International Baccalaureate Mathematics: Analysis and Approaches (SL & HL) Lecture Notes

INTERNATIONAL BACCALAUREATE MATHEMATICS: ANALYSIS AND APPROACHES (SL & HL) LECTURE NOTES

CONTENTS

• 2.1 LINES (or LINEAR FUNCTIONS)

• 2.2 QUADRATICS (or QUADRATIC FUNCTIONS)

• 2.3 FUNCTIONS, DOMAIN, RANGE, GRAPH

• 2.4 COMPOSITION OF FUNCTIONS: fog

• 2.5 THE INVERSE FUNCTION: f⁻¹

• 2.6 TRANSFORMATIONS OF FUNCTIONS

• 2.7 ASYMPTOTES

• 2.8 EXPONENTS - THE EXPONENTIAL FUNCTION a^x

• 2.9 LOGARITHMS - THE LOGARITHMIC FUNCTION y=log_a(x)

• 2.10 EXPONENTIAL EQUATIONS

• 2.11 POLYNOMIAL FUNCTIONS

• 2.12 SUM AND PRODUCT OF ROOTS

• 2.13 RATIONAL FUNCTIONS – PARTIAL FRACTIONS

• 2.14 POLYNOMIAL AND RATIONAL INEQUALITIES

• 2.15 MODULUS EQUATIONS AND INEQUALITIES

• 2.16 SYMMETRIES OF f(x) - MORE TRANSFORMATIONS

2.1 LINES (or LINEAR FUNCTIONS)

• Basic Notions on Coordinate Geometry:

  • Given two points A(x₁, y₁) and B(x₂, y₂):

    • The gradient or slope (m) of line segment AB is defined as: $m = rac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

      • Behavior based on the slope:

      • Increasing: m > 0

      • Decreasing: m < 0

      • Horizontal: $m = 0$

      • Vertical: Not defined.

  • Distance between A and B:
    d{AB} = ext{Distance} = ext{√}ig((x2 - x1)^2 + (y2 - y_1)^2ig)

  • Midpoint M(x, y) of line segment AB:
    Migg( rac{x1 + x2}{2}, rac{y1 + y2}{2}igg)

• Example 1:

  • Given points A(1,4) and B(7,12):

    • Slope: $m = rac{12-4}{7-1} = rac{8}{6} = rac{4}{3}$

    • Distance: d_{AB} = ext{√}ig((7-1)^2 + (12-4)^2ig) = ext{√}(36 + 64) = 10

    • Midpoint: Migg( rac{1 + 7}{2}, rac{4 + 12}{2}igg) = M(4, 8)

• The Equation of a Line:

  • Form: $y = mx + c$

    • Horizontal line equation: $y = c$

    • Vertical line equation: $x = c$ (not a function)

  • Slope-intercept form: $m = an(θ)$

• Example 2: Graphs of two lines:

  • Line L₁: $y = 2x$ (Slope = 2)

  • Line L₂: $y = -2x$ (Slope = -2)

• Parallel and Perpendicular Lines:

  • Parallel: Lines L₁ || L₂ if $m<em>1 = m</em>2$

  • Perpendicular: Lines L₁ ⊥ L₂ if $m<em>2 = - rac{1}{m</em>1}$

2.2 QUADRATICS (or QUADRATIC FUNCTIONS)

• The Simplest Quadratic: $y = x^2$

  • Table of values:

    • x: -3, -2, -1, 0, 1, 2, 3 → y: 9, 4, 1, 0, 1, 4, 9

  • Domain: $x ∈ ext{R}$

  • Range: $y ≥ 0$

  • Shape: Parabola.

• The Quadratic Function Form:
  $y = ax^2 + bx + c$, where $a ≠ 0$ .

  • Basic Characteristics:

  1. Concavity:

     • If a > 0, parabola opens upwards.

     • If a < 0, opens downwards.

  2. Discriminant: $ext{Δ} = b^2 - 4ac$ (Roots determination).

     • ext{Δ} > 0: 2 roots;

     • $ext{Δ} = 0$: 1 root;

     • ext{Δ} < 0: No real roots.

  3. Finding Roots (x-intercepts):

     • Use quadratic formula:
       $x_{1,2} = rac{-b ± ext{√Δ}}{2a}$.

  4. Y-intercept: For $x=0$, $y=c$.

  5. Axis of Symmetry: $x= rac{-b}{2a}$ (vertex coordinates).

• Example 1: Consider $y = 2x^2 - 12x + 10$:

  • $a=2$ (opens upwards)

  • Calculating Discriminant: ext{Δ} = (-12)^2 - 4(2)(10) = 64 > 0 (2 roots).

  • Roots are: $x_{1,2} = rac{12 ± 8}{4} = 1, 5$.

  • Y-intercept: $y = 10$.

  • Axis of symmetry: $x = rac{12}{4} = 3$; Vertex at (3, -8).

• Quadratic Inequalities in the form:
  ax^2 + bx + c > 0, ext{etc.}: Graph allows easy solution visualization based on sign analysis.

2.3 FUNCTIONS, DOMAIN, RANGE, GRAPH

• Definition:

  • A function f maps x to y.

    • Denoted as $f(x) = y$

    • Example: Let X={1, 2, 3} and Y={a, b, c, d}.

      • Mapping scheme:

      • $f(1) = a$

      • $f(2) = b$

      • $f(3) = d$

• Domain & Range:

  • Domain $D_f$: Set of all x values.

  • Range $R_f$: Set of all y values.

• Example 1:

  • For the function $f(x)=2x$, with domain $x∈R$, the range $y∈R$.

2.4 COMPOSITION OF FUNCTIONS: fog

• Definition:

  • $g(f(x)) = f ightarrow g$ is a composite function.

    • Formally, $f ext{g}(x) = f(g(x))$.

• Example:

  • Let function f(x) = x² and g(x) = 3x + 5:

    • Then, fog = f(g(x)) = (3x + 5)².

2.5 THE INVERSE FUNCTION: f⁻¹

• Definition:

  • Inverse function reverses mapping of f: if $f: X o Y$ then $f^{-1}: Y o X$.

  • $y=f(x) <br>ightarrow x=f^{-1}(y)$.

• Finding f⁻¹ Steps:

  1. Set $f(x) = y$

  2. Solve for x

  3. Replace y with x.

• Example:

  • For $f(x) = 3x + 5$:

    • Set $3x + 5 = y$, solve: $x = rac{y-5}{3}$, thus $f^{-1}(x) = rac{x-5}{3}$.

2.6 TRANSFORMATIONS OF FUNCTIONS

• Vertical Translations:

  • $f(x) + k$ moves graph up or down based on k.

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

• Horizontal Translations:

  • $f(x-h)$ shifts left or right by h.

2.7 ASYMPTOTES

• Vertical Asymptotes (VA):

  • Occur where the denominator is zero and the function is undefined.

• Horizontal Asymptotes (HA):

  • Describe behavior of function as x approaches infinity.

2.8 EXPONENTS - THE EXPONENTIAL FUNCTION a^x

• Definition:

  • $f(x)=a^x$ with a>0.

• Properties:

  • Always positive: f(x) > 0 for all x.

  • Increasing if a > 1, decreasing if 0<a<1.

2.9 LOGARITHMS - THE LOGARITHMIC FUNCTION log_a(x)

• Definition: The logarithm to the base a is given by:

  • $log_a(x) = y ext{ if } a^y = x$.

• Common Properties:

  • $log<em>a(a) = 1$, $log</em>a(1) = 0$, etc.

2.10 EXPONENTIAL EQUATIONS

• Solve equations where the variable is an exponent, using logarithms.

2.11 POLYNOMIAL FUNCTIONS

• Definition:

  • Polynomial functions take the form $f(x)=a<em>nx^n + a</em>{n-1}x^{n-1} + ext{…} + a<em>1x + a</em>0$.

2.12 SUM AND PRODUCT OF ROOTS

• Related to Vieta's formulas, which relate coefficients of a polynomial to sums and products of its roots.

2.13 RATIONAL FUNCTIONS - PARTIAL FRACTIONS

• These functions can be expressed as the ratio of two polynomial functions.

2.14 POLYNOMIAL AND RATIONAL INEQUALITIES

• Graph the functions to determine the intervals where the inequalities hold.

2.15 MODULUS EQUATIONS AND INEQUALITIES

• Solve absolute value equations and inequalities by considering cases.

2.16 SYMMETRIES OF f(x) - MORE TRANSFORMATIONS

• Even functions: symmetric with respect to the y-axis; Odd functions: symmetric about the origin.