Pre-sessionals - Maths 1
Contact
- Name: Domenico Mergoni
- Email: d.mergoni -at- lse.ac.uk
- Work: London School of Economics
Tip
Internet is a great resource. Use it. Some resources I like:
Introduction to Functions and Linear Equations
Introduction to Functions
- What is a function?
- Definition: A function is a relation between a set of inputs (domain) and a set of possible outputs (range), such that each input is related to exactly one output.
- Example: Temperature conversion function.
- Exercise: Identify whether given relations are functions or not.
- Domain and Range of a function
- Definition of domain and range.
- Example: Find the domain and range of the function \(f(x) = \sqrt{x}\).
- Exercise: Determine the domain and range of a given function.
- Notation and Terminology
- Notation: \(f(x)\) represents the output of function \(f\) for input \(x\).
- Terminology: Input, output, independent variable, dependent variable, etc.
- Exercise: Translate word problems into function notation.
- What is a function?
Linear Functions
- Definition of Linear Functions
- A linear function is a function whose graph is a straight line.
- Example: \(f(x) = 2x + 3\) is a linear function.
- Exercise: Determine whether given functions are linear or not.
- Graphing Linear Functions
- Plotting points and connecting with a line.
- Example: Graph the function \(f(x) = -0.5x + 2\).
- Exercise: Graph a set of linear functions.
- Slope and \(y\)-Intercept
- Definition of slope and \(y\)-intercept.
- Calculation of slope and \(y\)-intercept from an equation.
- Example: Find the slope and \(y\)-intercept of \(f(x) = 3x - 1\).
- Exercise: Calculate slope and \(y\)-intercept of given functions.
- Definition of Linear Functions
Polynomial Functions of Degree 2
- Definition of Polynomial Functions
- A polynomial function is a function consisting of terms with non-negative integer powers.
- Example: \(f(x) = 2x^2 - 4x + 1\) is a polynomial function.
- Exercise: Identify polynomial functions among given expressions.
- Quadratic Functions
- Definition and standard form: \(f(x) = ax^2 + bx + c\).
- Example: Identify coefficients of \(a\), \(b\), and \(c\) in \(f(x) = 5x^2 - 2x + 7\).
- Exercise: Write quadratic functions in standard form.
- Graphing Quadratic Functions
- Plotting quadratic curves.
- Finding vertex and axis of symmetry.
- Example: Graph \(f(x) = x^2 - 4x + 3\) and find its vertex.
- Exercise: Graph given quadratic functions and locate their vertices.
- Definition of Polynomial Functions
Applications of Functions
- Real-World Examples of Functions
- Distance-time and temperature-time functions.
- Example: Express the height of an object in terms of time.
- Exercise: Identify functions in everyday scenarios.
- Modeling with Linear and Quadratic Functions
- Using linear functions for proportional relationships.
- Using quadratic functions for parabolic motion.
- Example: Model the height of a ball thrown vertically upward.
- Exercise: Model a real-world scenario using a linear or quadratic function.
- Simple Problems Involving Functions
- Solving basic problems using functions.
- Example: Find the time when a car reaches a certain distance.
- Exercise: Solve problems involving linear and quadratic functions.
- Real-World Examples of Functions
Conclusion and Recap
- Summarize the key concepts covered in the lecture.
- Highlight the importance of functions in various fields.
Q&A Session