Mathematics Advanced Study Guide

Functions

MA-F1: Working with Functions

F1.1: Algebraic Techniques

• Prior Knowledge & Factorisation:

  • Difference of two squares: $(a + b)(a - b) = a^2 - b^2$

  • Methodical steps for factorising algebraic expressions:

  1. Always search for a common factor first.

  2. For expressions with $2$ terms, check for the difference of two squares.

  3. For expressions with $3$ terms where $a = 1$ (monic quadratic), apply the product and sum method.

  4. For expressions with $3$ terms where $a \neq 1$ (non-monic quadratic), apply the product and sum method.

  5. For expressions with $4$ terms, factorise by grouping in pairs.

• Index Laws and Surds:

  • Operations include standard index laws and arithmetic with surds, simplifying surds, and rationalising the denominator.

  • Note: Binomial denominators are outside the scope of this course.

• Solving Quadratic Equations:

  • Method 1: Factoring

  • Method 2: Completing the square

  • Method 3: Quadratic Formula:     $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Algebraic Fractions:

  • Always factorise the numerator of an algebraic fraction first before simplifying.

F1.2: Introduction to Functions

• Core Definitions and Notations:

  • Use function notation $f(x)$.

  • Understand domain (set of valid input values) and range (set of valid output values).

  • Express domain and range using interval notation, such as $[4, \infty)$.

  • Identify independent variables (inputs) and dependent variables (outputs).

• Types of Relations:

  • Relations can be categorized as one-to-one, one-to-many, many-to-one, and many-to-many.

  • Apply the vertical line test to identify whether a relation is a function (a line passing vertically through the graph must intersect it at most once).

  • Determine whether a function is one-to-one using horizontal line tests.

F1.3: Linear, Quadratic, and Cubic Functions

• Linear Functions:

  • Direct variation is given by $y = kx$, where $k$ represents the constant of variation. Direct variation yields a straight-line graph.

  • Gradient-intercept equation of a straight line: $y = mx + c$, where $m$ is the gradient and $c$ is the $y$--intercept.

  • Formula for the gradient of a straight line passing through $(x_1, y_1)$ and $(x_2, y_2)$:     $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$

  • Point-gradient equation of a line passing through $(x_1, y_1)$ with gradient $m$:     $y - y_1 = m(x - x_1)$

  • Parallel lines possess identical gradients ($m_1 = m_2$).

  • Two non-vertical lines with gradients $m_1$ and $m_2$ are perpendicular if:     $m_1 m_2 = -1$

• Quadratic Functions:

  • Recognize key features of a quadratic curve, including its parabolic nature and turning point.

• Concavity: Curves can open upwards (concave up) or downwards (concave down).

• Axis of symmetry calculation:

  • Take the average of the zeroes, or

  • Use the formula $x = -\frac{b}{2a}$

• Intercepts:

  • $y$--intercept: set $x = 0$, yielding $y = c$

  • $x$--intercepts (zeroes): set $y = 0$, factorise, and solve (verify existence using the discriminant).

• Factored form $y = a(x - \alpha)(x - \beta)$:

  • The $x$--intercepts (zeroes) are $x = \alpha$ and $x = \beta$

  • The axis of symmetry is $x = \frac{1}{2}(\alpha + \beta)$

• Vertex form $y = a(x - h)^2 + k$ (derived via completing the square):

  • The axis of symmetry is $x = h$

  • The vertex is $(h, k)$

• Discriminant analysis:     $\Delta = b^2 - 4ac$

  • If $\Delta > 0$, there are two real zeroes.

  • If $\Delta = 0$, there is exactly one real zero.

  • If $\Delta < 0$, there are no real zeroes.

  • Break-Even Analysis:

• The break-even point occurs when total costs equal total income.

• Relationship: $\text{profit} = \text{revenue} - \text{costs}$

• Income equation: $I = mx$

• Cost equation: $C = mx + b$

  • Cubic Functions:

• Basic form: $f(x) = kx^3$

• Shifted vertex form: $f(x) = k(x - b)^3 + c$

• Factored form with three distinct real roots: $f(x) = k(x - a)(x - b)(x - c)$

F1.4: Further Functions and Relations

• Standard functions and geometric relations to recognize and sketch:

  • Square root function:     $y = \sqrt{x}$

• Quartic polynomial:     $y = x^4$

• Circle centered at origin with radius $r$:     $x^2 + y^2 = r^2$

• Circle centered at $(a, b)$ with radius $r$:     $(x - a)^2 + (y - b)^2 = r^2$

• Transform expanded circle equations of the form $x^2 + y^2 + ax + by + c = 0$ into standard form $(x - a)^2 + (y - b)^2 = r^2$ by completing the square for both $x$ and $y$ terms.

• Upper semi-circle centered at origin:     $y = \sqrt{r^2 - x^2}$

• Lower semi-circle centered at origin:     $y = -\sqrt{r^2 - x^2}$

• Exponential functions:

• Hyperbolas:

• Polynomials in general: analyze key components including coefficients and degree.

• Absolute value function:     $y = |x|$

Graphing Techniques

MA-F2: Graphing Techniques

Transformations of Functions

• Translations:

  • Vertical shift: $y = f(x) + c$ translates $f(x)$ by $c$ units in the positive $y$--direction.

  • Horizontal shift: $y = f(x - b)$ translates $f(x)$ by $b$ units in the positive $x$--direction. Conversely, $y = f(x + b)$ represents a translation of $b$ units in the negative $x$--direction.

• Stretches (Dilations):

  • Vertical stretch: $y = k f(x)$ dilates the graph by a scale factor of $k$ in the $y$--direction, leaving the $x$--axis invariant.

  • Horizontal stretch: $y = f\left(\frac{x}{a}\right)$ dilates the graph by a scale factor of $a$ in the $x$--direction, leaving the $y$--axis invariant. Note that $y = f(ax)$ represents a horizontal stretch of scale factor $\frac{1}{a}$ in the $x$--direction.

• Reflections:

  • Reflection in the $y$--axis: $y = f(-x)$

  • Reflection in the $x$--axis: $y = -f(x)$

  • Rotation of $180^\circ$ about the origin: $y = -f(-x)$

• Even and Odd Symmetry:

  • A function $f(x)$ is even if $f(-x) = f(x)$, exhibiting line symmetry about the $y$--axis.

  • A function $f(x)$ is odd if $f(-x) = -f(x)$, exhibiting rotational symmetry of $180^\circ$ about the origin.

• Universal Transformation Sequence:

  • To transform $y = f(x)$ to the comprehensive form $y = k f\left(\frac{1}{a}(x - b)\right) + c$, apply the transformations in the following order:

  1. Stretch horizontally by a factor of $a$

  2. Shift right by $b$ units

  3. Stretch vertically by a factor of $k$

  4. Shift upwards by $c$ units

Asymptotes and Inequalities

• Testing for Asymptotes:

  • Vertical Asymptotes: Locate values of $x = a$ where the denominator equals zero while the numerator is non-zero.

  • Horizontal Asymptotes: Divide the numerator and denominator by the highest power of $x$ present in the denominator. Apply the limit property where $\frac{1}{x} \rightarrow 0$ as $x \rightarrow \pm\infty$. If $f(x)$ approaches a definite finite value $b$, then $y = b$ is a horizontal asymptote.

• Solving Inequalities:

  • Linear Inequalities: Solve algebraically. Remember to reverse the inequality direction whenever multiplying or dividing both sides by a negative quantity.

  • Quadratic Inequalities:

  1. Rearrange all terms to the left-hand side (LHS) so the right-hand side equals zero.

  2. Sketch the curve $y = \text{LHS}$, clearly identifying the $x$--intercepts.

  3. Read the solution set directly from the graph based on whether LHS is required to be positive ($> 0$) or negative ($< 0$).

  • Absolute Value Equations and Inequalities: Resolve algebraically or graphically by sketching the intersected boundary lines and identifying regions of satisfaction.

Trigonometric Functions

MA-T1: Trigonometric and Measure of Angles

T1.1: Trigonometry

• Right-Angled Triangle Ratios:   $\sin(A) = \frac{\text{opp}}{\text{hyp}}, \quad \cos(A) = \frac{\text{adj}}{\text{hyp}}, \quad \tan(A) = \frac{\text{opp}}{\text{adj}}$

• Sine Rule:   $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

  • Applies to any triangle. Take care when finding an angle to check for the ambiguous case when given two sides and a non-included angle.

• Cosine Rule:

  • Side length form:     $c^2 = a^2 + b^2 - 2ab\cos(C)$

  • Angle form:     $\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$

• Area of a Non-Right-Angled Triangle:   $A = \frac{1}{2}ab\sin(C)$

• Applications in 2D and 3D:

  • Angles of elevation (measured upwards from horizontal) and depression (measured downwards from horizontal).

  • True bearings (3-digit bearings measured clockwise from true North, e.g., $045^\circ$) and compass bearings (e.g., $N45^\circ E$).

T1.2: Radians

• Unit Circle and Radian Measure:

  • Convert between degrees and radians using $\pi\text{ rad} = 180^\circ$.

  • Recall exact trigonometric values for special angles in both degrees ($0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$) and radians ($0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$).

  • Evaluate trigonometric ratios for angles of any magnitude across all four quadrants.

  • Recognize and sketch parent graphs for $y = \sin(x)$, $y = \cos(x)$, and $y = \tan(x)$.

• Arc Length and Sector Area:

  • Arc length formula (where $\theta$ is in radians):     $l = r\theta$

  • Area of a circular sector formula (where $\theta$ is in radians):     $A = \frac{1}{2}r^2\theta$

MA-T2: Trigonometric Functions and Identities

Reciprocal Ratios and Pythagorean Identities

• Reciprocal Trigonometric Functions:   $\sec(A) = \frac{1}{\cos(A)}$   $\csc(A) = \frac{1}{\sin(A)}$   $\cot(A) = \frac{\cos(A)}{\sin(A)} = \frac{1}{\tan(A)}$

  • Be prepared to sketch the graphs of these reciprocal functions.

• Fundamental Quotient Identity:   $\tan(x) = \frac{\sin(x)}{\cos(x)}$

• Pythagorean Identities:   $\cos^2(x) + \sin^2(x) = 1$   $1 + \tan^2(x) = \sec^2(x)$   $1 + \cot^2(x) = \csc^2(x)$

• Simplifying and Proving Identities:

  • Prove trigonometric identities using fundamental and Pythagorean identities.

  • Evaluate expressions using angles of any magnitude and complementary angle relations (e.g., $\sin\left(\frac{\pi}{2} - x\right) = \cos(x)$).

  • Simplify complex expressions and solve trigonometric equations, including those reducible to quadratic form.

MA-T3: Trigonometric Functions and Graphs

Transformations of Sine, Cosine, and Tangent

• Amplitude:

  • For $y = k f(x)$, representing a vertical stretch by factor $k$.

  • The parent functions $y = \sin(x)$ and $y = \cos(x)$ both have an amplitude of $1$.

• Period:

  • For $y = f(nx)$, representing a horizontal dilation by a factor of $\frac{1}{n}$.

  • Parent functions $y = \sin(x)$ and $y = \cos(x)$ have a period of $2\pi$ (one full revolution).

  • Parent function $y = \tan(x)$ has a period of $\pi$ (half a revolution).

  • Transformed period for sine and cosine:     $\text{Period} = \frac{2\pi}{n}$

  • Transformed period for tangent:     $\text{Period} = \frac{\pi}{n}$

• Phase Shift:

  • Expressed as $y = f(x + b)$.

  • The phase angle corresponds to the argument value when $x = 0$.

  • Functions $y = \sin(x + b)$, $y = \cos(x + b)$, and $y = \tan(x + b)$ all have a phase of $b$, which shifts the original graph left by $b$ units.

• Vertical Shift and Mean Value:

  • Expressed as $y = f(x) + c$.

  • The mean value (centreline) of a wave is the average of its maximum and minimum values.

  • $y = \sin(x) + c$ and $y = \cos(x) + c$ both possess a mean value of $c$, representing a vertical translation upwards by $c$ units.

Calculus

MA-C1: Introduction to Differentiation

C1.1: Gradients of Tangents

• Distinguish clearly between continuous and discontinuous functions.

• The angle of inclination $\theta$ of a line or tangent with the positive $x$--axis is directly linked to its gradient $m$ by:   $\tan(\theta) = m$

C1.2: Difference Quotients

• Interpret and calculate the difference quotient as the average rate of change of $f(x)$ over an interval:   $\text{Average Rate of Change} = \frac{f(x + h) - f(x)}{h}$

• Interpret gradient functions across various practical contexts (e.g., slope on distance-time graphs equals velocity; slope on velocity-time graphs equals acceleration).

C1.3: Derivative Function from First Principles

• Define the derivative function $f'(x)$ rigorously using first principles:   $f'(x) = \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h}$

• Interpret the derivative evaluated at a specific point as the exact instantaneous rate of change of the function at that point.

C1.4: Differentiation Rules & Kinematics

• Power Rule:   $\frac{d}{dx}\left(x^n\right) = n x^{n-1}$

• Product Rule:   $\text{For } y = uv, \quad \frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$

• Quotient Rule:   $\text{For } y = \frac{u}{v}, \quad \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$

• Chain Rule:   $\text{For } y = g(u) \text{ where } u = f(x), \quad \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$

• Composite Power Rule:   $\text{For } y = [f(x)]^n, \quad \frac{dy}{dx} = n f'(x) [f(x)]^{n-1}$

• Tangents, Normals, and Kinematics:

  • Construct linear equations for tangents and normal lines ($m_{\text{normal}} = -\frac{1}{m_{\text{tangent}}}$) at specified points on a curve.

  • Displacement, Velocity, and Acceleration:

  • Velocity $v(t)$ is the rate of change of displacement $s(t)$ with respect to time $t$: $v = \frac{ds}{dt}$.

  • Acceleration $a(t)$ is the rate of change of velocity $v(t)$ with respect to time $t$: $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$.

MA-C2: Differential Calculus

Derivatives of Trigonometric, Exponential, and Logarithmic Functions

• Trigonometric Functions:   $\text{For } y = \sin(f(x)), \quad \frac{dy}{dx} = f'(x) \cos(f(x))$   $\text{For } y = \cos(f(x)), \quad \frac{dy}{dx} = -f'(x) \sin(f(x))$   $\text{For } y = \tan(f(x)), \quad \frac{dy}{dx} = f'(x) \sec^2(f(x))$

• General Exponential Functions:   $\frac{d}{dx}\left(a^x\right) = (\ln a) a^x \implies \text{For } y = a^{f(x)}, \quad \frac{dy}{dx} = (\ln a) f'(x) a^{f(x)}$

• Natural Logarithmic Functions:   $\frac{d}{dx}(\ln x) = \frac{1}{x} \implies \text{For } y = \ln(f(x)), \quad \frac{dy}{dx} = \frac{f'(x)}{f(x)}$

• Base-a Logarithmic Functions:   $\frac{d}{dx}\left(\log_a x\right) = \frac{1}{x \ln a} \implies \text{For } y = \log_a(f(x)), \quad \frac{dy}{dx} = \frac{f'(x)}{(\ln a) f(x)}$

• Base-e Exponential Functions:   $\text{For } y = e^{f(x)}, \quad \frac{dy}{dx} = f'(x) e^{f(x)}$

• Systematically apply product, quotient, and chain rules to combinations of these functions.

MA-C3: Applications of Differentiation

Curve-Sketching and Optimisation

• Utilize first derivatives $f'(x)$ to identify stationary points and intervals of increase/decrease.

• Utilize second derivatives $f''(x)$ to measure concavity and locate points of inflection.

• Systematic Curve-Sketching Menu:

  • 0. Preparation: Combine algebraic fractions into single terms; fully factorise expressions.

  • 1. Domain: Identify valid input values before sketching.

  • 2. Symmetry: Check if the function is even, odd, or neither.

  • 3A. Intercepts: Find the $y$--intercept (set $x=0$) and all $x$--intercepts/zeroes (set $y=0$).

  • 3B. Sign: Construct a table of signs to identify positive and negative regions.

  • 4A. Vertical Asymptotes: Inspect points of discontinuity for vertical asymptotes.

  • 4B. Horizontal Asymptotes: Evaluate end-behaviour as $x \rightarrow \infty$ and $x \rightarrow -\infty$.

  • 5A. First Derivative Zeroes: Find roots and discontinuities of $f'(x)$.

  • 5B. Slope Analysis: Test surrounding $f'(x)$ values to classify stationary points (local minimum, local maximum, stationary point of inflection).

  • 6A. Second Derivative Zeroes: Find roots and discontinuities of $f''(x)$.

  • 6B. Concavity Analysis: Test surrounding $f''(x)$ values to verify points of inflection and concavity changes.

• Optimisation Problems: Formulate expressions for physical quantities, differentiate, solve for stationary values ($f'(x) = 0$), and verify global maximum or minimum behavior using second derivative tests or sign tables.

MA-C4: Integral Calculus

Integration Formulas and Area Calculations

• Indefinite Integrals & Reverse Chain Rules:

  • Power rule:     $\int x^n \, dx = \frac{1}{n + 1} x^{n+1} + c$

  • Reverse chain rule for powers:     $\int f'(x) [f(x)]^n \, dx = \frac{1}{n + 1} [f(x)]^{n+1} + c$

  • Trigonometric integrals:     $\int f'(x) \sin(f(x)) \, dx = -\cos(f(x)) + c$     $\int f'(x) \cos(f(x)) \, dx = \sin(f(x)) + c$     $\int f'(x) \sec^2(f(x)) \, dx = \tan(f(x)) + c$

  • Exponential integrals:     $\int e^x \, dx = e^x + c$     $\int e^{ax+b} \, dx = \frac{1}{a} e^{ax+b} + c$     $\int f'(x) e^{f(x)} \, dx = e^{f(x)} + c$

  • Logarithmic integrals:     $\int \frac{1}{x} \, dx = \ln|x| + c$     $\int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + c$

  • General exponential integrals:     $\int a^x \, dx = \frac{a^x}{\ln a} + c$     $\int f'(x) a^{f(x)} \, dx = \frac{a^{f(x)}}{\ln a} + c$

• Areas and Numerical Approximations:

  • Determine exact areas under curves bounded by the $x$--axis using definite integrals.

  • Apply the Trapezoidal Rule to approximate area under curves using equal subintervals.

  • Calculate exact enclosed areas between two intersecting curves $y = f(x)$ and $y = g(x)$.

Exponential and Logarithmic Functions

MA-E1: Logarithms and Exponentials

Definition, Laws, and Graphs

• Logarithm Definition as Indices:   $y = a^x \iff x = \log_a y$

• Fundamental Inverse Properties:   $\log_a\left(a^x\right) = x \quad \text{and} \quad a^{\log_a x} = x$

• Logarithmic Laws:   $\log_a x + \log_a y = \log_a(xy)$   $\log_a x - \log_a y = \log_a\left(\frac{x}{y}\right)$   $\log_a\left(x^n\right) = n \log_a x$   $\log_a 1 = 0$   $\log_a a = 1$   $\log_a\left(\frac{1}{x}\right) = -\log_a x$   $\text{Change of base law: } \log_a x = \frac{\log_b x}{\log_b a}$

• Natural Logarithms and Exponential Derivatives:   $\frac{d}{dx}\left(e^x\right) = e^x \implies \text{For } y = e^{f(x)}, \quad \frac{dy}{dx} = f'(x) e^{f(x)}$

• Graphing Exponential and Logarithmic Transformations:

  • Exponential parent forms: $y = k a^x$, $y = k a^{-x}$, along with shifts $y = k a^x + c$ and $y = k a^{x+b}$.

  • Logarithmic parent form: $y = \log_a x$, along with shifts $y = k \log_a x + c$.

  • Solve exponential equations by taking logarithms of both sides to isolate index values.

Financial Mathematics

MA-M1: Modelling Financial Situations

M1.1 & M1.2: Compound Interest, Annuities, and Arithmetic Sequences

• Modelling Investments and Loans:

  • Solve compound interest problems.

  • Analyze present value and future value annuities using tables of interest factors.

• Arithmetic Sequences and Series:

  • Distinction: A sequence is an ordered list of numbers; a series is the sum of terms in a sequence.

  • Recursive definition: $T_n = T_{n-1} + d$, with initial condition $T_1 = a$.

  • Formula for the $n$th term of an arithmetic sequence:     $T_n = a + (n - 1)d$     where $a$ is the first term and $d$ is the common difference.

  • Formulas for the sum of the first $n$ terms of an arithmetic sequence:     $S_n = \frac{n}{2}(a + l) \quad \text{(where } l \text{ is the last term)}$     $S_n = \frac{n}{2}[2a + (n - 1)d]$

M1.3 & M1.4: Geometric Sequences and Financial Applications

• Geometric Sequences and Series:

  • Recursive definition: $T_n = r T_{n-1}$, with initial condition $T_1 = a$.

  • Formula for the $n$th term of a geometric sequence:     $T_n = a r^{n-1}$     where $a$ is the first term and $r$ is the common ratio.

  • Formulas for the sum of the first $n$ terms of a geometric sequence:     $S_n = \frac{a(1 - r^n)}{1 - r} = \frac{a(r^n - 1)}{r - 1}$

  • Limiting sum of an infinite geometric series (valid only when $|r| < 1$):     $S = \frac{a}{1 - r}$

• Financial Applications:

  • Model growth and decay phenomena using geometric sequences.

  • Calculate effective annual rate of interest to evaluate competing investment or loan options compounded daily, monthly, quarterly, or six-monthly.

  • Analyze reducing balance loans as compound interest accounts with periodic repayments, calculating balance outstanding step-by-step.

  • Calculate future and present values of annuities by generating geometric series expressions for compounded values and applying sum formulas.

Statistical Analysis

MA-S1: Probability and Discrete Probability Distributions

S1.1: Probability and Set Notation

• Probability Foundations:

  • Evaluate theoretical probability and relative frequency estimates.

  • Analyze multi-stage chance experiments using arrays and tree diagrams.

  • Set Notation:

  • Complement of event $A$: $\bar{A}$ (or $A'$ or $A^c$)

  • Intersection (both $A$ and $B$): $A \cap B$

  • Union (either $A$ or $B$ or both): $A \cup B$

  • Mutually exclusive events cannot occur simultaneously ($A \cap B = \emptyset$).

  • Basic Probability Rules:     $P(\bar{A}) = 1 - P(A)$     $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• Conditional Probability & Independence:

  • Formula for conditional probability of $A$ given $B$:     $P(A|B) = \frac{P(A \cap B)}{P(B)}$

  • Event $A$ is independent of event $B$ if $P(A|B) = P(A)$.

  • Multiplication law for independent events:     $P(A \cap B) = P(A) P(B)$

S1.2: Discrete Probability Distributions

• Categorize random variables as discrete (countable) or continuous (measurable).

• For any discrete random variable $X$, the sum of all individual probabilities must equal $1$:   $\sum P(X = x) = 1$

• Measures of Centre and Spread:

  • Mean or Expected Value:     $\text{E}(X) = \mu$

  • Variance:     $\text{Var}(X) = \text{E}\left[(X - \mu)^2\right] = \text{E}\left(X^2\right) - \mu^2$

  • Relation between variance and standard deviation:     $\text{Var}(X) = \sigma^2$

  • A sample mean $\bar{x}$ estimates the population mean $\mu$, and sample standard deviation $s$ estimates population standard deviation $\sigma$ (estimates improve as sample size increases).

MA-S2: Descriptive Statistics and Bivariate Data Analysis

S2.1: Summary Statistics and Outliers

• Classify, organize, and interpret data using frequency tables, cumulative distributions, Pareto charts, and two-way tables.

• Summarize five-number datasets using parallel box-plots: Lower Extreme, Lower Quartile ($Q_1$), Median ($Q_2$), Upper Quartile ($Q_3$), and Upper Extreme.

• Central Tendency: Mean ($\bar{x}$) and Median ($\tilde{x}$).

• Spread: Range, Quantiles (deciles, quartiles, percentiles), and Interquartile Range:   $\text{IQR} = Q_3 - Q_1$

• Outlier Rule:

  • A value is classified as an outlier if it satisfies either:     $\text{Score} < Q_1 - 1.5 \times \text{IQR} \quad \text{or} \quad \text{Score} > Q_3 + 1.5 \times \text{IQR}$

S2.2: Bivariate Data Analysis

• Interpret bivariate scatterplots in terms of form (linear/non-linear), direction (positive/negative), and strength (strong/moderate/weak).

• Distinguish between dependent and independent variables.

• Compute and interpret Pearson's correlation coefficient ($r$).

• Fit a least-squares regression line to make numerical predictions.

• Interpolation vs Extrapolation: Interpolation predicts within the dataset range; extrapolation predicts outside the range. Extrapolation far beyond the dataset boundaries can produce highly inaccurate predictions.

MA-S3: Continuous Random Variables & Normal Distribution

S3.1: Continuous Random Variables

• Properties of a Probability Density Function (PDF) $f(x)$: 1. $f(x) \ge 0$ for all $x$

  2. $\int_{-\infty}^{\infty} f(x) \, dx = 1$

• Probability defined as area under the curve:   $P(X \le x) = \int_{a}^{x} f(x) \, dx$

• Identify mode (peak of PDF) and analyze Cumulative Distribution Functions (CDF) to solve for median and percentiles.

S3.2: The Normal Distribution

• Standard normal $z$--score formula:   $z = \frac{x - \mu}{\sigma}$   where $\mu$ is the population mean and $\sigma$ is the population standard deviation.

• Empirical Rule ($68\%-95\%-99.7\%$ Rule):

  • Approximately $68\%$ of scores fall between $z = -1$ and $z = 1$ ($\mu \pm 1\sigma$).

  • Approximately $95\%$ of scores fall between $z = -2$ and $z = 2$ ($\mu \pm 2\sigma$).

  • Approximately $99.7\%$ of scores fall between $z = -3$ and $z = 3$ ($\mu \pm 3\sigma$).

• Use standard normal probability tables to calculate probabilities for any specified $z$--score.