Mathematics 309  the elementary geometry of wave motion
Part I  scaling and shifting
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.


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.
In order to see which ones,
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:
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