Comprehensive Study Notes for College Algebra: Numbers, Functions, Trigonometry, Induction, Binomial Theorem, and Conic Sections

Numbers, Inequalities, and Absolute Values

Sets

Understanding sets is foundational in algebra. A set is a well-defined collection of objects, called elements. Sets can be described by listing elements or using set-builder notation.

• Notation: Curly brackets are used, e.g., A = {1, 2, 3, 4}.

• Set-builder notation: S = {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: ∅ is the set with no elements.

Real Numbers

The real numbers (R) include:

• Natural numbers (N): {1, 2, 3, ...}

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

• Rational numbers (Q): Numbers of the form p/q where p, q ∈ Z, q ≠ 0

• Irrational numbers: Real numbers not rational, e.g., √2, π

Real numbers can be represented on the number line and ordered using <, >, ≤, ≥.

Intervals

Intervals are subsets of the real line:

• Open interval: (a, b) = {x ∈ R | a < x < b}

• Closed interval: [a, b] = {x ∈ R | a ≤ x ≤ b}

• Half-open intervals: [a, b), (a, b]

Inequalities

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

• Example: Solve 3x + 1 > 2x:

  • 3x + 1 > 2x ⇒ x > -1

• Compound inequalities: Use logical AND/OR to combine conditions.

• Express solutions: Use interval or set notation.

Absolute Values

The absolute value of a real number a is its distance from zero:

• Definition: |a| = a if a ≥ 0; |a| = -a if a < 0

• Key properties:

  • |a| ≥ 0

  • |a|^2 = a^2

  • |ab| = |a||b|

  • |a + b| ≤ |a| + |b| (Triangle Inequality)

  • |a| = √{a^2}

• Solving absolute value equations/inequalities: Split into cases based on the definition.

Functions

Definition and Basics

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

• Domain: Set of all possible inputs (dom(f)).

• Range: Set of all possible outputs (range(f)).

• Natural domain: All real numbers for which the formula makes sense.

Example: f(x) = 1/√{x-1} has domain (1, ∞).

Graphing Functions

• The graph of f is the set {(x, f(x)) | x ∈ 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

• Shifts: f(x + c) shifts left/right; f(x) + c shifts up/down.

• Scaling: f(cx) compresses/stretches horizontally; c f(x) vertically.

• Reflections: f(-x) reflects over y-axis; -f(x) over x-axis.

Even and Odd Functions

• Even: f(-x) = f(x) (symmetric about y-axis).

• Odd: f(-x) = -f(x) (symmetric about origin).

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

Classification and Combination of Functions

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

• Rational: g(x) = p(x)/q(x) where p, q are polynomials, q(x) ≠ 0

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

• Operations: Sums, differences, products, quotients, and composition (f∘g).

Inverse Functions

• One-to-one (injective): f(x_1) = f(x_2) ⇒ x_1 = x_2

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

• Graph: The graph of f^{-1} is the reflection of f across y = x.

Angles and Trigonometric Functions

Radian Measure

• Definition: The radian measure of an angle is the ratio of arc length s to radius r: $\alpha = \frac{s}{r}$

• Conversions:

  • Degrees to radians: multiply by $\frac{\pi}{180}$

  • Radians to degrees: multiply by $\frac{180}{\pi}$

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

• Area of sector: $A = \frac{1}{2} r^2 \alpha$

Trigonometric Functions

• Definitions (unit circle):

  • $\sin \theta = \frac{y}{r}$

  • $\cos \theta = \frac{x}{r}$

  • $\tan \theta = \frac{y}{x}$

  • $\csc \theta = \frac{r}{y}$

  • $\sec \theta = \frac{r}{x}$

  • $\cot \theta = \frac{x}{y}$

• Special angles: Know exact values for $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$.

• Periodicity:

  • $\sin, \cos, \sec, \csc$ have period $2\pi$

  • $\tan, \cot$ have period $\pi$

Trigonometric Identities

• Pythagorean: $\sin^2 \theta + \cos^2 \theta = 1$

• Quotient: $\tan \theta = \frac{\sin \theta}{\cos \theta}$, $\cot \theta = \frac{\cos \theta}{\sin \theta}$

• Sum and difference:

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

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

• Double angle:

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

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

• Product-to-sum and sum-to-product formulas (see notes for details).

Inverse Trigonometric Functions

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

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

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

• Domains and ranges: Know the principal values for each function.

Trigonometric Equations

• General solutions use periodicity:

  • $\sin \theta = x \implies \theta = \arcsin x + 2k\pi$ or $\theta = \pi - \arcsin x + 2k\pi$

  • $\cos \theta = x \implies \theta = \arccos x + 2k\pi$ or $\theta = -\arccos x + 2k\pi$

  • $\tan \theta = x \implies \theta = \arctan x + k\pi$

Polar Coordinates

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

• Conversion:

  • $x = r \cos \theta$

  • $y = r \sin \theta$

  • $r = \sqrt{x^2 + y^2}$

  • $\tan \theta = \frac{y}{x}$

• Common polar graphs: Circles ($r = c$), lines ($\theta = k$), spirals ($r = k\theta$), roses ($r = a \cos n\theta$ or $r = a \sin n\theta$), cardioids ($r = a + b \sin \theta$).

• Symmetry tests: Replace $\theta$ with $-\theta$, $\pi - \theta$, or $\pi + \theta$ to check for symmetry about x-axis, y-axis, or origin, respectively.

Expressing $a \cos x + b \sin x$ as $R \cos(x - \theta)$

• Let $a = R \cos \theta$, $b = R \sin \theta$

• Then $R = \sqrt{a^2 + b^2}$, $\tan \theta = \frac{b}{a}$

• So $a \cos x + b \sin x = R \cos(x - \theta)$

Mathematical Induction

Principle of Mathematical Induction

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

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

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

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

Example: Prove $1 + 2 + ... + n = \frac{n(n+1)}{2}$ for all $n$.

Sigma Notation and Binomial Theorem

Sigma Notation

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

• Properties:

  • Linearity: $\sum (a_j + b_j) = \sum a_j + \sum b_j$

  • Constants: $\sum k = nk$

Common Sums

• $\sum_{r=1}^n r = \frac{n(n+1)}{2}$

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

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

• Arithmetic series: $a + (a+d) + ... + [a + (n-1)d] = \frac{n}{2}[2a + (n-1)d]$

• Geometric series: $a + ar + ar^2 + ... + ar^{n-1} = a \frac{1 - r^n}{1 - r}$ (if $r \neq 1$)

Factorials and Binomial Coefficients

• Factorial: $n! = n \cdot (n-1) \cdot ... \cdot 1$, $0! = 1$

• 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

• Statement: $(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$

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

• Applications: Expanding powers, finding specific coefficients, and combinatorics.

Conic Sections

Quadratic Forms and Canonical Forms

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

• 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 $y^2$ with same sign, e.g., $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

• Circle: Special case of ellipse with $a = b$

• Hyperbola: $x^2$ and $y^2$ with opposite signs, e.g., $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Transformations

• Translation: Completing the square shifts the center.

• Rotation: Used to eliminate $xy$ term; principal axes are found by rotating through angle $\alpha$ where $\cot 2\alpha = \frac{A - C}{B}$.

Summary Table: Types of Conic Sections

Equation	Type	Key Features
$y^2 = 4ax$	Parabola	One squared term; opens right if $a > 0$
$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$	Ellipse	Both terms positive; closed curve
$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	Hyperbola	One term negative; two branches
$x^2 + y^2 = r^2$	Circle	Special ellipse; center at origin

Appendix: Mathematical Reasoning

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

• Theorem: A true statement proved from axioms and definitions.

• Axiom: A statement assumed true without proof.

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

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

• Contrapositive: "If not q then not p" ($\neg q \Rightarrow \neg p$)

• Equivalence: $p \Leftrightarrow q$ means both $p \Rightarrow q$ and $q \Rightarrow p$

• Proof by contradiction: Assume the opposite and show a contradiction arises.

Additional info: These notes are based on the MATH1034A Algebra manual and cover all foundational topics in college algebra, including sets, real numbers, inequalities, functions, trigonometry, induction, sigma notation, binomial theorem, and conic sections. For more detailed worked examples and exercises, refer to the original manual or a standard college algebra textbook.