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CIE A-Level Maths Study Notes

1.2.6 Transformations of Graphs

Graph transformations are a pivotal concept in offering a window into the dynamic nature of functions. This section provides an in-depth exploration of various graph transformations, including vertical and horizontal translations, stretches, compressions, reflections, and their combinations. These transformations are key to understanding how alterations in a function's equation reflect on its graph.

Graph Transformations

1. Translation

  • Vertical Translation:
    • Upward: y=f(x)+cy = f(x) + c (Graph moves up)
    • Downward: y=f(x)cy = f(x) - c (Graph moves down)
vertical translation

Image courtesy of cuemath

  • Horizontal Translation:
    • Rightward: y=f(xc)y = f(x - c) (Graph moves right)
    • Leftward: y=f(x+c)y = f(x + c) (Graph moves left)
horizontal translation

Image courtesy of cuemath

2. Stretch

  • Vertical Stretch:
    • y=af(x)y = af(x) (Stretches vertically)
vertical sketch

Image courtesy of openlibrary

  • Horizontal Stretch:
    • y=f(xa)y = f(\frac{x}{a}) (Stretches horizontally)
horizontal sketch

Image courtesy of openlibrary

3. Reflection

  • Over X-axis:
    • y=f(x)y = -f(x) (Flips over x-axis)
  • Over Y-axis:
    • y=f(x)y = f(-x) (Flips over y-axis)
Reflection illustration

Image courtesy of lumen

Examples

Example 1:

Transform y=x2y = x^2 by vertically stretching by 2, and translating 3 units up and 2 units right.

Solution:

  • New function: y=2(x2)2+3y = 2(x - 2)^2 + 3
  • The graph is stretched, moved right and up.
transformations of graphs

The blue curve represents the original quadratic function y=x2y = x^2. It is stretched vertically by a factor of 2, and translating 3 units up and 2 units to the right, then the red curve would represent the function y=2(x2)2+3y = 2(x - 2)^2 + 3.

Example 2:

Transform y=xy = \sqrt{x} by reflecting over the x-axis and stretching horizontally by 2.

Solution:

  • New function: y=x2y = -\sqrt{\frac{x}{2}}
  • The graph is reflected and stretched horizontally.
transformations of graphs
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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