STPM Mathematics T Functions Overview

Overview

This lecture covers STPM Mathematics T functions, focusing on types of basic graphs, domain and range, and tips for graph sketching, with a brief mention of upcoming lessons and class logistics.

Class and Lecture Logistics

• Mathematics T STPM lessons are held every Saturday at 4 pm via Academy YouTuber.
• Certificate link is shared at the end of class and is active for 30 minutes; password is "LIMAU".
• Students must use their own email to access certificates.
• Use the chat appropriately and keep attention during lessons.

Introduction to Functions in Form 6

• Functions in Form 6 are divided into four parts: basic functions, exponential/logarithmic functions, trigonometric functions, and polynomial/rational functions.
• Key subtopics: inverse function, composite function, one-to-one (injective) functions, and sketching graphs.
• Questions often mix functions, requiring understanding of their interactions (e.g., composite and inverse in a single question).

Graph Sketching: Basics and Types

• Sketching graphs is critical for finding domain and range and determining one-to-one functions.
• Main types of basic graphs: identity (straight line), quadratic (smile/frown), cubic, modulus (V-shape), reciprocal, square root, reciprocal square, and transformations.
• Always label axes, intersection points, and the function itself on sketches for full exam marks.

Determining Domain and Range

• Domain: possible x-values (horizontal extent of the graph).
• Range: possible y-values (vertical extent of the graph).
• Use interval notation (e.g., (-∞, ∞)) or set notation; check question requirements.
• For most basic graphs:
  • Straight line/cubic: domain and range are both (-∞, ∞).
  • Quadratic (y = x²): domain (-∞, ∞), range [0, ∞).
  • Modulus: domain (-∞, ∞), range [0, ∞).
  • Reciprocal: domain and range are (-∞, 0) ∪ (0, ∞), zero not included due to asymptotes.
  • Square root: domain [0, ∞), range [0, ∞).
• For transformed graphs, vertical/horizontal shifts change the starting points in domain/range.](streamdown:incomplete-link)

Transformation of Graphs

• f(x) + a: shifts graph up by a units.
• f(x) - a: shifts graph down by a units.
• f(x + a): shifts graph left by a units.
• f(x - a): shifts graph right by a units.

Exam Tips for Graph Sketching

• Curve shape (1 mark), intersection/details (1 mark), and correct labeling (1 mark) are required for full marks.
• Never let your reciprocal graph touch the axes/asymptotes.
• Label all critical points and function names.

Key Terms & Definitions

• Function — a relation assigning each input exactly one output.
• Inverse Function — reverses the action of a function.
• Composite Function — combining two functions, written as (f∘g)(x) = f(g(x)).
• Domain — set of all possible input (x) values.
• Range — set of all possible output (y) values.
• Asymptote — a line the graph approaches but never touches.
• Transformation — shifting or reflecting a graph.

Action Items / Next Steps

• Practice sketching basic graphs, labeling axes, and finding domain/range.
• Try the 2017 STPM function question before next class.
• Join the next lesson for exponential, logarithmic, and trigonometric graph sketches.
• Fill out the certificate form using your email within 30 minutes of class end.