Summary

Global Analytical Properties

These are the universal lenses used to analyze any graph. While every function is checked for these, the results (like “End Behavior”) look different depending on the family.

• Transformations: Horizontal/Vertical shifts, Vertical stretch/shrink, and Reflections ($x$ and $y$ axis).
• Domain & Range: The set of all possible inputs ($x$) and outputs ($y$).
• Intercepts: The points where the graph crosses the $x$-axis (roots/zeros) and $y$-axis.
• Rate of Change: Analyzed as a constant (Slope) or variable (Derivative).
• Inverses: The “undoing” function; requires a One-to-One relationship (often requiring Domain Restriction for Quadratics and Trig).
• Continuity & Discontinuity: Analyzing if the graph is a single unbroken curve or has holes/breaks.
• Symmetry: Checking if a function is Even ($y$-axis symmetry) or Odd (origin symmetry).
• Inequalities: Solving for intervals where $f(x) > 0$, $f(x) < 0$, or other relational conditions.
• Composition of Functions: Combining functions as $f(g(x))$; the output of one becomes the input of another.

Linear Functions

• Slope (Rate of Change): The constant steepness; the “m” in $y = mx + b$.
• Monotonicity: The function is strictly increasing or strictly decreasing for its entire domain.
• $x$ and $y$ Intercepts: The single points where the line crosses the axes.
• Parallel Lines: Share the same slope ($m_1 = m_2$).
• Perpendicular Lines: Slopes are negative reciprocals ($m_1 m_2 = -1$).
• Point-Slope Form: $y - y_1 = m(x - x_1)$ for writing equations from a point and slope.

Quadratic Functions

• Vertex (Global Max/Min): The absolute highest or lowest point of the parabola.
• Axis of Symmetry: The vertical line $x = h$ that splits the graph perfectly in half.
• Concavity: The graph is always either “concave up” (cup) or “concave down” (frown).
• Discriminant: Using $b^2 - 4ac$ to predict the number and type of roots (Real vs. Complex).
• Turning Points: Always has exactly one turning point.
• Factoring Methods: Greatest Common Factor (GCF), Difference of Squares ($a^2 - b^2$), Factor by Grouping, Trinomial Factoring.
• Completing the Square: Method to convert $ax^2 + bx + c$ to vertex form $a(x-h)^2 + k$.
• Product Formulas: Special patterns including $(a+b)^2 = a^2 + 2ab + b^2$, $(a-b)^2 = a^2 - 2ab + b^2$, and $(a+b)(a-b) = a^2 - b^2$.
• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for finding roots.
• Vertex Form: $y = a(x-h)^2 + k$ for identifying the vertex $(h, k)$ directly.

Polynomial Functions

• Degree ($n$): The highest exponent, which dictates the maximum number of roots and turning points.
• End Behavior: Determined by the Degree and Leading Coefficient (e.g., “up-up” or “down-up”).
• Multiplicity: Determines if the graph crosses the x-axis (odd multiplicity) or bounces (even multiplicity).
• Turning Points: Can have up to $n-1$ turning points.
• Inflection Points: Specific locations where the concavity switches from “up” to “down.”
• Polynomial Division: Long Division and Synthetic Division for dividing polynomials.
• Factor Theorem: $(x-c)$ is a factor of $P(x)$ if and only if $P(c) = 0$.
• Remainder Theorem: When $P(x)$ is divided by $(x-c)$, the remainder is $P(c)$.
• Complex Roots: When the discriminant is negative, roots occur in conjugate pairs ($a + bi$ and $a - bi$).
• Rational Root Theorem: Possible rational zeros are of the form $\frac{p}{q}$ where $p$ divides the constant term and $q$ divides the leading coefficient.

Rational Functions

• Vertical Asymptotes: “Invisible walls” caused by values that make the denominator zero.
• Horizontal/Slant Asymptotes: Describes the “End Behavior” as $x \to \infty$.
• Holes (Removable Discontinuities): Occur when a factor in the numerator and denominator cancels out.
• Branches: The separate, disconnected curves created by asymptotes.
• Domain Restrictions: Values where $Q(x) = 0$ make the function undefined and must be excluded.
• Simplifying Rational Expressions: Canceling common factors between numerator and denominator (while noting excluded values).

Exponential Functions

• Standard Form: $f(x) = a \cdot b^x$ where $a$ is the initial value and $b$ is the base.
• Growth vs. Decay: If $b > 1$, exponential growth; if $0 < b < 1$, exponential decay.
• Growth/Decay Rate Form: $f(t) = a(1 \pm r)^t$ where $r$ is the percentage rate.
• Natural Exponential: $f(x) = e^x$ where $e \approx 2.71828$ (Euler’s number).
• Horizontal Asymptote: Typically $y = 0$ (the x-axis) for standard exponential functions.
• Domain & Range: Domain is all real numbers $(-\infty, \infty)$; Range is $(0, \infty)$ for standard form.
• Applications: Compound interest, population growth, radioactive decay.
• One-to-One Property: Always passes the Horizontal Line Test; has an inverse (logarithmic function).

Logarithmic Functions

• Definition: $\log_b(x) = y \iff b^y = x$ (logarithm as the inverse of exponentiation).
• Natural Logarithm: $\ln(x) = \log_e(x)$ where the base is $e$.
• Domain Restriction: $x > 0$ (logarithms only defined for positive arguments).
• Range: All real numbers $(-\infty, \infty)$.
• Vertical Asymptote: $x = 0$ (the y-axis) for standard logarithmic functions.
• Logarithm Properties:
  • Product Rule: $\log_b(mn) = \log_b(m) + \log_b(n)$
  • Quotient Rule: $\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)$
  • Power Rule: $\log_b(m^p) = p \cdot \log_b(m)$
  • Zero Property: $\log_b(1) = 0$
  • Identity Property: $\log_b(b) = 1$
  • Inverse Property: $b^{\log_b(x)} = x$ and $\log_b(b^x) = x$
• Change of Base Formula: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$ for converting between bases.

Absolute Value Functions

• Definition: $|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$
• V-Shape Graph: Creates a sharp corner at the vertex.
• Standard Form: $f(x) = a|x - h| + k$ where $(h, k)$ is the vertex.
• Piecewise Nature: Can be rewritten as a piecewise function with two linear pieces.
• Solving Absolute Value Equations: $|x| = a$ yields two solutions: $x = a$ or $x = -a$ (if $a > 0$).
• Absolute Value Inequalities: Analyzed by considering both positive and negative cases.

Piecewise Functions

• Definition: Functions defined by different expressions on different intervals of the domain.
• Notation: Written with cases showing the formula and corresponding domain for each piece.
• Continuity Analysis: Check if the pieces connect smoothly or have jumps/breaks at boundary points.
• Evaluating: Determine which piece to use based on the input value’s interval.

Circles (Coordinate Geometry)

• Standard Form (Center-Radius): $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center and $r$ is the radius.
• Pythagorean Connection: Derived from the distance formula and Pythagorean theorem.
• Completing the Square: Used to convert general form to standard form.
• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ for finding the distance between two points.
• Midpoint Formula: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ for finding the point halfway between two endpoints.

Trigonometric Functions

• Periodicity: The repeating nature of the graph; the distance required to complete one full cycle.
• Amplitude: The vertical “reach” of the wave; the distance from the midline to the peak.
• Phase Shift: The trigonometric term for a horizontal shift.
• Midline (Vertical Shift): The horizontal center line ($y=d$) that the wave oscillates around.
• Domain Restrictions: Essential for defining Inverse Trig functions (Arccosine, Arcsine, etc.).
• Asymptotes: Periodic vertical breaks found in $\tan$, $\cot$, $\sec$, and $\csc$.
• Unit Circle: The foundational circle with radius 1 where $x^2 + y^2 = 1$; coordinates are $(\cos\theta, \sin\theta)$.
• Reference Angles: The acute angle formed with the x-axis; used to find trig values in any quadrant.
• Coterminal Angles: Angles that differ by full rotations ($360°$ or $2\pi$); share the same terminal side.
• Reciprocal Functions: $\csc\theta = \frac{1}{\sin\theta}$, $\sec\theta = \frac{1}{\cos\theta}$, $\cot\theta = \frac{1}{\tan\theta}$.
• Pythagorean Identities:
  • $\sin^2\theta + \cos^2\theta = 1$
  • $1 + \tan^2\theta = \sec^2\theta$
  • $1 + \cot^2\theta = \csc^2\theta$
• Quotient Identities: $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and $\cot\theta = \frac{\cos\theta}{\sin\theta}$.
• Even/Odd Identities: $\sin(-x) = -\sin(x)$ (odd), $\cos(-x) = \cos(x)$ (even).
• Cofunction Identities: $\sin\left(\frac{\pi}{2} - x\right) = \cos(x)$, $\cos\left(\frac{\pi}{2} - x\right) = \sin(x)$.
• Sum and Difference Formulas:
  • $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
  • $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
  • $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
• Double-Angle Formulas:
  • $\sin(2\theta) = 2\sin\theta\cos\theta$
  • $\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$
  • …
• Half-Angle Formulas (the $\pm$ depends on the quadrant of $\frac{\theta}{2}$):
  • $\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$
  • $\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}$
  • $\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{\sin\theta}$
• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ for solving oblique triangles when you know angle-side pairs.
• Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos(C)$ for solving triangles when you know SSS or SAS.
• Inverse Identities:
  • Composition: $\sin(\arcsin(x)) = x$, $\cos(\arccos(x)) = x$, $\tan(\arctan(x)) = x$
  • Complementary: $\arcsin(x) + \arccos(x) = \frac{\pi}{2}$
  • Negative Argument: $\arcsin(-x) = -\arcsin(x)$, $\arctan(-x) = -\arctan(x)$

Radicals and Rational Exponents

• Radical Notation: $\sqrt[n]{x}$ represents the $n$th root of $x$.
• Rational Exponent Form: $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$
• Properties of Radicals
• Simplifying Radicals: Factor out perfect powers to simplify expressions.

Complex Numbers

• Standard Form: $a + bi$ where $a$ is the real part and $bi$ is the imaginary part.
• Imaginary Unit: $i = \sqrt{-1}$ with the property $i^2 = -1$.
• Powers of $i$: Cycle every 4 powers ($i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$).
• Complex Plane: Real part on x-axis, imaginary part on y-axis; geometric representation.
• Operations: Add/subtract by combining like terms; multiply using distributive property and $i^2 = -1$.
• Complex Conjugate: $a - bi$ is the conjugate of $a + bi$; used for division.
• Geometric Interpretation: Multiplying by $i$ represents a 90° rotation in the complex plane.

Sequences and Series

• Arithmetic Sequence: $a_n = a_1 + (n - 1)d$ where $d$ is the common difference.
• Arithmetic Series: $S_n = \frac{n(a_1 + a_n)}{2}$; sums a fixed number of terms.
• Geometric Sequence: $a_n = a_1 \cdot r^{n-1}$ where $r$ is the common ratio.
• Geometric Series: $S_n = a_1\left(\frac{1 - r^n}{1 - r}\right)$; sums $n$ terms.
• Infinite Geometric Series: $S_\infty = \frac{a_1}{1 - r}$ when $|r| < 1$; the sum converges to a finite value.

Binomial Theorem

• Expansion: $(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$ for any positive integer $n$.
• Binomial Coefficient: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$; read as ”$n$ choose $k$.”
• Generalizes the product formulas $(a + b)^2$ and $(a + b)^3$ to any power.

Limits

• Definition: $\lim_{x \to a} f(x) = L$ means $f(x)$ gets arbitrarily close to $L$ as $x$ gets arbitrarily close to $a$.
• Convergence: A sequence or function settles toward a single finite value.
• Divergence: Grows without bound or oscillates without settling.
• Connection: The infinite geometric series $S_\infty$ is the limit of the partial sums as $n \to \infty$.