Linear and Quadratic Functions

MYP Unit Framework

Key Concept: RELATIONSHIPS Related Concepts: Generalization, Patterns, Representation Global Context: Scientific and Technical Innovation (How do mathematical models enable prediction and technological design?) Statement of Inquiry: Functions model real-world patterns and enable prediction and problem-solving.

Inquiry Questions

ATL Skills

• Thinking: Identify patterns and generalise relationships; evaluate models for their fit to real data
• Communication: Interpret and create graphs; communicate mathematical reasoning clearly
• Research: Collect real-world data and find functions that model it
• Self-Management: Organise work systematically; meet deadlines for investigations

1. Linear Functions

The General Form

A linear function has the form y = mx + c, where:

• m is the gradient (slope) — measures steepness
• c is the y-intercept — where the line crosses the y-axis

Gradient

Gradient = change in y / change in x = (y₂ - y₁) / (x₂ - x₁)

• Positive gradient: line slopes upward (left to right)
• Negative gradient: line slopes downward (left to right)
• Zero gradient: horizontal line
• Undefined gradient: vertical line

Finding the Equation of a Line

Given gradient and a point: y - y₁ = m(x - x₁)

Given two points: First calculate gradient, then use one point.

Sketching Linear Graphs

1. Identify the y-intercept (c)
2. Use the gradient to find another point (rise/run)
3. Draw the straight line through both points

Special Cases

• Horizontal lines: y = k (gradient = 0)
• Vertical lines: x = k (gradient undefined)
• Lines through the origin: y = mx (c = 0)

2. Systems of Linear Equations

Solving two linear equations simultaneously means finding the point (x, y) that satisfies BOTH equations.

Methods of Solution

Graphical Method: Plot both lines; the intersection point is the solution.

Substitution Method: Rearrange one equation to express one variable in terms of the other. Substitute into the second equation.

Elimination Method: Add or subtract equations to eliminate one variable.

Number of Solutions

• One solution: Lines intersect at one point (different gradients)
• No solution: Lines are parallel (same gradient, different intercept)
• Infinite solutions: Lines are coincident (same line)

3. Quadratic Functions

The General Form

A quadratic function has the form y = ax² + bx + c, where a does not equal 0.

Key Features of a Quadratic Graph (Parabola)

• Shape: U-shaped (if a > 0) or inverted U-shaped (if a < 0)
• Vertex: The turning point — minimum (a > 0) or maximum (a < 0)
• Axis of Symmetry: Vertical line through the vertex: x = -b / (2a)
• y-intercept: Point where x = 0 (this is c)
• x-intercepts: Points where y = 0 (roots or zeros)

Vertex Form

y = a(x - h)² + k, where (h, k) is the vertex.

Completing the square converts the general form to vertex form.

Factored Form

y = a(x - p)(x - q), where p and q are the x-intercepts.

4. Solving Quadratic Equations

Method 1: Factorisation

Express the quadratic as a product of two linear factors.

Example: x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0 → x = -2 or x = -3

Method 2: Quadratic Formula

x = [-b +/- sqrt(b² - 4ac)] / 2a

The discriminant (D = b² - 4ac) determines the number of solutions:

• D > 0: Two distinct real roots
• D = 0: One repeated real root
• D < 0: No real roots (complex roots)

Method 3: Completing the Square

Useful for finding the vertex and solving equations.

Example: x² + 6x + 5 = (x + 3)² - 4

Method 4: Graphical Solution

Find the x-intercepts of y = ax² + bx + c.

5. Applications of Functions

Linear Applications

• Distance-time relationships: Distance = speed x time
• Cost-revenue models: Total cost = fixed cost + (variable cost per unit x units)
• Temperature conversion: F = 9/5 C + 32
• Currency exchange: Amount in foreign currency = exchange rate x amount in local currency

Quadratic Applications

• Projectile motion: Height = -5t² + vt + h (under gravity)
• Area optimisation: Maximising area for a given perimeter
• Profit maximisation: Revenue - cost as a quadratic function
• Bridge design: Parabolic arches and suspension cables

Real-World Investigation

Collect data from a real situation (e.g., dropping a ball from different heights, measuring bounce height). Plot the data and determine whether a linear or quadratic model is more appropriate.

6. Transformations of Functions

Translations

y = f(x) + k shifts the graph UP by k units y = f(x + h) shifts the graph LEFT by h units

Reflections

y = -f(x) reflects across the x-axis y = f(-x) reflects across the y-axis

Stretches

y = af(x) vertical stretch by factor of a y = f(bx) horizontal compression by factor of 1/b

Summative Assessment

Task: Mathematical investigation (800-1000 words equivalent) applying linear and quadratic functions to a real-world context.

Criteria:

• A: Knowing and Understanding — Select and apply mathematical procedures correctly
• B: Investigating Patterns — Identify patterns, generalise relationships, and justify conclusions
• C: Communicating — Use mathematical language, notation, and representations clearly
• D: Applying Mathematics in Real-World Contexts — Apply functions to a real-world situation; evaluate the model's limitations

Option 1: Investigate the relationship between the drop height and bounce height of a ball. Model with a linear function. Discuss limitations.

Option 2: Model the profit of a business given fixed and variable costs and a demand function. Find the production level that maximises profit.

Option 3: Investigate a parabolic shape in the real world (e.g., a bridge arch, a fountain). Determine its quadratic equation and discuss applications.

Formative Assessment

• Skills practice: finding equations of lines, solving quadratics
• Graphing activities: using technology (Desmos, GeoGebra) to explore transformations
• Real-world problem sets: applying functions to authentic scenarios
• Peer assessment: evaluating each other's solutions and reasoning
• Quick quizzes: gradient, intercept, vertex, discriminant

Interdisciplinary Connections

• Physics: Projectile motion — quadratic functions model height over time
• Economics: Supply and demand; break-even analysis; profit maximisation
• Engineering: Structural design — parabolic arches, cable-stayed bridges
• Geography: Population growth models; linear and exponential models

Service as Action

• Financial Literacy Workshop: Create resources for younger students about budgeting, using linear models to track income and expenses.
• Community Mapping: Survey and graph data about a local issue (traffic flow, park usage) and present findings to the local council.

IB Learner Profile

• Inquirers: Investigate patterns in the world and express them mathematically
• Thinkers: Apply logical reasoning to solve problems and evaluate models
• Communicators: Express mathematical ideas clearly using multiple representations
• Reflective: Consider the limitations of mathematical models and their assumptions

Self-Test

1. What is the gradient of the line passing through (1, 3) and (4, 11)?
2. Find the equation of a line with gradient 2 passing through (3, 7).
3. Solve the system: 2x + y = 7 and x - y = 2.
4. What is the vertex of y = x² - 4x + 5?
5. Solve x² - 7x + 10 = 0 by factorisation.
6. What does the discriminant tell you about a quadratic equation?
7. Write the quadratic formula.
8. Convert y = x² + 6x + 11 to vertex form.
9. Describe how the graph of y = x² changes to become y = (x - 3)² + 2.
10. Give a real-world example of a quadratic relationship and explain why it is quadratic.

This unit aligns with IB MYP Mathematics guide, developed for Year 4 (Class 9) students.