Algebraic Reasoning and Functions
MYP Unit Framework
Key Concept: RELATIONSHIPS Related Concepts: Generalization. Patterns. Representation. Global Context: Scientific and Technical Innovation (How can algebraic models help us understand, predict, and optimise complex systems?) Statement of Inquiry: Algebraic functions model relationships between variables, enabling prediction, optimisation, and understanding of change.
Inquiry Questions
| Type | Question |
|---|---|
| Factual | What is a function? What is the difference between linear, quadratic, and exponential functions? How do you find the vertex of a parabola? What is a transformation of a function? |
| Conceptual | How can a mathematical EQUATION REPRESENT a real-world relationship? What makes a model 'good enough' for prediction? |
| Debatable | Can mathematical models EVER capture the full complexity of real-world systems — or do they always DISTORT reality in dangerous ways? Should we base policy decisions (e.g., interest rates, climate targets) on mathematical models? |
1. Functions — The Fundamental Idea
What Is a Function?
'A function is a RELATIONSHIP between two sets — the INPUT (domain) and the OUTPUT (range) — where EACH input has EXACTLY ONE output. A function is like a MACHINE: you put something in, and the machine transforms it into something else according to a fixed RULE.'
f(x) notation: 'f(x) is read as "f of x." It means "the value of the function f when the input is x." If f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11.'
Domain and Range
- Domain: The set of all POSSIBLE inputs. 'For f(x) = 1/x, the domain is ALL REAL NUMBERS EXCEPT 0 — because division by zero is UNDEFINED.'
- Range: The set of all POSSIBLE outputs. 'For f(x) = x², the range is ALL NON-NEGATIVE real numbers — because a square can NEVER be negative.'
Function Families
'Functions can be organised into FAMILIES based on their GENERAL FORM. Each family has DISTINCTIVE properties — shape, behaviour, key features.'
2. Linear Functions — Constant Rate of Change
The Form: f(x) = mx + c
'm' is the GRADIENT (slope) — the RATE OF CHANGE. 'c' is the Y-INTERCEPT — the value of f(x) when x = 0.
Rate of Change
'The gradient tells you how MUCH y changes for EVERY ONE UNIT increase in x. m = Δy/Δx = (y₂ − y₁)/(x₂ − x₁). A POSITIVE gradient means y INCREASES as x increases. A NEGATIVE gradient means y DECREASES as x increases.'
Real-World Applications
'Linear functions model CONSTANT RATE OF CHANGE. Examples: Distance = speed × time (if speed is constant). Cost = fixed cost + (variable cost × quantity). Converting between Celsius and Fahrenheit: F = 1.8C + 32.'
3. Quadratic Functions — Curved Change
The Form: f(x) = ax² + bx + c
'The graph of a quadratic function is a PARABOLA — a symmetric U-shape (or upside-down U-shape).'
Key features:
- Vertex: The TURNING POINT — maximum or minimum. 'The vertex is at x = −b/(2a). The y-coordinate of the vertex is f(−b/(2a)).'
- Axis of symmetry: The vertical line through the vertex: x = −b/(2a).
- Y-intercept: The point where the parabola crosses the y-axis: (0, c).
- X-intercepts (roots): The solutions to ax² + bx + c = 0. Found using FACTORISATION, the QUADRATIC FORMULA: x = [−b ± √(b² − 4ac)]/(2a), or COMPLETING THE SQUARE.
The Discriminant: D = b² − 4ac
'The discriminant tells you how MANY real roots the quadratic has:
- D > 0: TWO distinct real roots. The parabola crosses the x-axis TWICE.
- D = 0: ONE repeated real root. The parabola TOUCHES the x-axis at one point.
- D < 0: NO real roots. The parabola NEVER crosses the x-axis.'
Real-World Applications
'Quadratic functions model ACCELERATION. The height of a projectile: h(t) = −4.9t² + v₀t + h₀ (on Earth). Profit = −ax² + bx + c (finding the price that MAXIMISES profit). Area optimisation: what dimensions give the MAXIMUM area for a given perimeter?'
4. Exponential Functions — Growth and Decay
The Form: f(x) = a · bˣ
'a' is the INITIAL VALUE (when x = 0). 'b' is the GROWTH (b > 1) or DECAY (0 < b < 1) factor.
Key Difference from Linear and Quadratic
'Linear functions change by a CONSTANT AMOUNT. Quadratic functions change by a CONSTANTly CHANGING amount. EXPONENTIAL functions change by a CONSTANT PERCENTAGE. This is the MOST POWERFUL pattern in mathematics — because it produces EXPLOSIVE growth or RAPID decay.
Example: 'A population of 100 bacteria, doubling every hour: f(t) = 100 × 2ᵗ. After 1 hour: 200. After 2 hours: 400. After 10 hours: 102,400. After 24 hours: 1.6 BILLION.'
Real-World Applications
- Population growth: f(t) = P₀ × eʳᵗ
- Compound interest: A = P(1 + r/n)ⁿᵗ
- Radioactive decay: N(t) = N₀ × e^(−λt)
- Viral spread: Early stages of a pandemic follow exponential growth
5. Transformations of Functions
Moving and Stretching Graphs
'Understanding transformations allows you to sketch ANY function by starting from a BASIC parent function and applying SHIFTS, REFLECTIONS, and STRETCHES.'
| Transformation | Effect on graph | Formula |
|---|---|---|
| Vertical shift | Moves the graph UP or DOWN | f(x) + k |
| Horizontal shift | Moves the graph LEFT or RIGHT | f(x − h) |
| Reflection in x-axis | Flips the graph VERTICALLY | −f(x) |
| Reflection in y-axis | Flips the graph HORIZONTALLY | f(−x) |
| Vertical stretch/compression | Makes graph TALLER or SHORTER | a·f(x) |
| Horizontal stretch/compression | Makes graph WIDER or NARROWER | f(b·x) |
Composite Transformations
'When MULTIPLE transformations are applied, the ORDER MATTERS. The general transformed function is: y = a·f(b(x − h)) + k.'
6. Modelling with Functions
The Modelling Cycle
- Identify the problem and the variables.
- Choose an appropriate function family based on the pattern of change.
- Use DATA to find the PARAMETERS of the function.
- Verify the model — does it fit the data REASONABLY well?
- Use the model to PREDICT or OPTIMISE.
- Evaluate the model's LIMITATIONS.
Choosing the Right Model
'Look at the PATTERN of change: Constant difference → LINEAR. Constant second difference → QUADRATIC. Constant ratio → EXPONENTIAL.
'But real-world data is MESSY. No model is PERFECT. The question is not "Is the model TRUE?" — it is "Is the model USEFUL?"'
Your Summative Assessment — The Modelling Project
Task: Choose a REAL-WORLD PHENOMENON that can be modelled by a linear, quadratic, or exponential function. Collect or obtain DATA. Find the equation of the BEST-FITTING function using algebraic methods. Use your model to make a PREDICTION. Analyse: How WELL does your model fit the data? What are the LIMITATIONS of your model? What are the IMPLICATIONS of using your model to make decisions? Write a STRUCTURED REPORT (1000–1200 words) with an introduction, methodology, results, discussion, and conclusion.
'This project MIRRORS the IB DP Mathematics Internal Assessment (IA). Developing these skills NOW will give you a SIGNIFICANT advantage in the Diploma Programme.'
ATL Skills
| Skill | Focus |
|---|---|
| Critical Thinking | Evaluating model validity. Distinguishing between correlation and causation. |
| Research | Collecting and analysing real-world data. |
| Communication | Writing a structured mathematical report with clear explanations. |
| Information Literacy | Using technology (graphing calculators, spreadsheets) to analyse data. |
Formative Assessments
| Assessment | Focus |
|---|---|
| Function identification | Given a table of values, identify the function family and find the equation. |
| Graph sketching | Sketch the graphs of f(x), f(x) + 3, f(x − 2), and 2f(x) for a given function. |
| Quadratic problem set | Solve a set of problems involving finding roots, vertex, and y-intercept. |
| Exponential applications | Calculate compound interest, population growth, and radioactive decay problems. |
Interdisciplinary Connections
- Physics: Kinematics (quadratic), radioactive decay (exponential).
- Economics: Supply and demand, compound interest, profit optimisation.
- Biology: Population growth models, bacterial growth, drug concentration.
- TOK: Can a mathematical model be 'true'? What is the relationship between mathematics and reality?
Service as Action
- Tutoring: Tutor younger students in algebraic concepts.
- Data analysis for a local NGO: Help a community organisation analyse their data (e.g., donor trends, service usage).
- Financial literacy workshop: Design and deliver a workshop on compound interest and saving for fellow students.
IB Learner Profile Attributes
| Attribute | How This Unit Develops It |
|---|---|
| Thinkers | Students analyse problems, identify patterns, and construct mathematical models. |
| Inquirers | Students explore real-world phenomena through mathematical investigation. |
| Knowledgeable | Students build deep understanding of function families and their properties. |
| Reflective | Students evaluate the strengths and limitations of their models. |
Self-Test Questions
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What is a function? Explain the difference between domain and range.
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A line passes through (2, 5) and (6, 13). Find the equation of the line.
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For f(x) = 2x² − 8x + 6, find: the vertex, the axis of symmetry, and the roots.
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The population of a town grows from 10,000 to 12,100 in two years. Assuming exponential growth, find the annual growth rate and predict the population after 5 years.
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Explain how the graph of f(x) = x² is transformed to g(x) = −(x − 3)² + 4.
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Describe the MODELLING CYCLE and apply it to a real-world example of your choice.
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'A model is never right, but sometimes it is useful.' Explain what this statement means and give an example.
