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Understanding Graphical Transformations: Translations, Reflections, and Stretches

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Explore the fundamentals of graphical transformations in mathematics, including vertical and horizontal translations, reflections across axes, and both vertical and horizontal stretches and shrinks. Learn how translations shift graphs horizontally and vertically, how reflections alter the orientation of graphs, and how stretching affects their width. We will also discuss the importance of the order of transformations when combining them for accurate graph representations. Examples will be provided to illustrate each concept clearly for better understanding.

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Understanding Graphical Transformations: Translations, Reflections, and Stretches

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  1. Section 1-5 Graphical Transformations

  2. Section 1-5 • vertical and horizontal translations • reflections across the axes • vertical and horizontal stretches and shrinks • combining transformations

  3. Vertical and Horizontal Translations • horizontal translations • y = f (x – c) shifts right by c units examples: • y = f (x + c) shifts left by c units examples:

  4. Vertical and Horizontal Translations • vertical translations • y = f (x ) – c shifts down by c units examples: • y = f (x) + c shifts up by c units examples:

  5. Reflections Across Axes • reflection across the x-axis: y = – f (x) examples: • reflection across the y-axis: y = f (– x) examples:

  6. Vertical Stretches and Shrinks • a vertical stretch makes the graph narrower • a vertical shrink makes the graph wider • vertical stretch/shrink: • examples:

  7. Horizontal Stretches and Shrinks • a horizontal stretch makes the graph wider • a horizontal shrink makes the graph narrower • horizontal stretch/shrink: • examples:

  8. Combining Transformations • the order makes a difference • reflections are always applied first, but sometimes it is easier to sketch the graph by ignoring the reflection until the end • shift the graph into the correct position and then when you sketch it, make it narrower/wider and with any reflections

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