College Algebra: Core Concepts, Functions, Trigonometry, Induction, and Conic Sections

Numbers, Inequalities, and Absolute Values

Sets

Sets are fundamental collections of objects, called elements. Sets can be described by listing elements or by set-builder notation.

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

• Set-builder: 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 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 and Intervals

• Natural numbers (N): N = {1, 2, 3, ...}

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

• Rational numbers (Q): Q = {p/q | p, q ∈ Z, q ≠ 0}

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

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

Intervals are subsets of the real line:

• Open: $(a, b) = \{x \in \mathbb{R} | a < x < b\}$

• Closed: $[a, b] = \{x \in \mathbb{R} | a \leq x \leq b\}$

Inequalities

Inequalities describe the order of real numbers and solution sets can be written in set or interval notation.

• Solving: Manipulate as with equations, but reverse the inequality when multiplying/dividing by a negative.

• Example: Solve $3x + 1 > 2x$:

$3x + 1 > 2x \implies x > -1$ Solution: $(-1, \infty)$

Absolute Value

The absolute value of $a$ is its distance from zero:

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

• Key properties:

• $|a| \geq 0$

• $|a|^2 = a^2$

• $|ab| = |a||b|$

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

• 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 $Y$ assigns each $x \in D$ a unique $f(x) \in Y$.

• Domain: Set of all $x$ for which $f(x)$ is defined.

• Range: Set of all $f(x)$ values as $x$ varies over the domain.

• Example: $f(x) = \frac{1}{\sqrt{x-1}}$ has domain $(1, \infty)$.

Graphing Functions

• The graph of $f$ is the set $\{(x, f(x)) | x \in D\}$.

• 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 intervals.

Transformations

• Shifts: $y = f(x + c)$ (left/right), $y = f(x) + c$ (up/down)

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

• Reflections: $y = f(-x)$ (about $y$-axis), $y = -f(x)$ (about $x$-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: $g(x) = \frac{p(x)}{q(x)}$ where $p, q$ are polynomials, $q(x) \neq 0$

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

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

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

Inverse Functions

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

• Inverse: $f^{-1}(y) = x$ such that $f(x) = y$

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

• Example: $f(x) = 3x + 5$ has inverse $f^{-1}(x) = \frac{x-5}{3}$

Angles and Trigonometric Functions

Radian Measure

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

• 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:

  • $\sin \theta = \frac{y}{r}$

  • $\cos \theta = \frac{x}{r}$

  • $\tan \theta = \frac{y}{x}$

  • $\csc \theta = \frac{r}{y}$

  • $\sec \theta = \frac{r}{x}$

  • $\cot \theta = \frac{x}{y}$

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

• Periodicity: $\sin$ and $\cos$ have period $2\pi$; $\tan$ has period $\pi$.

Trigonometric Identities

• $\sin^2 \theta + \cos^2 \theta = 1$

• $1 + \tan^2 \theta = \sec^2 \theta$

• $\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta$

• $\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$

• Double angle: $\sin 2\theta = 2\sin \theta \cos \theta$, $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$

Inverse Trigonometric Functions

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

• 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 $\tan y = x$

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 and Conversion

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

• Conversion:

  • $x = r \cos \theta$

  • $y = r \sin \theta$

  • $r = \sqrt{x^2 + y^2}$

  • $\tan \theta = \frac{y}{x}$ (with quadrant considerations)

Polar Equations and Graphs

• Equations can be converted between Cartesian and polar forms.

• Common polar graphs:

• Circle: $r = c$

• Line through origin: $\theta = k$

• Spiral: $r = k\theta$

• Rose: $r = a \cos n\theta$ or $r = a \sin n\theta$

• Cardioid: $r = a + b \sin \theta$

Symmetry Tests

• $x$-axis: Replace $\theta$ with $-\theta$

• $y$-axis: Replace $\theta$ with $\pi - \theta$

• Origin: Replace $\theta$ with $\pi + \theta$

Mathematical Induction

Principle of Mathematical Induction

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

• Base case: Prove $P(1)$ is true.

• Inductive step: Assume $P(k)$ is true, prove $P(k+1)$ is true.

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

Examples

• Sum of first $n$ odd numbers: $1 + 3 + 5 + ... + (2n-1) = n^2$

• Sum of squares: $1^2 + 2^2 + ... + n^2 = \frac{1}{6} n(n+1)(2n+1)$

Sigma Notation and Binomial Theorem

Sigma Notation

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

• Arithmetic series: $\sum_{j=0}^{n-1} (a + jd) = \frac{n}{2}[2a + (n-1)d]$

• Geometric series: $\sum_{j=0}^{n-1} ar^j = a \frac{1 - r^n}{1 - r}$

• Sum of first $n$ integers: $\sum_{r=1}^n r = \frac{n(n+1)}{2}$

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

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

Factorials and Binomial Coefficients

• Factorial: $n! = n \cdot (n-1) \cdots 2 \cdot 1$, $0! = 1$

• 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

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

• Example: Expand $(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$

Conic Sections

Quadratic Forms and Canonical Forms

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

• By completing the square and/or rotating axes, can be reduced to 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

Form	Curve	Key Features
$y^2 = 4ax$	Parabola	One quadratic, one linear term
$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$	Ellipse	Both quadratic terms, same sign
$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	Hyperbola	Quadratic terms, opposite signs

Change of Axes

• Translation: Shifts the origin to simplify the equation.

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

Example:

• Given $x^2 + 2x + 2y^2 - 8y + 7 = 0$, complete the square to get $(x+1)^2 + 2(y-2)^2 = 2$ (ellipse).

Appendix: Basic Mathematical Notions

• 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" ($p \implies q$)

• Equivalence: "p if and only if q" ($p \iff q$)

• Contrapositive: The contrapositive of $p \implies q$ is $\neg q \implies \neg p$ (logically equivalent).

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

Additional info: This summary covers the foundational topics in college algebra, including sets, numbers, inequalities, functions, trigonometry, induction, sigma notation, binomial theorem, and conic sections, as outlined in the provided lecture manual. For more detailed examples and exercises, refer to the original manual or standard college algebra textbooks.