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# Function transformations

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### Function transformations

1. 1. Function Transformations
2. 2. Objectives: <ul><li>To interpret the meaning of the symbolic representations of functions and operations on functions including: </li></ul><ul><ul><li>a·f(x), </li></ul></ul><ul><ul><li>f(|x|), </li></ul></ul><ul><ul><li>f(x) + d, </li></ul></ul><ul><ul><li>f(x – c), </li></ul></ul><ul><ul><li>f(b·x), and </li></ul></ul><ul><ul><li>|f(x)|. </li></ul></ul><ul><li>To explore the following basic transformations as applied to functions: </li></ul><ul><ul><li>Translations, </li></ul></ul><ul><ul><li>Reflections, and </li></ul></ul><ul><ul><li>Dilations. </li></ul></ul>
3. 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. 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. 5. Translations • • • • • 2 0 -6 4 -4 0 -2 6 0 4 • • • • •
6. 6. Translations • • • • • 2 0 -2 4 0 0 2 6 4 8 • • • • •
7. 7. Translations • • • • • 6 4 2 8 4 4 0 10 6 0 • • • • •
8. 8. Translations • • • • • -4 -6 2 -2 4 -6 0 0 6 0 • • • • •
9. 9. Reflections • • • • • 6 10 4 0 2 4 0 2 -2 -4 • • • • •
10. 10. Reflections across the y-axis : <ul><li>y = f(-x)           </li></ul><ul><li>Take f(x) and draw its mirror image across the y -axis (reflects the graph left to right and right to left). </li></ul><ul><ul><li>This is called an EVEN function. </li></ul></ul><ul><ul><li>To test if a function is even, show that </li></ul></ul><ul><ul><ul><li>f(-x) = f(x) . </li></ul></ul></ul>
11. 11. Reflections • • • • • -10 - 6 10 - 2 2 - 4 4 0 0 6 • • • • •
12. 12. Reflections across the x-axis : <ul><li>y = - f(x)          </li></ul><ul><li>Take f(x) and draw its mirror image across the x -axis (turns the graph upside down). </li></ul><ul><li>y = |f(x)|          </li></ul><ul><li>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. </li></ul>
13. 13. Compressions • • • • 0 4 -8 -2 -4 -6 0 2 2 6 • • • • • •
14. 14. Dilations • • • • 0 8 0 -4 -2 4 2 -8 -4 4 • • • • • •
15. 15. Homework <ul><li>Tonight’s homework (and last night’s) illustrates these transformations and some combinations of them. </li></ul><ul><li>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. </li></ul><ul><li>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. </li></ul>
16. 16. Absolute Value
17. 17. Symmetry around the origin : <ul><li>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. </li></ul><ul><li>This is called an ODD function </li></ul><ul><li>To test if a function is even, show that f(-x) = -f(x) </li></ul>
18. 18. Reflection across the line y = x : <ul><li>x = f(y)             </li></ul><ul><li>Take f(x) and draw its mirror image across the line y = x (the two functions are inverses of each other). </li></ul>