Chapter 1 Functions 【1.1 Functions - Part 3 Graph of Functions】中文讲解 STPM Maths T Sem 1

Introduction

This tutorial focuses on understanding the graph of functions, specifically addressing reciprocal functions as part of the STPM Mathematics Semester 1 syllabus. By the end of this guide, you will grasp how to graph reciprocal functions and interpret their characteristics, enabling you to tackle related problems in mathematics.

Step 1: Understanding Functions

• A function is a relationship where each input has a unique output.
• The graph of a function is a visual representation of these relationships.
• Familiarize yourself with basic function types, including linear, quadratic, and reciprocal functions.

Step 2: Introduction to Reciprocal Functions

• A reciprocal function is defined as ( f(x) = \frac{1}{x} ).
• This function is characterized by:
  • A vertical asymptote at ( x = 0 ) (the function is undefined at zero).
  • A horizontal asymptote at ( y = 0 ) (the function approaches zero as ( x ) approaches infinity).

Step 3: Sketching the Graph of a Reciprocal Function

1. Identify Key Features:

   • Vertical asymptote at ( x = 0 ).
   • Horizontal asymptote at ( y = 0 ).
   • The function will never touch the axes.

2. Plot Points:

   • Choose values for ( x ) to calculate corresponding ( y ) values:
     • For ( x = 1 ), ( f(1) = 1 )
     • For ( x = -1 ), ( f(-1) = -1 )
     • For ( x = 2 ), ( f(2) = 0.5 )
     • For ( x = -2 ), ( f(-2) = -0.5 )

3. Draw the Graph:

   • Plot the points on a Cartesian plane.
   • Draw curves approaching the asymptotes.
   • Ensure the curves are in the first and third quadrants for positive and negative values of ( x ).

Step 4: Analyzing the Graph

• Observe the behavior of the function as ( x ) approaches the asymptotes.
• Note that as ( x ) gets larger, ( y ) gets closer to zero.
• Understand that the function never crosses the axes, reinforcing the concept of asymptotes.

Step 5: Practice Problems

• To reinforce learning, solve problems related to other reciprocal functions by altering the basic form:
  • ( f(x) = \frac{1}{x - a} ) introduces a horizontal shift.
  • ( f(x) = \frac{a}{x} ) scales the function vertically.

Conclusion

Reciprocal functions are foundational in understanding more complex mathematical concepts. By mastering their graphs and characteristics, you can effectively approach various functions in your studies. Practice sketching and analyzing different forms of reciprocal functions to solidify your understanding and prepare for future topics in mathematics. For further learning, consider exploring the related videos for more in-depth discussions on different types of functions.