Transformations of Trigonometric Graphs

Introduction

In this lecture, we cover the transformations of trigonometric graphs. Before watching this, it’s helpful to review other videos on:

• Transformations of graphs (4 transformations needed at GCSE level)
• Trigonometric graphs: y = sin(x), y = cos(x), and y = tan(x)

Recap of Transformations

Reflections

• y = -f(x): Reflects the graph in the x-axis. Points above x-axis go below and vice versa. Points on the x-axis remain invariant.
• y = f(-x): Reflects the graph in the y-axis. Points on the right move to the left and vice versa. Points on the y-axis remain invariant.

Translations

• y = f(x) + a (outside the brackets): Moves the graph a units upwards if a > 0, downwards if a < 0.
• y = f(x + a): Moves the graph a units to the left.

Recap of Trig Graphs

y = sin(x)

• Sketched between 0 and 360 degrees
• Starts at the origin: 0,0 -> 90°,1 -> 180°,0 -> 270°,-1 -> 360°,0

y = cos(x)

• Starts at 0,1 -> 90°,0 -> 180°,-1 -> 270°,0 -> 360°,1

y = tan(x)

• Has asymptotes where the graph never reaches (e.g., 90°, 270°)
• Starts at the origin, curves upwards to the first asymptote, continues through this pattern

Examples of Transformations

Example 1: y = -sin(x)

• Reflects the sine graph in the x-axis.
• Key Points:
  • 0° remains 0°
  • 90°,1 reflects to 90°,-1
  • -90°,-1 reflects to -90°,1

Example 2: y = sin(-x)

• Reflects the sine graph in the y-axis.
• Key Points:
  • 0° remains 0°
  • 90°,1 reflects to -90°,1
  • -90°,-1 reflects to 90°,-1
  • 180°,0 reflects to -180°,0

Example 3: y = cos(x) + 1

• Moves cos(x) graph one unit upwards.
• Key Points:
  • 0,1 moves to 0,2
  • 90°,0 moves to 90°,1
  • 180°,-1 moves to 180°,0
  • 270°,0 moves to 270°,1
  • 360°,1 moves to 360°,2

Example 4: y = cos(x) - 3

• Moves cos(x) graph three units downwards.
• Key Points:
  • 0,1 moves to 0,-2
  • 90°,0 moves to 90°,-3
  • 180°,-1 moves to 180°,-4
  • 270°,0 moves to 270°,-3
  • 360°,1 moves to 360°,-2

Example 5: y = cos(x + 180°)

• Moves cos(x) graph 180° to the left.
• Key Points:
  • 180°,-1 moves to 0°,-1
  • 0,1 moves to -180°,1 (adjust other points similarly)
  • 270°,0 moves to 90°,0
  • Continue pattern to sketch correct curve

Example 6: y = cos(x - 90°)

• Moves cos(x) graph 90° to the right.
• Key Points:
  • 0,1 moves to 90°,1
  • 90°,0 moves to 180°,0
  • 180°,-1 moves to 270°,-1
  • 270°,0 moves to 360°,0

Example 7: y = -tan(x)

• Reflects tan(x) graph in the x-axis.
• Key Points:
  • Points on the x-axis remain 0
  • Points above x-axis go below and vice versa
  • Sketch the reflected curves through these points

Summary

• Know your 4 transformations: vertical reflection, horizontal reflection, vertical translation, and horizontal translation.
• Familiarize yourself with the key points of the basic trig graphs.
• When applying transformations, focus on how these key points move.

Watching the recap videos again could be helpful.