You may wonder how each of these transformations works. How does attaching the specified parameter re-shape the rule  f  in the manner indicated by the graphs?

The transformations aren't very difficult to understand, and they provide good practice in thinking about functions as very general input-to-output processes.

Suppose we are given  f . Graphically, we obey the rule  y = f(x)  as follows:

First, find the input  x  on the x-axis. Next, compute the output value  y  using the rule for  y = f(x) . Finally, plot a single point at a distance  y  directly above (or below, if the output  y  is negative) the position of the input  x  on the x-axis.

This process produces one point on the graph of  y = f(x)  for each input  x .

Here's how the process changes when we transform the rule with parameters:

  • Horizontal Shifts  f(x ± a)
    &  Stretches/Crunches  f(ax)

    We obey these rules as follows:

    First, find the input  x  on the x-axis.

    Next, move along the x-axis to a new position: either  x ± a  or  ax , depending on the case.

    At this "new" place on the x-axis, compute the output value  y  using the input-output rule for  f .

    Finally, plot a point at a distance  y  directly above (or below) the position of the original input  x — not above (or below) the "new"  x  value.

    The net effect of this process is to grab the output value at the "new"  x  and "pull it over", so that it is graphed above the original input  x .

    The only difference between a shift and a stretch/crunch is that for a shift, all values get "pulled over" by the same amount, and for a stretch/crunch they get "pulled over" by an amount that is proportional to the size of the input  x.

  • Vertical Shifts  f(x) ± a
    &  Stretches/Crunches  af(x)

    The situation is very similar to the cases just described, except that the "pulling" of values is vertical rather than horizontal.

    We obey these rules as follows:

    First, find the input  x  on the x-axis.

    Next, compute the output value  y = f(x)  using the input-output rule for  f .

    Don't plot this  y  value above (or below)  x ...

    Instead, locate a new  y  value at a distance of  y ± a  or  ay  above (or below)  x , depending on the case. Plot this point.

    The net effect of this process is to grab the original output value and "pull it up" (or down) to a "new" output value.

    The only difference between a shift and a stretch/crunch is that for a shift, all values get "pulled up" by the same amount, and for a stretch/crunch they get "pulled up" by an amout that is proportional to the size of the original output  y.

Get it? If you do, you probably have a very good understanding of functional input-output rules. Thinking in these terms will help you to use transformations to modify "known" functions into rules that describe entire families of "similar" situations.

 
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