## 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 y-values 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(2x-1)+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(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:
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.