Math Problem Statement
Solution
The image shows six different transformations applied to the graphs of functions. Below is how each transformation affects the equation of a general function :
-
Dilation of factor from the -axis:
- This type of transformation compresses the graph horizontally. It is represented by modifying the input variable in the function.
- New equation: .
-
Dilation of factor from the -axis:
- This is a horizontal compression by a factor of , meaning the input is multiplied by .
- New equation: .
-
Dilation of factor from the -axis:
- This is a vertical stretch by a factor of , meaning the output of the function is multiplied by .
- New equation: .
-
Dilation of factor from the -axis:
- This is a vertical stretch by a factor of , meaning the output of the function is multiplied by .
- New equation: .
-
Reflection in the -axis:
- This transformation reflects the graph over the -axis, meaning the output of the function is negated.
- New equation: .
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Reflection in the -axis:
- This transformation reflects the graph over the -axis, meaning the input variable is negated.
- New equation: .
Let me know if you'd like further clarification or details on any of these transformations!
Here are five related questions for further exploration:
- How does a vertical dilation differ from a horizontal dilation in terms of the function's equation?
- What effect does a reflection in the -axis have on the domain of a function?
- How would you combine a reflection in the -axis with a dilation of factor from the -axis?
- Can dilations and reflections change the range of a function? Why or why not?
- How can we graphically interpret a dilation of a function from the -axis versus from the -axis?
Tip:
For horizontal transformations (like dilations from the -axis or reflections in the -axis), changes happen inside the function’s argument , while vertical transformations affect the whole function output.
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Math Problem Analysis
Mathematical Concepts
Transformations of Functions
Dilations
Reflections
Graph Transformations
Formulas
Horizontal Dilation: f(ax) where a is a factor from the y-axis
Vertical Dilation: af(x) where a is a factor from the x-axis
Reflection in x-axis: -f(x)
Reflection in y-axis: f(-x)
Theorems
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Suitable Grade Level
Grades 10-12
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