Composite Functions Using Translations -- Example #2

In this section, we will use what we know of translations to graph complex functions from simple ones. We will still use the y = x2 function, but we will apply multiple transformations all at the same time. The graphs following will show the transformations as they are applied one at a time. A greyed out graph shows the previous step and a blue graph shows the current step. Try to follow along as best you can.

General Principles
When you decompose a function and turn it into the sum of its translations, the best strategy is to work from the outside in. Start with the original function,and work with what is in the brackets, and then work outward. The first changes are always related to the x-coodinate; the y coodinate changes come last. The following examples should make it more clear. Dont' forget that f (x) = x2.

Graph of the Function f (2x-10) + 3

 In this case, we have a pair of composition functions working on the x-coodinate. We resolve this by factoring of the middle terms into f (2x-10) + 3  = f [2 (x-5) ] + 3. In this case, we apply the transformation that is furthest in the middle, which happens to be x+5. This results in a shift of 5 units right. Now we apply the change in the x co-ordinate scaling by compressing the graph by 2. The shape of the curve has changed. We apply the last transformation, which is the + 3 that occurs after the function. This shifts the graph a total of 3 units upward Here is another comparison of the original graph with the new one.

On to Composite Function Example Number 3
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