 # Horizontal and Vertical Graph Transformations

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Examples, solutions, videos, worksheets, and activities to help PreCalculus students learn about horizontal and vertical graph transformations.

How to graph horizontal and vertical stretches and compressions?
Vertical Stretch and Vertical Compression
y = af(x), a > 1, will stretch the graph f(x) vertically by a factor of a.
y = af(x), 0 < a < 1, will stretch the graph f(x) vertically by a factor of a.

Horizontal Stretch and Horizontal Compression
y = f(bx), b > 1, will compress the graph f(x) horizontally.
y = f(bx), 0 < b < 1, will stretch the graph f(x) horizontally.

How to graph horizontal and vertical translations?
Horizontal Shift
y = f(x + c), will shift f(x) left c units.
y = f(x - c), will shift f(x) right c units.

Vertical Shift
y = f(x) + d, will shift f(x) up d units.
y = f(x) - d, will shift f(x) down d units.

The following diagrams show horizontal and vertical transformations of functions and graphs. Scroll down the page for more examples and solutions on horizontal and vertical transformations. Function Transformations: Horizontal and Vertical Stretches and Compressions
This video explains how to graph horizontal and vertical stretches and compressions in the form a×f(b(x-c))+d.
This video looks at how a and b affect the graph of f(x).

Function Transformations: Horizontal and Vertical Translations
This video explains how to graph horizontal and vertical translation in the form a*f(b(x-c))+d. This video looks at how c and d affect the graph of f(x). Functions Transformations: A Summary
This video reviews function transformation including stretches, compressions, shifts left, shifts right, and reflections across the x and y axes. Horizontal and Vertical Graph Transformations
9 full examples as well as the basic outline of doing horizontal and vertical translations of graphs are shown.

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