2.7: Use Absolute Value Functions and Transformations Objectives: 1.To graph an absolute value...
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![Page 1: 2.7: Use Absolute Value Functions and Transformations Objectives: 1.To graph an absolute value function by performing transformations on the parent 2.To.](https://reader036.fdocuments.us/reader036/viewer/2022072008/56649d6f5503460f94a51799/html5/thumbnails/1.jpg)
2.7: Use Absolute Value Functions and 2.7: Use Absolute Value Functions and TransformationsTransformations
Objectives:
1.To graph an absolute value function by performing transformations on the parent
2.To apply transformations to graphing any function
![Page 2: 2.7: Use Absolute Value Functions and Transformations Objectives: 1.To graph an absolute value function by performing transformations on the parent 2.To.](https://reader036.fdocuments.us/reader036/viewer/2022072008/56649d6f5503460f94a51799/html5/thumbnails/2.jpg)
Building the Absolute Value Building the Absolute Value FunctionFunction
The absolute value function is defined by f (x) = |x|.
The graph of the absolute value function is similar to the linear parent function, except it must always be positive.
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Building the Absolute Value Building the Absolute Value FunctionFunction
The absolute value function is defined by f (x) = |x|.
So we just take the negative portion of the graph and reflect it across the x-axis making that part positive.
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![Page 4: 2.7: Use Absolute Value Functions and Transformations Objectives: 1.To graph an absolute value function by performing transformations on the parent 2.To.](https://reader036.fdocuments.us/reader036/viewer/2022072008/56649d6f5503460f94a51799/html5/thumbnails/4.jpg)
Building the Absolute Value Building the Absolute Value FunctionFunction
The absolute value function is defined by f (x) = |x|.
This is the absolute value parent functionparent function.
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f x = x
![Page 5: 2.7: Use Absolute Value Functions and Transformations Objectives: 1.To graph an absolute value function by performing transformations on the parent 2.To.](https://reader036.fdocuments.us/reader036/viewer/2022072008/56649d6f5503460f94a51799/html5/thumbnails/5.jpg)
Parent FunctionParent Function
• V-shape
• It is symmetric about the y-axis
• The vertexvertex is the minimum point on the graph
![Page 6: 2.7: Use Absolute Value Functions and Transformations Objectives: 1.To graph an absolute value function by performing transformations on the parent 2.To.](https://reader036.fdocuments.us/reader036/viewer/2022072008/56649d6f5503460f94a51799/html5/thumbnails/6.jpg)
TranslationTranslation
A translationtranslation is a transformation that shifts a graph horizontally or vertically, but doesn’t change the overall shape or orientation.
![Page 7: 2.7: Use Absolute Value Functions and Transformations Objectives: 1.To graph an absolute value function by performing transformations on the parent 2.To.](https://reader036.fdocuments.us/reader036/viewer/2022072008/56649d6f5503460f94a51799/html5/thumbnails/7.jpg)
TranslationTranslation
The graph of
y = |x – h| + k
is the graph of y = |x| translated h horizontal units and y vertical units.
• The new vertex is at (h, k)
![Page 8: 2.7: Use Absolute Value Functions and Transformations Objectives: 1.To graph an absolute value function by performing transformations on the parent 2.To.](https://reader036.fdocuments.us/reader036/viewer/2022072008/56649d6f5503460f94a51799/html5/thumbnails/8.jpg)
Stretching and ShrinkingStretching and Shrinking
The graph of y = a|x| is graph of y = |x| vertically stretched or shrunk depending on the |a|.
The value of a acts like the slope.
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ReflectionReflection
The graph of y = a|x| is graph of y = |x| reflected across the x-axis when a < 0.
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f x = x
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f x = - x
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Multiple TransformationsMultiple Transformations
In general, the graph of an absolute value function of the form y = a|x – h| + k can involve translations, reflections, stretches or shrinks.
To graph an absolute value function, start by identifying the vertex.
![Page 11: 2.7: Use Absolute Value Functions and Transformations Objectives: 1.To graph an absolute value function by performing transformations on the parent 2.To.](https://reader036.fdocuments.us/reader036/viewer/2022072008/56649d6f5503460f94a51799/html5/thumbnails/11.jpg)
Graphing Absolute Value FunctionsGraphing Absolute Value Functions
Graphing y = a|x – h| + k is easy:
• Plot the vertex (h, k). (note…if +h inside that means h is negtive, if – h inside that means h is positive)
• Use the a value as slope to plot more points. Remember you have to do positive and negative slope to get points on both sides of the V
• Connect the dots in a V-shape.
![Page 12: 2.7: Use Absolute Value Functions and Transformations Objectives: 1.To graph an absolute value function by performing transformations on the parent 2.To.](https://reader036.fdocuments.us/reader036/viewer/2022072008/56649d6f5503460f94a51799/html5/thumbnails/12.jpg)
Exercise 2Exercise 2
Graph the following functions without making a table.
1. y = |x – 2| + 3 This graph will go right 2 and up 3 so from the origin go right 2 and up 3. This is the vertex (2, 3). Now from that point use the positive and negative slope (a = 1 here) to get more points.
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f x = x
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Exercise 2Exercise 2
Graph the following functions without making a table.
1.y = (1/2)|x| This function does not have an “h” or “k” so the vertex is (0, 0). Since a = ½ the slope is ½. Go up 1 and right 2 then up one and left 2. 4
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f x = x
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Exercise 2Exercise 2
Graph the following functions without making a table.
1.f (x) = -3|x + 1| – 2 This graph will go left 1 and down two so the vertex will be (-1, -2). Since “a” is negative the graph will open down. Since the value of “a” is 3 the slope will be 3 and -3 (just remember to go down.)
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f x = x
![Page 15: 2.7: Use Absolute Value Functions and Transformations Objectives: 1.To graph an absolute value function by performing transformations on the parent 2.To.](https://reader036.fdocuments.us/reader036/viewer/2022072008/56649d6f5503460f94a51799/html5/thumbnails/15.jpg)
Transformations in GeneralTransformations in General
You can perform transformations on the graph of any function in manner similar to transformations on the absolute value function.