Course Outline
Unit 1: Understanding Quadratic Functions
Students are introduced to quadratic functions and the characteristic shape of their graphs. They learn to recognise quadratic relationships in equations, tables, and graphs and identify the important features of a parabola.
Subtopics:
- What is a Quadratic Function?
- Features of a Parabola
- Intercepts
- Domain and Range
- Graphing Quadratic Functions
Key Concepts:
- Quadratic functions and degree
- Parabolas
- Vertices and axes of symmetry
- Maximum and minimum values
- x-intercepts and y-intercepts
- Domain and range
- Tables of values and graph construction
Unit 2: Forms of Quadratic Equations
Students explore the three main forms of quadratic functions and learn how each form reveals different graphical and algebraic information. They practise converting between forms and selecting the most useful representation for a given problem.
Subtopics:
- Standard Form
- Vertex Form
- Factored Form
- Converting Between Forms
- Comparing Different Forms
Key Concepts:
- Standard form: (y=ax^2+bx+c)
- Vertex form: (y=a(x-h)^2+k)
- Factored form: (y=a(x-r_1)(x-r_2))
- Coefficients and graph features
- Roots, intercepts, and vertices
- Expanding, factoring, and completing the square
- Equivalent quadratic expressions
Unit 3: Solving Quadratic Equations
Students develop a range of methods for solving quadratic equations. They learn to choose an efficient method, verify their solutions, and connect algebraic solutions to the x-intercepts of a graph.
Subtopics:
- Solving by Factoring
- Solving by Completing the Square
- Solving Using the Quadratic Formula
- The Discriminant
- Choosing an Appropriate Method
Key Concepts:
- Zero Product Property
- Factoring quadratic expressions
- Completing the square
- Quadratic formula
- Exact and approximate solutions
- Repeated and distinct roots
- The discriminant
- Connections between roots and graphs
Unit 4: Transformations of Quadratic Functions
Students investigate how changes to the equation of a quadratic function transform its graph. They learn to describe, predict, and graph translations, reflections, stretches, and compressions.
Subtopics:
- Vertical Translations
- Horizontal Translations
- Reflections
- Stretching and Compressing
- Combining Transformations
Key Concepts:
- Parent function (y=x^2)
- Horizontal and vertical translations
- Reflections across the x-axis
- Vertical stretches and compressions
- The effect of coefficient (a)
- Transformations in vertex form
- Combining multiple transformations
Unit 5: Applications of Quadratic Functions
Students apply quadratic functions to practical situations and mathematical models. They use graphs, equations, and technology to solve problems involving motion, area, maximum and minimum values, and real-world data.
Subtopics:
- Modelling Real-World Situations
- Projectile Motion
- Optimisation Problems
- Quadratic Modelling with Technology
- Review and Problem Solving
Key Concepts:
- Creating quadratic models
- Interpreting variables and parameters
- Projectile paths
- Maximum height and time of flight
- Area and optimisation problems
- Maximum and minimum values
- Quadratic regression
- Model limitations and predictions
- Multi-step problem solving
Skills Developed
Throughout the course, students will develop their ability to:
- Recognise and represent quadratic relationships.
- Interpret the features of parabolic graphs.
- Convert between standard, vertex, and factored forms.
- Solve quadratic equations using multiple methods.
- Select efficient strategies for different problems.
- Describe and graph transformations.
- Use quadratic functions to model practical situations.
- Interpret solutions within real-world contexts.
- Use graphing technology effectively.
- Communicate mathematical reasoning clearly and accurately.