Edexcel IAL - Pure Maths 3- 1.4 Combinations of transformations- Study notes - New syllabus
Edexcel IAL – Pure Maths 3- 1.4 Combinations of transformations -Study notes- New syllabus
Edexcel IAL – Pure Maths 3- 1.4 Combinations of transformations -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- 1.4 Combinations of transformations
Combinations of Transformations of Graphs
Starting from the graph of \( y = f(x) \), new graphs can be obtained by applying one or more transformations.
The syllabus requires understanding and combining the following transformations:
- \( y = af(x) \)
- \( y = f(x) + a \)
- \( y = f(x + a) \)
- \( y = f(ax) \)
Individual Transformations (Recap)
| Form | Effect on Graph |
| \( y = af(x) \) | Vertical stretch if \( |a|>1 \), compression if \( 0<|a|<1 \); reflection if \( a<0 \) |
| \( y = f(x) + a \) | Vertical translation up by \( a \) units |
| \( y = f(x + a) \) | Horizontal translation left by \( a \) units |
| \( y = f(ax) \) | Horizontal compression by factor \( a \) if \( a>1 \); stretch if \( 0<a<1 \) |
Order of Transformations
When more than one transformation is applied, the order is important.
Recommended order:
- Transform inside the function first (horizontal changes)
- Then transform outside the function (vertical changes)
For example, in \( y = 2f(3x) \):
- Apply \( y = f(3x) \) first (horizontal compression)
- Then apply \( y = 2f(x) \) (vertical stretch)
Even and Odd Reflections
- \( y = f(-x) \) reflects the graph in the y-axis
- \( y = -f(x) \) reflects the graph in the x-axis
Trigonometric Transformations
Transformations apply in the same way to trigonometric functions.
- Amplitude changes come from coefficients outside the function
- Period changes come from coefficients inside the function
- Phase shifts come from additions inside the function
Important Syllabus Note
The graph of \( y = f(ax + b) \) is not required.
Only transformations of the form:
\( y = af(x),\quad y = f(x) + a,\quad y = f(x + a),\quad y = f(ax) \)
and their combinations are examined.
Example
Describe the transformations from \( y = f(x) \) to:
\( y = 2f(3x) \)
▶️ Answer / Explanation
- \( y = f(3x) \): horizontal compression by factor 3
- \( y = 2f(x) \): vertical stretch by factor 2
Result: Compress horizontally, then stretch vertically.
Example
Describe the transformations from \( y = f(x) \) to:
\( y = f(-x) + 1 \)
▶️ Answer / Explanation
- \( y = f(-x) \): reflection in the y-axis
- \( y = f(x) + 1 \): translation up by 1 unit
Result: Reflect in y-axis, then move up.
Example
Sketch the graph of:
\( y = 3 + \sin 2x \)
▶️ Answer / Explanation
- \( y = \sin 2x \): horizontal compression by factor 2
- \( y = \sin x + 3 \): translation up by 3 units
Amplitude remains 1, midline is \( y = 3 \).