Richard Hsu
Victor Jitlin
Arthur Krumins
Mathematics 309 - the elementary geometry of wave motion
Part I - scaling and shifting (note that this section is an edited and expanded version of prof. Casselman's page) One way of obtaining one graph from
another is by scaling and shifting either x or y or both, so
that the graph y=f(x) becomes the graph y = af(bx + c) + d.
The way to see what happens to the graph is to understand what happens in
simple steps with just one change at a time. The Blue text shows general principles.
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y = f(x)
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y = 2f(x) All y-values on the graph are scaled
by 2. In subsituting cf for f, vertical
distances are scaled by c. |
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y = f(x)
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y = f(x)+1 1 is added to all y values. In subsituting f(x) + c for f(x), The graph is moved up by c.
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y = f(x)
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y = f(2x) The height at x on the new graph is
equal to the height at 2x on the old one. The new
graph is obtained by compressing the old one horizontally by
2. In substituting cx for x, horizontal
distances are scaled by 1/c. |
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y = f(x)
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y = f(x+1) The height at x on the new graph
is equal to the height at x+1 on the old one. The new
graph is obtained by shifting the old one 1 to the left. In substituting x+c for x, the graph is shifted horizontally c to the left.
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y = f(x) |
y = f(2x-1)+1 |
Only the last one is tricky, since it involves a sequence of
substitutions. In order to see which ones, we unravel the process.
- The function f(2x-1)+1 is obtained from f(2x-1) by
adding 1.
- The function f(2x-1) is obtained from f(x-1) by
substituting 2x for x.
- The function f(x-1) is obtained from f(x) by
substituting x-1 for x.
To get the graph, we go
backwards. We start with the graph of f(x).
- Substitute x-1 for x, y = f(x-1):
- Substitute 2x for x, y = f(2x-1):
- Add 1 to y, y = f(2x-1)+1:
Additional Resources
this page here has a few little tools for analyzing functions, including one that will plot a graph of f(x) alongside f(x) + c, f(x+c), and f(cx) in colour. Changing the values of c will demonstrate the shifts discussed.
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