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.

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.
In subsituting f(x) + c for f(x),
The graph is moved up by c.
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.
In substituting x+c for x,
the graph is shifted horizontally c to the left.
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.