function transformations. objectives: to interpret the meaning of the symbolic representations of...
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Function Transformations
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.
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.
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).
Translations
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2
0
-6
4-4
0-2
60
4
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Translations
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2
0
-2
4 0
0 2
64
8
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Translations
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6
4
2
8 4
4 0
106
0
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Translations
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-4
-6
2
-2 4
-6 0
06
0
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Reflections
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6
10
4
0 2
4 0
2-2
-4
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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).
Reflections
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-10
- 6
10
- 2 2
- 4 4
0 0
6
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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.
Compressions
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0
4
-8
-2 -4
-6 0
2 2
6
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Dilations
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0
8
0
-4 -2
4 2
-8-4
4
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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.
Absolute Value
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)
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).