Comprehensive Study Notes: College Algebra (MATH1034A Lecture Manual)

Numbers, Inequalities, and Absolute Values

Sets

Sets are fundamental objects in mathematics, representing collections of distinct elements. Understanding set notation and operations is essential for algebraic reasoning.

• Set: A well-defined collection of objects, called elements.

• Notation: Curly brackets list elements, e.g., {1, 2, 3}.

• Set-builder notation: {x | A(x)} means the set of all x for which A(x) is true.

• Subset: A ⊂ B if every element of A is also in B.

• Intersection: A ∩ B = {x | x ∈ A and x ∈ B}.

• Union: A ∪ B = {x | x ∈ A or x ∈ B}.

• Empty set: ∅, the set with no elements.

Example: The set of all natural numbers between 18 and 956: {x ∈ N | 18 < x < 956}

Real Numbers

The real number system is built from several subsets:

• Natural numbers (N): {1, 2, 3, ...}

• Integers (Z): {..., -3, -2, -1, 0, 1, 2, 3, ...}

• Rational numbers (Q): Numbers expressible as $\frac{p}{q}$, where $p, q \in Z$, $q \neq 0$.

• Irrational numbers: Real numbers not rational, e.g., $\sqrt{2}$, $\pi$.

• Real numbers (R): All points on the number line, including both rationals and irrationals.

Ordering: For $a, b \in R$, $b > a$ if $b$ is to the right of $a$ on the number line.

• $a \leq b$ means $a < b$ or $a = b$.

• $a \geq b$ means $a > b$ or $a = b$.

Intervals

Intervals are subsets of the real line, often used to describe solution sets.

Notation	Description
$(a, b)$	Open interval: $\{x \in R | a < x < b\}$
$[a, b]$	Closed interval: $\{x \in R | a \leq x \leq b\}$
$(a, b]$	Half-open: $\{x \in R | a < x \leq b\}$
$[a, b)$	Half-open: $\{x \in R | a \leq x < b\}$
$(a, \infty)$	All $x > a$
$(-\infty, b]$	All $x \leq b$

Inequalities

Solving inequalities involves finding all real numbers that satisfy a given relation.

• Example: Solve $3x + 1 > 2x$:

  • $3x + 1 > 2x \implies x > -1$

  • Solution: $(-1, \infty)$

• Compound inequalities: $2x < 3x + 1 \leq 5x - 2$

  • Break into two: $3x + 1 > 2x$ and $3x + 1 \leq 5x - 2$

  • Find intersection of solution sets.

Absolute Value

The absolute value of $a$ is its distance from zero on the real number line.

• Definition: $|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}$

• Key properties:

  • $|a| \geq 0$ for all $a$

  • $|a|^2 = a^2$

  • $|a| = \sqrt{a^2}$

  • $|ab| = |a||b|$

  • $|a + b| \leq |a| + |b|$ (triangle inequality)

  • $|a| < b \iff -b < a < b$ (for $b \geq 0$)

  • $|a| > b \iff a < -b$ or $a > b$ (for $b \geq 0$)

Example: Solve $|2x + 4| < 1$:

• $-1 < 2x + 4 < 1 \implies -5 < 2x < -3 \implies -\frac{5}{2} < x < -\frac{3}{2}$

Functions

Definition and Basics

A function $f$ from set $D$ to set $Y$ assigns to each $x \in D$ a unique $f(x) \in Y$.

• Domain: The set $D$ of all possible inputs.

• Range: The set $\{f(x) | x \in D\}$ of all possible outputs.

• Real-valued function: $D \subset R$, $Y = R$.

Example: $f(x) = \frac{1}{\sqrt{x-1}}$ has domain $(1, \infty)$ (since $x-1 > 0$ and denominator $\neq 0$).

Graphing Functions

• Graph: The set of points $(x, f(x))$ for $x \in D$.

• Vertical Line Test: A curve is the graph of a function if no vertical line intersects it more than once.

• Piecewise functions: Defined by different formulas on different parts of the domain.

Transformations of Graphs

• Shifting: $y = f(x + c)$ (horizontal), $y = f(x) + c$ (vertical)

• Scaling: $y = f(cx)$ (horizontal), $y = cf(x)$ (vertical)

• Reflection: $y = f(-x)$ (about $y$-axis), $y = -f(x)$ (about $x$-axis)

Even and Odd Functions

• Even: $f(-x) = f(x)$ for all $x$ in domain. Graph is symmetric about $y$-axis.

• Odd: $f(-x) = -f(x)$ for all $x$ in domain. Graph is symmetric about the origin.

• Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

Classification and Combination of Functions

• Polynomial: $p(x) = a_nx^n + ... + a_0$

• Rational: $g(x) = \frac{p(x)}{q(x)}$, $q(x) \neq 0$

• Algebraic: Built from polynomials using roots and rational operations.

• Operations: $(f+g)(x) = f(x) + g(x)$, $(fg)(x) = f(x)g(x)$, $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ (where $g(x) \neq 0$)

• Composite: $(f \circ g)(x) = f(g(x))$

Inverse Functions

• One-to-one (injective): $f(x_1) = f(x_2) \implies x_1 = x_2$

• Inverse: If $f$ is one-to-one, $f^{-1}(y) = x$ where $y = f(x)$

• Horizontal Line Test: A function is one-to-one if every horizontal line intersects its graph at most once.

• Graph of inverse: Reflection of $f$'s graph across the line $y = x$.

Angles and Trigonometric Functions

Radian Measure

• Definition: The radian measure of an angle is the ratio $\frac{s}{r}$, where $s$ is arc length and $r$ is radius.

• Arc length: $s = r\alpha$ (with $\alpha$ in radians)

• Conversion: $1$radian$= \frac{180}{\pi}$degrees; $1$ degree $= \frac{\pi}{180}$ radians

Area of a Sector

• Formula: $A = \frac{1}{2} r^2 \alpha$ (with $\alpha$ in radians)

Trigonometric Functions

• $\sin \theta = \frac{y}{r}$

• $\cos \theta = \frac{x}{r}$

• $\tan \theta = \frac{y}{x}$

• $\csc \theta = \frac{r}{y}$

• $\sec \theta = \frac{r}{x}$

• $\cot \theta = \frac{x}{y}$

Special Angles Table

$\theta$	$0$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
Radians	$0$	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$
$\sin \theta$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos \theta$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$
$\tan \theta$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	undefined

Trigonometric Identities

• $\sin^2 \theta + \cos^2 \theta = 1$

• $1 + \tan^2 \theta = \sec^2 \theta$

• $\sin(\theta \pm \phi) = \sin \theta \cos \phi \pm \cos \theta \sin \phi$

• $\cos(\theta \pm \phi) = \cos \theta \cos \phi \mp \sin \theta \sin \phi$

• $\sin(2\theta) = 2 \sin \theta \cos \theta$

• $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$

Inverse Trigonometric Functions

• arcsin: $y = \arcsin x$ is the unique $y \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ with $x = \sin y$

• arccos: $y = \arccos x$ is the unique $y \in [0, \pi]$ with $x = \cos y$

• arctan: $y = \arctan x$ is the unique $y \in (-\frac{\pi}{2}, \frac{\pi}{2})$ with $x = \tan y$

Trigonometric Equations

• General solution for $\sin \theta = x$ ($|x| \leq 1$): $\theta = \arcsin x + 2k\pi$ or $\theta = \pi - \arcsin x + 2k\pi$, $k \in Z$

• General solution for $\cos \theta = x$ ($|x| \leq 1$): $\theta = \arccos x + 2k\pi$ or $\theta = -\arccos x + 2k\pi$

• General solution for $\tan \theta = x$: $\theta = \arctan x + k\pi$

Polar Coordinates

• Definition: $(r, \theta)$ where $r \geq 0$ is the distance from the origin, $\theta$ is the angle from the positive $x$-axis.

• Conversion: $x = r \cos \theta$, $y = r \sin \theta$; $r = \sqrt{x^2 + y^2}$, $\tan \theta = \frac{y}{x}$

• Polar equations: Equations in $r$ and $\theta$ describe curves in the plane (e.g., $r = 2$ is a circle of radius 2).

Expressions of the Form $a \cos x + b \sin x$

• Can be rewritten as $R \cos(x - \theta)$, where $R = \sqrt{a^2 + b^2}$, $\tan \theta = \frac{b}{a}$

• Example: $\cos x + \sqrt{3} \sin x = 2 \cos(x - \frac{\pi}{3})$

Mathematical Induction

Principle of Mathematical Induction

• To prove a statement $P(n)$ for all $n \in N$:

  1. Show $P(1)$ is true (base case).

  2. Assume $P(k)$ is true; show $P(k+1)$ is true (inductive step).

• If both steps hold, $P(n)$ is true for all $n \in N$.

Example: Prove $1 + 3 + 5 + ... + (2n-1) = n^2$ for all $n \in N$ by induction.

Sigma Notation and Binomial Theorem

Sigma Notation

• Definition: $\sum_{j=1}^n a_j = a_1 + a_2 + ... + a_n$

• Properties:

  • $\sum_{i=r}^n (a_i + b_i) = \sum_{i=r}^n a_i + \sum_{i=r}^n b_i$

  • $\sum_{i=r}^n k a_i = k \sum_{i=r}^n a_i$

Summation Formulas

• $\sum_{r=1}^n r = \frac{n(n+1)}{2}$

• $\sum_{r=1}^n r^2 = \frac{n(n+1)(2n+1)}{6}$

• $\sum_{r=1}^n r^3 = \frac{n^2(n+1)^2}{4}$

Factorials and Binomial Coefficients

• Factorial: $n! = 1 \cdot 2 \cdot ... \cdot n$, $0! = 1$

• Binomial coefficient: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

• Properties:

  • $\binom{n}{0} = \binom{n}{n} = 1$

  • $\binom{n}{k} = \binom{n}{n-k}$

  • $\binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k}$

Binomial Theorem

• For $n \in N$, $(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$

• Pascal's Triangle: The coefficients $\binom{n}{k}$ form the entries of Pascal's triangle.

• Example: $(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$

Conic Sections

Quadratic Forms and Canonical Forms

• General quadratic form: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

• Canonical forms:

  • Parabola: $y^2 = 4ax$

  • Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

  • Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Classification

• Parabola: One squared term, e.g., $y^2 = 4ax$

• Ellipse: Both $x^2$ and $y^2$ terms, same sign

• Hyperbola: Both $x^2$ and $y^2$ terms, opposite signs

Change of Axes

• Translation: Completing the square to shift the origin

• Rotation: Removing the $xy$ term by rotating axes through angle $\alpha$ where $\cot 2\alpha = \frac{A-C}{B}$

Summary Table: Conic Sections

Equation	Type	Key Features
$y^2 = 4ax$	Parabola	One squared term, axis of symmetry
$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$	Ellipse	Both terms positive, bounded
$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	Hyperbola	One term negative, two branches, asymptotes

Appendix: Mathematical Reasoning

Logic and Proof

• Statement: An expression that is either true or false.

• Implication: "If p then q" ($p \Rightarrow q$)

• Converse: $q \Rightarrow p$

• Contrapositive: $\neg q \Rightarrow \neg p$ (logically equivalent to the original implication)

• Equivalence: $p \Leftrightarrow q$ (both $p \Rightarrow q$ and $q \Rightarrow p$ are true)

• Direct proof: Prove $p \Rightarrow q$ directly.

• Proof by contradiction: Assume the negation and derive a contradiction.

• Counterexample: A single example disproving a statement.

Properties of Real Numbers

• Associative, commutative, distributive laws for addition and multiplication

• Order properties: $a \leq b$ or $b \leq a$; if $a \leq b$ and $b \leq a$, then $a = b$

• Intervals: See earlier table for notation and types

Additional info: This summary covers all major topics from the provided lecture manual, including definitions, properties, examples, and key formulas, and is structured for exam preparation in a college algebra course.