1. Special Functions

2. We have come a long way in this module and covered a lot of material dealing with graphs of polynomials. In this lesson we will look at parent functions and transformations. As you have seen in previous lessons, there is a lot to remember so take good notes and use them as you work through the lesson.

3. Parent functions are the main function of a function family. We are very familiar with y = mx + b, but this is not the parent function. The parent function for linear functions is y = x. Some texts say y = mx + b is the parent function with m =1 and b = 0, it is easier to recognize y = x as the parent function and you will see why later. y = x goes through the origin and has a slope of 1. When we add or subtract a constant (b) we are actually just taking the graph of y = mx and moving it up or down on the coordinate plane, which we call transforming. We can also change the slope, with or without changing the y- intercept, and this is a transformation also. Let’s look at some examples.

4. Parent function: Various Transformations of y = mx + b: Regardless of what we do to the graph – flip it, spin it, move it up or down, it is still a line resembling the parent function.

5. Parent function: Various Transformations of y = a|x - h| + k:

6. Parent function: Various Transformations of y = x 2 :

7. Parent function: Various Transformations of y = x 3 :

8. Now you should understand that the parent function is the main function and the changes we make to that equation are what transforms the graph. This is true for all parent functions. Here are a few more just so you can see the function in it’s simplest form. **See if you notice anything about the even and odd exponent parent functions

9. Parent functions:

11. And there are more…but the point is that once we start altering the equations of parent functions we tend to get more and more unique graphs. The combination of values that can be used make the number of transformations is endless for any one parent function. How do we keep it all straight? Rules…of course! (and many of them!)

12. k +k  Causes the graph to move up d units -k  Causes the graph to move down d units k +k  Causes the graph to move up d units -k  Causes the graph to move down d units Many of these may seem the same, but the differences are in the placement of the changes (before, inside, or after the parenthesis, absolute value bars, or radicals) ! a a>1  Causes a vertical stretch by the factor of a 0<a<1  Causes a vertical compression/ shrink by the factor of a -a  A negative in front of a causes a reflection over the x-axis a a>1  Causes a vertical stretch by the factor of a 0<a<1  Causes a vertical compression/ shrink by the factor of a -a  A negative in front of a causes a reflection over the x-axis h +h  Causes the graph to move left c units -h  Causes the graph to move right c units h +h  Causes the graph to move left c units -h  Causes the graph to move right c units b b>1  Causes a horizontal compression by 1/b b<1  Causes a horizontal stretch by 1/b -b  Causes reflection over the y-axis b b>1  Causes a horizontal compression by 1/b b<1  Causes a horizontal stretch by 1/b -b  Causes reflection over the y-axis

13. Still the same for absolute value functions a a>1  Causes a vertical stretch by the factor of a 0<a<1  Causes a vertical compression/ shrink by the factor of a -a  A negative in front of a causes a reflection over the x-axis a a>1  Causes a vertical stretch by the factor of a 0<a<1  Causes a vertical compression/ shrink by the factor of a -a  A negative in front of a causes a reflection over the x-axis b b>1  Causes a horizontal compression by 1/b b<1  Causes a horizontal stretch by 1/b -b  Causes reflection over the y-axis b b>1  Causes a horizontal compression by 1/b b<1  Causes a horizontal stretch by 1/b -b  Causes reflection over the y-axis k +k  Causes the graph to move up d units -k  Causes the graph to move down d units k +k  Causes the graph to move up d units -k  Causes the graph to move down d units h +h  Causes the graph to move left c units -h  Causes the graph to move right c units h +h  Causes the graph to move left c units -h  Causes the graph to move right c units

14. Still the same for radical functions a a>1  Causes a vertical stretch by the factor of a 0<a<1  Causes a vertical compression/ shrink by the factor of a -a  A negative in front of a causes a reflection over the x-axis a a>1  Causes a vertical stretch by the factor of a 0<a<1  Causes a vertical compression/ shrink by the factor of a -a  A negative in front of a causes a reflection over the x-axis b b>1  Causes a horizontal compression by 1/b b<1  Causes a horizontal stretch by 1/b -b  Causes reflection over the y-axis b b>1  Causes a horizontal compression by 1/b b<1  Causes a horizontal stretch by 1/b -b  Causes reflection over the y-axis k +k  Causes the graph to move up d units -k  Causes the graph to move down d units k +k  Causes the graph to move up d units -k  Causes the graph to move down d units h +h  Causes the graph to move left c units -h  Causes the graph to move right c units h +h  Causes the graph to move left c units -h  Causes the graph to move right c units

15. Some clarification: Any change in a or k (outside the main function) results in change on the y-axis. Any change in b or h (inside the main function) results in a change on the x-axis. From here – you just have to practice and refer back to this info as you do!