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Mathematics 309
Part I  The elementary geometry of wave motion
1. Scaling and shifting of graphs
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.
Simple steps
The way to see what happens
to the graph is to understand what happens in simple steps
with just one change at a time.


y = f(x)

y = 2f(x)
All yvalues on the graph are scaled by 2.
In subsituting cf for f,
vertical distances are scaled by c.



y = f(x)

y = f(x)+1
1 is added to all y values.



y = f(x)

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.



y = f(x)

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.



y = f(x)

y = f(2x1)+1

Only the last one is tricky,
since it involves
a sequence of substitutions.
Composites
In order to see how to deal with the general case,
we unravel the process.

The function f(2x1)+1 is obtained
from f(2x1) by adding 1.

The function f(2x1) is obtained
from f(x1) by substituting 2x
for x.

The function f(x1) is obtained
from f(x) by substituting x1
for x.
To get the graph, we go backwards. We start with the graph
of f(x).

Substitute x1
for x, y = f(x1):

Substitute 2x
for x, y = f(2x1):

Add 1 to y, y = f(2x1)+1:
The end result can be rather hard to visualize from the initial image,
and it is more important to
understand the steps involved
than it is to see everything explicitly.
Exercises
