1. Function Transformations

2. Objectives: To interpret the meaning of the symbolic representations of functions and operations on functions including: a·f(x), f(|x|), f(x) + d, f(x – c), f(b·x), and |f(x)|. To explore the following basic transformations as applied to functions: Translations, Reflections, and Dilations.

3. Definitions: Transformation – Operations that alter the form of a function. The common transformations are: translation (slide), reflection (or flip), compression (squeeze), dilation (stretch). Translation (slide) – a “sliding” of the graph to another location without altering its size or orientation. Reflection (flip) – the creation of the mirror image of a function across a line called the axis of reflection. Horizontal Compression (squeeze) – the squeezing of the graph towards the y-axis. Vertical Compression – the squeezing of the graph towards the x-axis. Horizontal Dilation (stretch) – the stretching of the graph away from the y-axis. Vertical Dilation – the stretching of the graph away from the x-axis.

4. Meaning of the notation: a · f(x) – multiply “f(x)” by “a” (multiply the “y-value” by “a”) f(|x|) – wherever the “x-value” is negative, make it positive. f(x) + d – add “d” to “f(x)” (add “d” to the “y-value”) f(x – c) – subtract “c” from the “x-value” and calculate f f(b·x) – multiply the “x-value” and “b” and calculate f. |f(x)| – wherever the function is negative, make it positive. (Wherever y is negative, make it positive).

5. Translations 2 0 -6 4-4 0 -2 60 4

6. Translations 2 0 -2 4 0 0 2 6 4 8

7. Translations 6 4 2 8 4 4 0 10 6 0

8. Translations -4 -6 2 -2 4 -6 0 0 6 0

9. Reflections 6 10 4 0 2 4 0 2 -2 -4

10. Reflections across the y-axis: y = f(-x) Take f(x) and draw its mirror image across the y- axis (reflects the graph left to right and right to left). –This is called an EVEN function. –To test if a function is even, show that f(-x) = f(x).

11. Reflections -10 - 6 10 - 2 2 - 4 4 0 0 6

12. Reflections across the x-axis: y = - f(x) Take f(x) and draw its mirror image across the x- axis (turns the graph upside down). y = |f(x)| Take the parts of f(x) that are under the x-axis and draw their mirror images above the x-axis. Leave the parts of f(x) that are above the x-axis where they are.

13. Compressions 0 4 -8 -2 -4 -6 0 2 2 6

14. Dilations 0 8 0 -4 -2 4 2 -8 -4 4

15. Homework Tonight’s homework (and last night’s) illustrates these transformations and some combinations of them. Once you’ve completed the work, take a few minutes to reflect on what you’ve done. Note the effect of the parameter changes on each function. You should see what we’ve seen here today. Tomorrow we’ll see how these ideas – these patterns – help us understand the graphs and the algebra behind many common functions as we apply transformations to parent functions.

16. Absolute Value

17. Symmetry around the origin: A function is symmetric around a point if a line can drawn through the point and extended until it reaches the function on both sides so that the line is bisected by the point. This is called an ODD function To test if a function is even, show that f(-x) = -f(x)

18. Reflection across the line y = x: x = f(y) Take f(x) and draw its mirror image across the line y = x (the two functions are inverses of each other).