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.