This section describes the analytic interpretation of what makes a transformation and how to use the function notation to perform (or read) a transformation quickly and easily.

The Analytic View

Here is a video!

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Transformations are about stretching and compressing and typically one discusses stretching or compressing by a factor. This is a hint that the analytic way to attain stretching/compressing is to use multiplication/division and not adding/subtracting.

Just like with translations, when analytically representing a transformation of a function , it is best to name the result of the transformation and define it as a manipulation of . Only, in the case of transformations, we will be using multiplication/division instead of addition/subtraction. Thus we have the following general form: Also like the rigid translations case, the transformations work the way we would expect for the values and the opposite of the way we would think for the values. Thus when we multiply by a larger number we would expect the values to get bigger (stretch), and if we multiply by smaller values (ie values less than one), we would expect the resulting values to shrink (compress). This is true for the values, but switched for the values. Consider the following graph of ;


Figure 1: The original

If we look at the transformation we would expect that it gets about 3 times as tall, ie to stretch to three times it’s height and we would be correct! The following is the graph of :


Figure 2: The

Next, if we multiplied the variable in our original by 4 we would ‘expect’ the x values to stretch out to the sides but of course the transformations (like the rigid translations) do the exact opposite of what we’d expect. Consider the following graph of below.


Figure 3:

Again, we could counter this attack on our intuition by rewriting the ‘form’ we use to counter this effect. Specifically let’s consider the form; Here plugging in the ‘intuitive’ value will yield our expected result. So in our example, if we wanted to be 4 times as wide, we could use in this form and have the following graph (Notice the -axis values in comparison to the original );


Figure 4: The

Alternatively we could use our mantra that “everything about is backwards”; thus making the values 4 times wider means we would divide by 4, rather than multiply by 4.

Try messing around with this interactive graph to get a feel for how transformations work. Make sure to notice what happens when you have values less than one, or when values are negative.

1 : In order to stretch a graph horizontally...
You multiply the -value before function evaluation by: a number larger than one to make it larger (stretched); a number smaller than one to make it smaller (shrink) and a negative to flip it. You multiply the -value before function evaluation by: a number smaller than one to make it larger (stretched); a number larger than one to make it smaller (shrink) and a negative to flip it. You multiply the -value before function evaluation by: a number smaller than one to make it larger (stretched); a number larger than one to make it smaller (shrink) and a positive number to flip it. You multiply the -value before function evaluation by: a number larger than one to make it larger (stretched); a number smaller than one to make it smaller (shrink) and a positive number to flip it. You multiply the -value after function evaluation by: a number larger than one to make it larger (stretched); a number smaller than one to make it smaller (shrink) and a negative to flip it. You multiply the -value after function evaluation by: a number smaller than one to make it larger (stretched); a number larger than one to make it smaller (shrink) and a negative to flip it. You multiply the -value after function evaluation by: a number smaller than one to make it larger (stretched); a number larger than one to make it smaller (shrink) and a positive number to flip it. You multiply the -value after function evaluation by: a number larger than one to make it larger (stretched); a number smaller than one to make it smaller (shrink) and a positive number to flip it.
2 : In order to stretch a graph vertically...
You multiply the -value before function evaluation by: a number larger than one to make it larger (stretched); a number smaller than one to make it smaller (shrink) and a negative to flip it. You multiply the -value before function evaluation by: a number smaller than one to make it larger (stretched); a number larger than one to make it smaller (shrink) and a negative to flip it. You multiply the -value before function evaluation by: a number smaller than one to make it larger (stretched); a number larger than one to make it smaller (shrink) and a positive number to flip it. You multiply the -value before function evaluation by: a number larger than one to make it larger (stretched); a number smaller than one to make it smaller (shrink) and a positive number to flip it. You multiply the -value after function evaluation by: a number larger than one to make it larger (stretched); a number smaller than one to make it smaller (shrink) and a negative to flip it. You multiply the -value after function evaluation by: a number smaller than one to make it larger (stretched); a number larger than one to make it smaller (shrink) and a negative to flip it. You multiply the -value after function evaluation by: a number smaller than one to make it larger (stretched); a number larger than one to make it smaller (shrink) and a positive number to flip it. You multiply the -value after function evaluation by: a number larger than one to make it larger (stretched); a number smaller than one to make it smaller (shrink) and a positive number to flip it.