Functions Formulae AA SL & AA HL: Complete IB Mathematics Guide
Welcome to the definitive guide for Functions Formulae covering IB Mathematics Analysis and Approaches at both Standard Level and Higher Level. This comprehensive resource encompasses all fundamental functions concepts that form the mathematical foundation for both AA SL and AA HL students, including linear functions and straight line equations, gradient calculations, quadratic functions with axis of symmetry, the quadratic formula and discriminant analysis, exponential and logarithmic functions, domain and range determination, composite and inverse functions, and function transformations. These core functions topics are essential building blocks for success in IB Mathematics AA at any level, providing the analytical framework necessary for calculus, algebra, statistics applications, and real-world mathematical modeling. Whether you're an AA SL student aiming for top marks or an AA HL student building foundations before advanced topics, mastering these formulae is absolutely critical for examination success and mathematical fluency.
Understanding Functions in IB Mathematics AA
Functions represent one of the most fundamental concepts in all of mathematics, serving as the language through which we describe relationships between quantities, model real-world phenomena, and solve complex problems across disciplines. In IB Mathematics AA, functions provide essential tools for calculus (understanding rates of change and accumulation), algebra (solving equations and manipulating expressions), statistics (modeling distributions and data patterns), and applied mathematics (physics, engineering, economics applications). Both AA SL and AA HL curricula emphasize deep conceptual understanding of functions alongside computational proficiency, requiring students to work fluently with multiple representations (algebraic, graphical, numerical, verbal), recognize function families and their characteristic properties, and apply functions to authentic problem-solving contexts that prepare for university-level mathematics and quantitative disciplines.
Linear Functions and Straight Lines AA SL & HL
Forms of Straight Line Equations
Understanding the various forms of straight line equations is fundamental to working with linear functions. Each form has particular advantages depending on the information given and the problem context.
where:
m = gradient (slope) of the line
c = y-intercept (where line crosses y-axis)
Most useful when: You know the gradient and y-intercept
where a, b, d are constants
Most useful when: Working with simultaneous equations or perpendicular lines
Note: Can convert to slope-intercept: \( y = -\frac{a}{b}x - \frac{d}{b} \)
where:
(x₁, y₁) = a point on the line
m = gradient of the line
Most useful when: You know one point and the gradient
Gradient Formula
where (x₁, y₁) and (x₂, y₂) are two points on the line
Remember: "Rise over run" - change in y divided by change in x
Interpretation:
- m > 0: line rises from left to right (positive slope)
- m < 0: line falls from left to right (negative slope)
- m = 0: horizontal line (no rise)
- m undefined: vertical line (no run, x₂ = x₁)
Find the equation of the line passing through points A(2, 5) and B(6, 13).
Solution:
Step 1: Calculate gradient using \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
\( m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2 \)
Step 2: Use point-slope form with point A(2, 5)
\( y - 5 = 2(x - 2) \)
Step 3: Expand and simplify to slope-intercept form
\( y - 5 = 2x - 4 \)
\( y = 2x + 1 \)
Answer: The equation is \( y = 2x + 1 \) (or \( 2x - y + 1 = 0 \) in general form)
Parallel and Perpendicular Lines
Parallel Lines: Have equal gradients: \( m_1 = m_2 \)
Example: Lines \( y = 3x + 2 \) and \( y = 3x - 7 \) are parallel (both have gradient 3)
Perpendicular Lines: Product of gradients equals -1: \( m_1 \times m_2 = -1 \)
Equivalently: \( m_2 = -\frac{1}{m_1} \) (negative reciprocal)
Example: Lines \( y = 2x + 1 \) (m₁ = 2) and \( y = -\frac{1}{2}x + 5 \) (m₂ = -1/2) are perpendicular
Quadratic Functions AA SL & HL
General Form and Key Features
where a, b, c are constants and \( a \neq 0 \)
Graph: Parabola
If a > 0: Parabola opens upward (∪ shape) - has minimum value
If a < 0: Parabola opens downward (∩ shape) - has maximum value
c value: y-intercept (where graph crosses y-axis)
Axis of Symmetry
For quadratic \( f(x) = ax^2 + bx + c \)
This vertical line passes through the vertex (turning point)
The parabola is symmetric about this line
To find vertex coordinates:
x-coordinate: \( x = -\frac{b}{2a} \)
y-coordinate: Substitute x-value into \( f(x) \)
where vertex is at point (h, k)
Relationship to general form:
\( h = -\frac{b}{2a} \) and \( k = f(h) \)
Most useful when: You need to identify vertex quickly
Solving Quadratic Equations
Solves \( ax^2 + bx + c = 0 \) where \( a \neq 0 \)
The ± symbol means two solutions (roots):
\( x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \) and \( x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \)
Works for all quadratic equations (even when factoring is impossible)
The Discriminant
The discriminant determines the nature of roots for \( ax^2 + bx + c = 0 \)
Three cases:
- Δ > 0: Two distinct real roots (parabola crosses x-axis twice)
- Δ = 0: One repeated real root (parabola touches x-axis once at vertex)
- Δ < 0: No real roots, two complex conjugate roots (parabola doesn't touch x-axis)
Perfect Square: If Δ is a perfect square, roots are rational
Without solving, determine the nature of roots for \( 2x^2 - 5x + 3 = 0 \)
Solution:
Identify coefficients: a = 2, b = -5, c = 3
Calculate discriminant: \( \Delta = b^2 - 4ac \)
\( \Delta = (-5)^2 - 4(2)(3) \)
\( \Delta = 25 - 24 = 1 \)
Since Δ = 1 > 0: The equation has two distinct real roots
Additionally: Since 1 is a perfect square, the roots are rational (the quadratic can be factored)
Check: \( 2x^2 - 5x + 3 = (2x - 3)(x - 1) = 0 \), so \( x = \frac{3}{2} \) or \( x = 1 \) ✓
| Discriminant Value | Nature of Roots | Graph Behavior |
|---|---|---|
| Δ > 0 (positive) | Two distinct real roots | Parabola crosses x-axis at two points |
| Δ = 0 (zero) | One repeated real root (double root) | Parabola touches x-axis at vertex only |
| Δ < 0 (negative) | No real roots (two complex roots) | Parabola does not intersect x-axis |
| Δ is perfect square | Rational roots (can factor) | x-intercepts are rational numbers |
Exponential and Logarithmic Functions AA SL & HL
Exponential Functions
Connection to natural exponential:
\[ a^x = e^{x \ln a} \]where e ≈ 2.71828... (Euler's number)
Key Properties:
- Domain: all real numbers
- Range: y > 0 (always positive)
- y-intercept: (0, 1) since a⁰ = 1
- If a > 1: exponential growth (increasing)
- If 0 < a < 1: exponential decay (decreasing)
- Horizontal asymptote: y = 0
Logarithmic Functions
where a > 0, a ≠ 1, x > 0
"The logarithm base a of x equals y" means "a raised to power y equals x"
Inverse Relationship:
\[ a^{\log_a x} = x \quad \text{and} \quad \log_a(a^x) = x \]Special Logarithms:
- Common logarithm: \( \log x = \log_{10} x \) (base 10)
- Natural logarithm: \( \ln x = \log_e x \) (base e)
Product Law:
\[ \log_a(xy) = \log_a x + \log_a y \]Quotient Law:
\[ \log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y \]Power Law:
\[ \log_a(x^n) = n \log_a x \]Change of Base Formula:
\[ \log_a x = \frac{\log_b x}{\log_b a} = \frac{\ln x}{\ln a} \](a) Convert to logarithmic form: \( 3^4 = 81 \)
Solution: \( \log_3 81 = 4 \)
(b) Convert to exponential form: \( \log_2 32 = 5 \)
Solution: \( 2^5 = 32 \)
(c) Solve: \( \log_5 x = 3 \)
Solution: Convert to exponential: \( 5^3 = x \)
Therefore: \( x = 125 \)
- \( \log_a 1 = 0 \) (since \( a^0 = 1 \))
- \( \log_a a = 1 \) (since \( a^1 = a \))
- \( \log_a(a^n) = n \)
- \( a^{\log_a x} = x \)
- \( \ln e = 1 \) and \( e^{\ln x} = x \)
Domain and Range AA SL & HL
Understanding Domain and Range
Domain: The set of all possible input values (x-values) for which the function is defined
Range: The set of all possible output values (y-values) that the function can produce
Notation:
Interval notation: [a, b] includes endpoints, (a, b) excludes endpoints
Set builder notation: {x | condition on x}
| Function Type | Domain | Range |
|---|---|---|
| Linear: f(x) = mx + c | All real numbers (ℝ) | All real numbers (ℝ) |
| Quadratic: f(x) = ax² + bx + c (a > 0) | All real numbers (ℝ) | y ≥ k (where k is minimum value) |
| Quadratic: f(x) = ax² + bx + c (a < 0) | All real numbers (ℝ) | y ≤ k (where k is maximum value) |
| Square root: f(x) = √x | x ≥ 0 | y ≥ 0 |
| Rational: f(x) = 1/x | x ≠ 0 | y ≠ 0 |
| Exponential: f(x) = aˣ (a > 0) | All real numbers (ℝ) | y > 0 |
| Logarithmic: f(x) = log_a(x) | x > 0 | All real numbers (ℝ) |
- Division by zero: Denominator cannot equal zero
- Even roots: Cannot take square root (or even root) of negative number (in real numbers)
- Logarithms: Argument must be positive (cannot take log of zero or negative)
- Context restrictions: Real-world problems may impose additional constraints (e.g., time ≥ 0, distance > 0)
Composite and Inverse Functions AA SL & HL
Composite Functions
Read as "f composed with g" or "f of g of x"
Process:
- Apply function g to x (inner function)
- Apply function f to the result of g(x) (outer function)
Important: Order matters! Generally \( f(g(x)) \neq g(f(x)) \)
Given \( f(x) = 2x + 1 \) and \( g(x) = x^2 \), find:
(a) \( (f \circ g)(x) = f(g(x)) \)
Solution:
\( f(g(x)) = f(x^2) = 2(x^2) + 1 = 2x^2 + 1 \)
(b) \( (g \circ f)(x) = g(f(x)) \)
Solution:
\( g(f(x)) = g(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1 \)
Note: \( f(g(x)) \neq g(f(x)) \) in this case!
Inverse Functions
If f is a function, its inverse f⁻¹ satisfies:
\[ f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x \]Finding Inverse (algebraically):
- Replace f(x) with y
- Swap x and y
- Solve for y
- Replace y with f⁻¹(x)
Properties:
- Domain of f = Range of f⁻¹
- Range of f = Domain of f⁻¹
- Graphs of f and f⁻¹ are reflections in line y = x
Find the inverse of \( f(x) = 3x - 5 \)
Solution:
Step 1: Replace f(x) with y: \( y = 3x - 5 \)
Step 2: Swap x and y: \( x = 3y - 5 \)
Step 3: Solve for y:
\( x + 5 = 3y \)
\( y = \frac{x + 5}{3} \)
Step 4: Replace y with f⁻¹(x):
\( f^{-1}(x) = \frac{x + 5}{3} \)
Verify: \( f(f^{-1}(x)) = f\left(\frac{x+5}{3}\right) = 3\left(\frac{x+5}{3}\right) - 5 = x + 5 - 5 = x \) ✓
Function Transformations AA SL & HL
Vertical Translations:
\( f(x) + k \): shift up by k units
\( f(x) - k \): shift down by k units
Horizontal Translations:
\( f(x - h) \): shift right by h units
\( f(x + h) \): shift left by h units
Vertical Stretches/Compressions:
\( a \cdot f(x) \) where a > 1: vertical stretch by factor a
\( a \cdot f(x) \) where 0 < a < 1: vertical compression
Horizontal Stretches/Compressions:
\( f(bx) \) where b > 1: horizontal compression by factor 1/b
\( f(bx) \) where 0 < b < 1: horizontal stretch
Reflections:
\( -f(x) \): reflect in x-axis (flip vertically)
\( f(-x) \): reflect in y-axis (flip horizontally)
Interactive Quadratic Formula Calculator
Quadratic Equation Solver
Solve quadratic equations of form ax² + bx + c = 0
Study Strategies for Functions Success
Mastering Linear Functions
- Know all three forms cold: Practice converting between y=mx+c, ax+by+d=0, and point-slope form
- Visualize gradients: Understand positive/negative/zero/undefined slopes graphically
- Master parallel and perpendicular: Remember m₁ = m₂ for parallel, m₁ × m₂ = -1 for perpendicular
- Practice finding equations: From two points, from point and gradient, from graph
Mastering Quadratics
- Memorize the formulas: Quadratic formula, axis of symmetry, discriminant
- Use discriminant first: Before solving, check nature of roots
- Complete the square: Practice converting to vertex form
- Sketch graphs: Always identify vertex, axis of symmetry, y-intercept, and x-intercepts
Mastering Exponentials and Logarithms
- Understand the inverse relationship: Exponential and logarithmic are inverses
- Memorize log laws: Product, quotient, power rules are essential
- Practice conversions: Switch between exponential and logarithmic forms fluently
- Use change of base: Convert to natural logs for calculator work
Common Mistakes and How to Avoid Them
| Common Error | Correct Approach | Example |
|---|---|---|
| Confusing gradient formula order | Always use (y₂-y₁)/(x₂-x₁) consistently | For (1,3) and (4,9): m = (9-3)/(4-1) = 2, not (3-9)/(1-4) |
| Wrong sign in axis of symmetry | Remember the negative: x = -b/(2a) | For x² + 6x + 5: axis is x = -6/2 = -3, not +3 |
| Forgetting ± in quadratic formula | Always write both + and - solutions | x = (-b ± √Δ)/(2a) gives TWO roots (usually) |
| Discriminant sign errors | Calculate carefully: Δ = b² - 4ac | For x² - 4x + 5: Δ = 16 - 20 = -4 (negative!) |
| Log of negative or zero | Domain restriction: log(x) only defined for x > 0 | log(-5) is undefined in real numbers |
| Misapplying log laws | log(a+b) ≠ log(a) + log(b) | log(xy) = log(x) + log(y) but log(x+y) cannot simplify |
Real-World Applications
Linear Functions in Real Life
- Finance: Simple interest, currency conversion, cost functions
- Physics: Constant velocity motion, Ohm's law (V = IR)
- Business: Revenue models, break-even analysis, supply-demand equilibrium
- Everyday: Temperature conversion (Celsius to Fahrenheit), pricing with fixed costs
Quadratic Functions in Real Life
- Physics: Projectile motion, falling objects under gravity
- Engineering: Parabolic arch bridges, satellite dish design
- Business: Profit maximization (when cost/revenue are quadratic)
- Optimization: Finding maximum area with fixed perimeter
Exponential and Logarithmic Functions in Real Life
- Population Growth: Bacteria colonies, human populations
- Finance: Compound interest, investment growth
- Chemistry: Radioactive decay, pH calculations
- Medicine: Drug concentration over time
- Technology: Moore's Law (computing power growth)
- Acoustics: Decibel scale (logarithmic sound intensity)
Exam Preparation Checklist
- ✓ Write and recognize all three forms of straight line equations
- ✓ Calculate gradients accurately and interpret their meaning
- ✓ Find equations of parallel and perpendicular lines
- ✓ Calculate axis of symmetry for any quadratic
- ✓ Apply quadratic formula correctly (watch signs!)
- ✓ Use discriminant to determine nature of roots
- ✓ Sketch quadratic graphs with all key features labeled
- ✓ Convert fluently between exponential and logarithmic forms
- ✓ Apply all logarithm laws correctly
- ✓ Solve exponential and logarithmic equations
- ✓ Determine domain and range for various function types
- ✓ Find composite functions in correct order
- ✓ Find inverse functions algebraically
- ✓ Apply and combine function transformations
- ✓ Complete past paper questions under timed conditions
RevisionTown Resources for Functions
Expand your functions mastery with these essential RevisionTown resources:
- Functions Formulae AA HL Only - Advanced HL-exclusive topics
- Number and Algebra Formulae AI SL & AI HL - Alternative pathway algebra
- Prior Learning Formulae AA SL & AA HL - Foundation skills review
- Calculus Formulae AA SL & AA HL - Apply functions to calculus
- Quadratic Equation Calculator - Interactive solver
- Grade Calculator - Track your progress
- IB Diploma Points Calculator - Calculate final IB score
- IB Mathematics AA vs AI Guide - Choose right pathway
Technology and GDC Skills
- Graphing: Plot functions to visualize behavior, identify key features
- Table function: Generate value tables to see patterns
- Intersection: Find where graphs cross (solve equations graphically)
- Zero/Root finder: Solve equations numerically
- Maximum/Minimum: Find vertex of quadratics, optimize functions
- Trace: Explore function values at specific points
- Regression: Fit exponential or logarithmic models to data
Connecting to Other IB Math Topics
Functions integrate with virtually every other area of IB Mathematics AA:
- Calculus: Differentiate and integrate all function types studied here
- Algebra: Solve polynomial equations, work with rational expressions
- Trigonometry: Sine, cosine, tangent are functions with specific properties
- Sequences and Series: Arithmetic sequences are linear functions, geometric sequences use exponentials
- Statistics: Probability distributions are functions, regression finds best-fit function models
- Vectors: Vector equations of lines use parametric function forms
Conclusion
Mastering functions formulae is absolutely essential for success in IB Mathematics Analysis and Approaches at both Standard Level and Higher Level. The core functions concepts covered in this comprehensive guide—linear functions and straight line equations in multiple forms, gradient calculations and geometric interpretation, quadratic functions with vertex analysis and axis of symmetry, the quadratic formula and discriminant for solving and classifying roots, exponential and logarithmic functions with their inverse relationship, domain and range determination across function families, composite and inverse functions, and systematic function transformations—provide the fundamental mathematical framework upon which the entire AA curriculum is built.
Success in AA functions requires more than formula memorization. You must develop deep conceptual understanding of what functions represent (relationships between variables), fluency in moving between different representations (algebraic expressions, graphs, tables, verbal descriptions), recognition of function families and their characteristic behaviors, and the ability to select appropriate techniques for different problem types. Whether calculating a gradient between two points, determining the nature of quadratic roots using the discriminant, converting between exponential and logarithmic forms, or finding composite and inverse functions, systematic approach and careful attention to algebraic detail are paramount.
Regular practice with IB past papers, consistent work with your GDC to reinforce graphical understanding, systematic review of all formulae and their correct application contexts, and connection of abstract functions concepts to real-world applications (finance, physics, population dynamics, optimization) will build the comprehensive functions mastery necessary for top grades. Both AA SL and AA HL students benefit from complete command of these foundational topics, as they recur throughout Paper 1 and Paper 2, integrate with calculus and statistics content, and provide essential problem-solving tools.
Continue building your IB Mathematics expertise through RevisionTown's extensive collection of IB resources, practice with our interactive calculators, and connect functions knowledge to other mathematical areas. Master these essential formulae and techniques, develop strong algebraic and graphical reasoning skills, and you'll be thoroughly prepared for IB examinations and the mathematical challenges that await in university studies, STEM careers, and quantitative fields. Functions are the language of mathematics—speak it fluently!
