+, -, \times, \div$ functions together is simple: just substitute. But...
When you are subtracting and multiplying functions with two or more terms, do not forget parenthesis.
composite function
Given two functions $f$ and $g$, the composite function $f \circ g$ is defined by \[(f\circ g)(x) = f(g(x))\]
Given two functions $f$ and $g$, the composite function $f \circ g$ is defined by \[(f\circ g)(x) = f(g(x))\]
2.8: One-to-One Functions and Inverses
A function $f(x)$ with domain $A$ is called one-to-one if no two elements of $A$ are sent to the same value.
In other words, if $x_1\neq x_2$, then it is always the case that $f(x_1) \neq f(x_2)$.
In other words, if $x_1\neq x_2$, then it is always the case that $f(x_1) \neq f(x_2)$.
Horizontal Line Test
A function is one-to-one if and only if no horizontal line intersects the graph of $f(x)$ more than once.
A function is one-to-one if and only if no horizontal line intersects the graph of $f(x)$ more than once.
codomain
The codomain of a function $f$ is the set of all possible outputs when $f$ is evaluated on the domain.
The codomain of a function $f$ is the set of all possible outputs when $f$ is evaluated on the domain.
inverse function, $f^{-1}$
Let $f$ be a one-to-one function with domain $A$, codomain $B$. Then the inverse function, denoted $f^{-1}$, has domain $B$ and codomain $A$ and is defined by \[f^{-1}(y) = x \text{ if and only if } f(x) = y\]
Let $f$ be a one-to-one function with domain $A$, codomain $B$. Then the inverse function, denoted $f^{-1}$, has domain $B$ and codomain $A$ and is defined by \[f^{-1}(y) = x \text{ if and only if } f(x) = y\]
$f^{-1}$ is not equivalent to $\frac{1}{f}$. Don't confuse it with a negative exponent!
Inverse Function Property
Let $f$ be a one-to-one function with domain $A$ and codomain $B$. The inverse function $f^{-1}$ satisfies the following properties \begin{align} f^{-1}(f(x))=x \qquad &\text{ for every } x \text{ in } A\\ f(f^{-1}(x))=x \qquad &\text{ for every } x \text{ in } B \end{align}
Let $f$ be a one-to-one function with domain $A$ and codomain $B$. The inverse function $f^{-1}$ satisfies the following properties \begin{align} f^{-1}(f(x))=x \qquad &\text{ for every } x \text{ in } A\\ f(f^{-1}(x))=x \qquad &\text{ for every } x \text{ in } B \end{align}
Given a function $f$, here are the steps to find the inverse $f^{-1}$:
- Check $f$ is an one-to-one function. If so, proceed. If not, $f$ does not have an inverse.
- Write $y = f(x)$. If there is a domain restriction, convert the domain restriction into a codomain restriction.
- Solve this equation for $x$ in terms of $y$. Meaning, isolate $x$.
- Interchange $x$ and $y$, including the codomain restriction into a domain restriction if there was one. The resulting equation is $y = f^{-1}(x)$.
3.1: Quadratic Functions
quadratic function
A quadratic function is a polynomial function of degree 2, with form \[f(x) = ax^2 + bx + c \qquad a\neq 0\]
A quadratic function is a polynomial function of degree 2, with form \[f(x) = ax^2 + bx + c \qquad a\neq 0\]
To complete the square for $x^2 + bx$, add and subtract $\left(\dfrac{b}{2}\right)^2$.
standard form, vertex
A quadratic function $f(x) = ax^2 + bx + c$ can be expressed in the standard form \[f(x) = a(x - h)^2 + k\] by completing the square.
The vertex of the parabola is $(h, k)$.
A quadratic function $f(x) = ax^2 + bx + c$ can be expressed in the standard form \[f(x) = a(x - h)^2 + k\] by completing the square.
The vertex of the parabola is $(h, k)$.
Week 7
3.2: Polynomial Functions and Their Graphs
polynomial function
A polynomial function of degree $n$ is a function of the form \[P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \qquad a_n \neq 0\] where $n$ is an integer.
A polynomial function of degree $n$ is a function of the form \[P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \qquad a_n \neq 0\] where $n$ is an integer.
- $a_n, a_{n-1}, \dots, a_1, a_0$ are called the coefficients.
- $a_0$ is called the constant coefficient or constant term.
- $a_n$ is called the leading coefficient.
- $a_nx^n$ is called the leading term.
Monomials $P(x) = x^n$ have the following graphs:
$n = 1$
$n$ is even
$n$ is odd
$n = 1$
$n$ is even
$n$ is odd
$n$ is even
$n$ is odd
Informal Definition: continuity
A graph is continuous if you can draw it without lifting your pencil.
A graph is continuous if you can draw it without lifting your pencil.
end behavior
The end behavior of a polynomial depends on the degree $n$ and the leading coefficient $a_n$:
zeros
Suppose $P$ is a polynomial and $c$ is a real number. Then the following are equivalent:
Suppose $P$ is a polynomial and $c$ is a real number. Then the following are equivalent:
- $c$ is a zero of $P$.
- $c$ is a $x$-intercept of the graph of $P$.
- $x = c$ is a solution of the equation $P(x) = 0$. Meaning, $P(c) = 0$.
- $(x - c)$ is a factor of $P(x)$.
zero of multiplicity $m$
$c$ is a zero of multiplicity $m$ if $(x - c)^m$ appears in the factorization of $P$.
$c$ is a zero of multiplicity $m$ if $(x - c)^m$ appears in the factorization of $P$.
Graph Shape Near Zeros of Multiplicity $m$
If $c$ is a zero of multiplicity $m$, then the graph shape near $c$ corresponds is as follows:
If $c$ is a zero of multiplicity $m$, then the graph shape near $c$ corresponds is as follows:
$n$ odd, $n \neq 1$
or
$n$ even
or
or
$n$ even
or
$n$ odd,
$n \neq 1$
or
$n$ even
or
$n \neq 1$
or
$n$ even
or
Week 8
3.3: Real Zeros of Polynomials
division algorithm
$P(x)$ and $D(x)$ are polynomials, with $D(x) \neq 0$. Then there exist unique polynomials $Q(x)$ and $R(x)$, where the degree of $R(x)$ is 0 or less than the degree of $P(x)$ where: \[\dfrac{P(x)}{D(x)} = Q(x) + \dfrac{R(x)}{D(x)} \qquad \text{ or } \qquad P(x) = D(x)\cdot Q(x) + R(x)\]
$P(x)$ and $D(x)$ are polynomials, with $D(x) \neq 0$. Then there exist unique polynomials $Q(x)$ and $R(x)$, where the degree of $R(x)$ is 0 or less than the degree of $P(x)$ where: \[\dfrac{P(x)}{D(x)} = Q(x) + \dfrac{R(x)}{D(x)} \qquad \text{ or } \qquad P(x) = D(x)\cdot Q(x) + R(x)\]
- $P(x)$ is the dividend (what you are dividing)
- $D(x)$ is the divisor (what you are dividing by)
- $Q(x)$ is the quotient
- $R(x)$ is the remainder.
Remainder Theorem
If $P(x)$ is divided by $x - c$, then the remainder is $P(c)$.
If $P(x)$ is divided by $x - c$, then the remainder is $P(c)$.
Factor Theorem
$c$ is a zero of $P$ if and only if $(x-c)$ is a factor of $P(x)$.
$c$ is a zero of $P$ if and only if $(x-c)$ is a factor of $P(x)$.
Complete Factorization of $P(x)$ over $\mathbb{R}$
A complete factorization of a polynomial $P(x)$ over $\mathbb{R}$ is one where the resultant factors only have real coefficients.
A complete factorization of a polynomial $P(x)$ over $\mathbb{R}$ is one where the resultant factors only have real coefficients.
Complete Factorization of $P(x)$ over $\mathbb{R}$
For each zero $c$:
Do not factor irreducibles, leave them alone.
For each zero $c$:
- Setup:Convert into a factor $(x - c)$.
- Divide: Use the division algorithm to divide $P(x) = (x - c)Q(x)$.
- Multiplicity: Check if $c$ is a zero of $Q(x)$. If so, repeat and keep dividing $Q(x)$ until $c$ is no longer a zero. Move on to the next zero.
Do not factor irreducibles, leave them alone.
irreducible
An irreducible polynomial is a quadratic polynomial with no real zeros.
An irreducible polynomial is a quadratic polynomial with no real zeros.
An irreducible's discriminant $b^2 - 4ac < 0$ due to the quadratic formula \[x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
The $\sqrt{b^2 - 4ac}$ is a root of a negative number, so the zeros can't be real!
Real Factorization Theorem
Suppose $P(x)$ is a polynomial with real coefficients. A complete factorization of $P(x)$ over $\mathbb{R}$ will break down into linear (degree 1) factors and irreducible quadratics.
There are three possibilities: \begin{align} P(x) &= (\text{linear factors}) \\P(x) &= (\text{linear factors}) \cdot (\text{irreducible factors}) \\P(x) &= (\text{irreducible factors}) \end{align}3.5: Complex Zeros
Fundamental Theorem of Algebra
Every polynomial \[P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \qquad a_n \neq 0\] with complex coefficients has at least one complex zero.
Every polynomial \[P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \qquad a_n \neq 0\] with complex coefficients has at least one complex zero.
Complex Factorization Theorem
Suppose $P(x)$ is a degree $n$ polynomial with complex coefficients. Then:
Suppose $P(x)$ is a degree $n$ polynomial with complex coefficients. Then:
- there exists complex numbers $a, c_1, c_2, \dots, c_n$ where \[P(x) = a(x-c_1)(x-c_2)\cdots(x-c_{n-1})(x-c_{n})\] and
- $P(x)$ has exactly $n$ zeros, provided a zero of multiplicity $k$ is counted $k$ times.
To find a complete factorization of $P(x)$ over $\mathbb{C}$, take the complete factorization over $\mathbb{R}$ and factor the irreducibles into linear factors.
complex conjugate
The complex conjugate of $a + bi$ is $a - bi$.
The complex conjugate of $a + bi$ is $a - bi$.
Conjugate Pairs Theorem
If $P(x)$ has real coefficients and $a + bi$ is a zero, then $a - bi$ must also be a zero.
If $P(x)$ has real coefficients and $a + bi$ is a zero, then $a - bi$ must also be a zero.
Week 9
3.6: Rational Functions
rational function
A rational function has the form \[r(x) = \dfrac{P(x)}{Q(x)}\]where $P$ and $Q$ are polynomials.
A rational function has the form \[r(x) = \dfrac{P(x)}{Q(x)}\]where $P$ and $Q$ are polynomials.
There are two ways to approach a number $a$: from the left and from the right.
Notation
Visualization
$x\rightarrow a^-$
(from the left)
$x\rightarrow a^+$
(from the right)
(from the left)
$x\rightarrow a^+$(from the right)
Notation
Visualization
$x\rightarrow a^-$
(from the left)
$x\rightarrow a^+$
(from the right)
(from the left)
$x\rightarrow a^+$(from the right)
The line $x = a$ is a vertical asymptote if $y \rightarrow \pm \infty$ as $x \rightarrow a^+$ or $x \rightarrow a^-$.
The line $y = b$ is a horizontal asymptote if $y \rightarrow b$ as $x \rightarrow \pm \infty$.
The line $y = b$ is a horizontal asymptote if $y \rightarrow b$ as $x \rightarrow \pm \infty$.
Suppose you have a rational function \[r(x) = \dfrac{a_nx^n + \cdots}{b_mx^m + \cdots}\]
- To find vertical asymptotes, set denominator $= 0$ and solve for $x$.
- To find horizontal asymptotes, there are three cases, depending on the leading terms:
- If $n < m$, then $y = 0$ is the horizontal asymptote.
- If $n = m$, then $y = \dfrac{a_n}{b_m}$.
- If $n > m$, there are no horizontal asymptotes.
4.1: Exponential Functions
exponential function
The exponential function with base $a$ has the form \[f(x) = a^x\] where $a > 0$ and $a \neq 1$.
The exponential function with base $a$ has the form \[f(x) = a^x\] where $a > 0$ and $a \neq 1$.
The exponential function $f(x) = a^x$ has two possible graph shapes:
In general:
$a > 1$
$0 < a < 1$
$a > 1$
$0 < a < 1$
- Domain: $\mathbb{R}$
- Codomain (range): $(0, \infty)$
- Asymptotes: $y = 0$
simple interest
Interest applied only once against the principal.
Interest applied only once against the principal.
principal
The money invested/borrowed initially.
The money invested/borrowed initially.
compound interest
Compound interest is calculated by \[A(t) = P\left(1 + \dfrac{r}{n}\right)^{nt}\] where
Compound interest is calculated by \[A(t) = P\left(1 + \dfrac{r}{n}\right)^{nt}\] where
- $A(t) = $ amount after $t$ years
- $P = $ principal
- $r = $ interest rate per year
- $n = $ number of times interest is compounded per year
- $t = $ number of years.
Annual Percentage Yield (APY)
The simple interest rate in one year which accounts for compounding.
The simple interest rate in one year which accounts for compounding.
4.2: The Natural Exponential Function
e
The number $e$ is defined to be \[\left(1 + \dfrac{1}{n}\right)^n \rightarrow e \quad \text{as} \quad n \rightarrow \infty\]
The number $e$ is defined to be \[\left(1 + \dfrac{1}{n}\right)^n \rightarrow e \quad \text{as} \quad n \rightarrow \infty\]
natural exponential function
Exponential function with base $e$: \[f(x) = e^x\]
Exponential function with base $e$: \[f(x) = e^x\]
continuously compounded interest
Continuously compounded interest is calculated by \[A(t) = Pe^{rt}\] where
Continuously compounded interest is calculated by \[A(t) = Pe^{rt}\] where
- $A(t) = $ amount after $t$ years
- $P = $ principal
- $r = $ interest rate per year
- $t = $ number of years.
logarithmic function
Suppose $a > 0, a \neq 1$. The logarithmic function with base $a$ is defined by \[\log_a x = y \qquad \text{means} \qquad a^y = x\]
Suppose $a > 0, a \neq 1$. The logarithmic function with base $a$ is defined by \[\log_a x = y \qquad \text{means} \qquad a^y = x\]
4.3: Logarithmic Functions
Properties of Logarithms
- $\log_a 1 = 0$
- $\log_a a = 1$
- $\log_a a^x = x$
- $a^{\log_a x} = x$
common logarithm
Base 10 logarithm, the base is omitted: \[\log x = \log_{10} x\]
Base 10 logarithm, the base is omitted: \[\log x = \log_{10} x\]
natural logarithm
Base $e$ logarithm, denoted by $\ln$: \[\ln x = \log_{e} x\]
Base $e$ logarithm, denoted by $\ln$: \[\ln x = \log_{e} x\]
Properties of Natural Logarithms
- $\ln 1 = 0$
- $\ln e = 1$
- $\ln e^x = x$
- $e^{\ln x} = x$
The function $f(x) = \log_a x$ has the general shape
In general:
In general:
- Domain: $(0, \infty)$
- Codomain (range): $\mathbb{R}$
- Asymptotes: $x = 0$
Week 10
4.4: Laws of Logarithms
Laws of Logarithms
Let $a > 0, a \neq 1$ and $A,B,C \in \mathbb{R}$ where $A > 0$ and $B > 0$.
Let $a > 0, a \neq 1$ and $A,B,C \in \mathbb{R}$ where $A > 0$ and $B > 0$.
- $\log_a(AB) = \log_a A + \log_a B$
- $\log_a\left(\dfrac{A}{B}\right) = \log_a A - \log_a B$
- $\log_a(A^C) = C\cdot\log_a A$
4.5: Exponential and Logarithmic Equations
Solving Exponential Equations
- Isolate the exponential expression.
- If there are two exponential expressions, put one on each side.
- Take the logarithm of each side. Bring down the exponent with Laws of Logarithms.
- Solve for the variable.
Solving Logarithmic Equations
- Isolate the logarithmic expression (combine with Laws of Logarithms if necessary).
- If there are two logarithmic expressions, put one on each side.
- Write the logarithm in exponential form, or exponentiate both sides with the base.
- Solve for the variable.