Mathematics 309 - the elementary geometry of wave motion

Part II - Scaling and Shifting

It is prudent to learn how a graph looks when you change its parameters. For example, a function y = f(x) changes shape when you change its parameters.


  • f(x) -> cf(x) (for some parameter c)
    • All y-values are multiplied by c, and the vertical distances are scaled by c.
  • f(x) -> f(x) + a (for some parameter a)
    		•
    All y-values have a added to them, meaning you shift the graph vertically by a.
  • f(x) -> f(cx)
    • The value of y at x on the new graph is equal to cx on the old graph. The graph is scaled horizontally by 1/c.
  • f(x) -> f(x + a)
    		•
    The value of y at x on the new graph is equal to x + a on the old graph. The graph is shifted left by a.



    This is the graphical representation of the above with a particular function y = f(x).


    The function y = f(x)
    The function y = 2f(x)
    y = f(2x)
    y = f(x) + 1
    y = f(x + 1)


    All images courtesy of MSPaint. By Alfred Chan