Mathematics Analysis and Approaches Standard Level Study Guide

Algebra and Equations

• Equations and Formulae

  • An equation is a statement equating two algebraic expressions. Values that make the statement true are solutions or roots, forming the solution set.

  • An identity is an equation true for all possible values of the variable.

  • A formula (plural: formulae) contains multiple variables and constants/parameters. Solving for one variable in terms of others is "changing the subject of the formula."

• Equations and Graphs

  • The graph of an equation is a visual representation of its solution set.

  • One-variable linear equations ($ax + b = 0, a \neq 0$) have one solution: $x = -\frac{b}{a}$. Their graph is a point on a number line.

  • Two-variable linear equations graph as lines on a 2D coordinate plane.

• Linear Equations and Slopes

  • Gradient/Slope ($m$) of a non-vertical line: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1} = \frac{\text{vertical change}}{\text{horizontal change}}$. Vertical lines have undefined slopes.

  • Forms of Linear Equations:

    • General form: $ax + by + c = 0$

    • Slope-intercept form: $y = mx + c$ (where $c$ is the y-intercept)

    • Point-slope form: $y - y<em>1 = m(x - x</em>1)$

    • Horizontal line: $y = c$ (slope is zero)

    • Vertical line: $x = c$ (slope is undefined)

  • Parallel and Perpendicular Lines:

    • Parallel if $m<em>1 = m</em>2$.

    • Perpendicular if $m<em>1 = -\frac{1}{m</em>2}$ or $m<em>1 \cdot m</em>2 = -1$

• Distance and Midpoints

  • Distance formula: Between $(x<em>1, y</em>1)$ and $(x<em>2, y</em>2)$ is $d = \sqrt{(x<em>1 - x</em>2)^2 + (y<em>1 - y</em>2)^2}$.

  • Midpoint formula: The midpoint of the segment joining $(x<em>1, y</em>1)$ and $(x<em>2, y</em>2)$ is $(\frac{x<em>1+x</em>2}{2}, \frac{y<em>1+y</em>2}{2})$.

Function Basics

• Definition of a Function

  • A correspondence between sets $X$ (domain) and $Y$ (range) where each element in $X$ maps to exactly one element in $Y$.

  • Vertical Line Test: A vertical line intersects a function's graph at most once.

  • Notation: $y = f(x)$ or $f: x \to f(x)$. $f(x)$ is the image of $x$ under $f$.

  • Relations: Correspondences that fail the function definition (e.g., $x^2 + y^2 = 1$).

• Domain and Range

  • Domain: The set of all real numbers for which the expression is defined (no division by zero, no square root of negatives).

  • Range: The set of all outputs. It is often found via algebraic and graphical analysis.

  • Asymptotes identify gaps: a vertical asymptote ($x=c$) occurs where the function is undefined; a horizontal asymptote ($y=c$) is the value the function approaches as $x \to \pm\infty$.

• Composite Functions

  • Created by substituting one function into another: $(g \circ f)(x) = g(f(x))$.

  • The function closest to $x$ is applied first (the "inside" function).

  • Order matters: $(f \circ g)(x) \neq (g \circ f)(x)$, except in special cases like inverses.

  • Domain of $g \circ f$: The set of all $x$ in the domain of $f$ such that $f(x)$ is in the domain of $g$.

• Inverse Functions

  • Function $g$ is the inverse of $f$ ($f^{-1}$) if $(f \circ f^{-1})(x) = x$ and $(f^{-1} \circ f)(x) = x$.

  • Existence: A function has an inverse only if it is one-to-one (passes the Horizontal Line Test).

  • Graphing: The graph of $f^{-1}$ is a reflection of $f$ over the line $y = x$.

  • Finding the analytical inverse:

    1. Determine if one-to-one.

    2. Replace $f(x)$ with $y$.

    3. Solve for $x$ in terms of $y$.

    4. Swap $x$ and $y$.

    5. Replace $y$ with $f^{-1}(x)$.

Transformations of Functions

• Common Functions:

  • Constant: $f(x) = c$

  • Identity: $f(x) = x$

  • Absolute Value: $f(x) = |x|$

  • Squaring: $f(x) = x^2$

  • Square Root: $f(x) = \sqrt{x}$

  • Cubing: $f(x) = x^3$

  • Reciprocal: $f(x) = \frac{1}{x}$

  • Inverse Square: $f(x) = \frac{1}{x^2}$

• Rigid Transformations:

  • Vertical translation: $y = f(x) + k$ (k>0 moves it up; k<0 moves it down).

  • Horizontal translation: $y = f(x - h)$ (h>0 shifts right; h<0 shifts left).

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

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

• Non-Rigid Transformations:

  • Vertical stretch (a>1) or shrink ($0:$y = a f(x)$. Affects y-coordinates.</strong></p></li><li><p><strong>Horizontal shrink ($a>1$) or stretch ($0: $y = f(ax)$. Affects x-coordinates by scale factor $\frac{1}{a}$.

Quadratic Functions and Inequalities

• Quadratic Basics

  • General form: $f(x) = ax^2 + bx + c, a \neq 0$.

  • Graph is a parabola. Opens up if a > 0 (minimum vertex) and down if a < 0 (maximum vertex).

  • Axis of symmetry: $x = -\frac{b}{2a}$.

• Vertex and Roots

  • Vertex form: $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

  • Factorized form: $f(x) = a(x - p)(x - q)$, where $p, q$ are roots.

  • The Discriminant ($\Delta = b^2 - 4ac$):

    • \Delta > 0: Two distinct real roots.

    • $\Delta = 0$: One shared real root (tangent to x-axis).

    • \Delta < 0: No real roots (imaginary solutions).

Sequences and Series

• General Sequences

  • A list of numbers in a definite order. Can be defined explicitly ($a<em>n = f(n)$) or recursively (defining $a</em>n$ based on $a_{n-1}$, e.g., Fibonacci sequences).

• Arithmetic Sequences and Series

  • Arithmetic Sequence: Terms have a common difference ($d$). $a<em>n = a</em>1 + (n - 1)d$.

  • Arithmetic Series Sum ($S<em>n$): $S</em>n = \frac{n}{2}(a<em>1 + a</em>n) = \frac{n}{2}(2a_1 + (n - 1)d)$.

• Geometric Sequences and Series

  • Geometric Sequence: Terms have a common ratio ($r$). $a<em>n = a</em>1 r^{n-1}$.

  • Geometric Series Sum ($S<em>n$): $S</em>n = a_1 \frac{1-r^n}{1-r}$.

  • Convergent Infinite Series: If |r| < 1, the sum reaches a limit: $S<em>\infty = \frac{a</em>1}{1-r}$.

• Sigma Notation

  • $\sum<em>{i=m}^{n} a</em>i$ represents the sum from lower limit $m$ to upper limit $n$.

• Compound Interest and Annuities

  • Compound Interest: $A = P(1 + \frac{r}{n})^{nt}$.

  • Continuous Compounding: $A = P e^{rt}$.

  • Future Value of Annuity (periodic payments $R$): $FV = R \frac{(1+i)^m - 1}{i}$.

The Binomial Theorem

• Pascal's Triangle: Provides coefficients for expanding $(x+y)^n$. Each entry is the sum of the two above it.

• Factorial: $n! = n \times (n-1) \times … \times 1$. ($0! = 1$).

• Binomial Coefficient: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$.

• Binomial Expansion Formula: $(x+y)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} y^r$.

Trigonometry

• Radian Measure

  • Radian is the central angle subtending an arc equal to the radius. $2\pi \text{ radians} = 360^\circ$.

  • Conversion: $1^\circ = \frac{\pi}{180} \text{ rad}$; $1 \text{ rad} = \frac{180}{\pi} \text{ deg}$.

  • Arc Length: $s = r \theta$ ($\theta$ in radians).

  • Sector Area: $A = \frac{1}{2} r^2 \theta$ ($\theta$ in radians).

• The Unit Circle

  • Circle coordinates $(x, y)$ map to $(\cos \theta, \sin \theta)$. $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

  • Signs in Quadrants: (I: all +), (II: sine +), (III: tangent +), (IV: cosine +).

• Trigonometric Identities

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

  • Double Angle:

    • $\sin 2x = 2 \sin x \cos x$

    • $\cos 2x = \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x$

• Trigonometric Rules for Triangles

  • Area of Triangle: $A = \frac{1}{2} ab \sin C$.

  • Sine Rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. (Note: Ambiguous Case for SSA).

  • Cosine Rule: $c^2 = a^2 + b^2 - 2ab \cos C$.

Introduction to Calculus

• Limits: Describe the behavior of a function as it approaches a certain value ($\lim_{x \to c} f(x)$).

• Differentiation

  • Derivative as Gradient: $f'(x)$ gives the slope of the tangent at point $x$.

  • Rules:

    • Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$.

    • Constants: $\frac{d}{dx} c = 0$.

    • Sine/Cosine: $\frac{d}{dx} \sin x = \cos x$; $\frac{d}{dx} \cos x = -\sin x$.

    • e/ln: $\frac{d}{dx} e^x = e^x$; $\frac{d}{dx} \ln x = \frac{1}{x}$.

    • Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$.

    • Product Rule: $\frac{d}{dx} (uv) = u \frac{dv}{dx} + v \frac{du}{dx}$.

    • Quotient Rule: $\frac{d}{dx} (\frac{u}{v}) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$.

• Curve Analysis

  • Stationary points occur where $f'(x) = 0$.

  • First Derivative Test: Sign change in $f'(x)$ determines max ($+$ to $-$) or min ($-$ to $+$).

  • Inflection points occur where $f''(x) = 0$ and concavity changes.

  • Kinematics: Displacement $s(t)$, Velocity $v(t) = s'(t)$, Acceleration $a(t) = v'(t)$.

• Integration

  • The reverse of differentiation (antiderivative).

  • Fundamental Theorem of Calculus: $\int_{a}^{b} f(x) dx = F(b) - F(a)$.

  • Area: Bounded by curve and x-axis is $\int_{a}^{b} f(x) dx$ (signed area).

  • Total Distance: $\int_{a}^{b} |v(t)| dt$.