Downloaded 66 times
Uploaded byTerry Gastauer
Function transformations
The document discusses function transformations including translations, reflections, dilations, and compressions. It defines these transformations and provides examples of how they affect the graph of a function. Translations slide the graph left or right without changing its shape or orientation. Reflections create a mirror image of the graph across an axis, flipping it. Compressions squeeze the graph towards or away from an axis. Dilations stretch or shrink the graph away from an axis. The document explains how to interpret various function notation and applies the transformations to example graphs.
In this document
Powered by AI
Introduction to function transformations and their symbolic representation. Objectives include interpreting functions and exploring transformations: translations, reflections, and dilations.
Defines transformation concepts such as translation, reflection, compression, and dilation. Explanation of notation related to altering functions and their properties.
Visual representations of different translations exemplified through data points to illustrate function shifts.
Visual presentations of reflections in functions including mirror images across the y-axis and x-axis, and the distinction between even and odd functions.
Data examples demonstrating compressions and dilations of functions, showcasing vertical and horizontal transformations.
Homework assignment focusing on understanding transformations and their parameters, emphasizing their application to parent functions.
Introduction to absolute value functions and their symmetry characteristics relating to odd functions.
Details on reflecting functions about the line y=x and its implications regarding inverse functions.
More Related Content
Similar to Function transformations
Function transformations
- 1.
- 2. Objectives: To interpretthe 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 thenotation: 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 6 0 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 they-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 thex-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.
- 17. Symmetry around theorigin : 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 theline 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).