Transformations of Trigonometric Graphs

Jun 22, 2024

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.