MATHEMATICS LAB// LAB-2//SHIFTING OF GRAPHS

Introduction

This tutorial provides a step-by-step guide on how to shift graphs in mathematics, as demonstrated in the video "MATHEMATICS LAB// LAB-2//SHIFTING OF GRAPHS" by Albin Kattakada. Understanding how to shift graphs is essential for visualizing functions and understanding transformations in mathematics. This guide will help you grasp the concepts of horizontal and vertical shifts effectively.

Step 1: Understanding the Basic Function

• Begin with a basic function, such as f(x) = x² (a simple quadratic function).
• Plot the original graph of the function on a coordinate system.
• Note the vertex point and the general shape of the graph (a parabola opening upwards).

Step 2: Performing Vertical Shifts

• To shift the graph vertically, you will modify the function by adding or subtracting a constant.
• Example:
  • For an upward shift, use f(x) = x² + k (where k > 0).
  • For a downward shift, use f(x) = x² - k (where k > 0).
• Plot the new graphs and observe how the vertex moves up or down based on the value of k.

Step 3: Performing Horizontal Shifts

• To shift the graph horizontally, you will alter the input variable (x).
• Example:
  • For a rightward shift, use f(x) = (x - h)² (where h > 0).
  • For a leftward shift, use f(x) = (x + h)² (where h > 0).
• Again, plot the new graphs and see how the vertex moves left or right according to the value of h.

Step 4: Combining Shifts

• You can combine both vertical and horizontal shifts in a single function.
• Example:
  • Combine shifts as follows: f(x) = (x - h)² + k.
• Analyze how the graph changes with both shifts and plot accordingly.

Step 5: Real-World Applications

• Understanding graph shifts is crucial for various applications, including physics (motion graphs), economics (cost functions), and engineering (design models).
• Practice by applying shifts to different types of functions, such as linear, exponential, and trigonometric functions, to see how shifts affect their graphs.

Conclusion

Shifting graphs is a fundamental concept in mathematics that enhances your understanding of functions and their transformations. By mastering vertical and horizontal shifts, as well as combining them, you can analyze and interpret graphical data more effectively. Next, consider exploring different types of functions or practicing the shifts with real-world data for deeper insights.