Math Problem Statement

Write down the equation of the image obtained when the graph of each of the functions below is transformed by: 1. A dilation of factor 1/2 from the y-axis 2. A dilation of factor 5 from the y-axis 3. A dilation of factor 3/2 from the x-axis 4. A dilation of factor 4 from the x-axis 5. A reflection in the x-axis 6. A reflection in the y-axis

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 f(x)f(x):

  1. Dilation of factor 12\frac{1}{2} from the yy-axis:

    • This type of transformation compresses the graph horizontally. It is represented by modifying the input variable xx in the function.
    • New equation: f(2x)f(2x).
  2. Dilation of factor 55 from the yy-axis:

    • This is a horizontal compression by a factor of 15\frac{1}{5}, meaning the input xx is multiplied by 55.
    • New equation: f(15x)f\left(\frac{1}{5}x\right).
  3. Dilation of factor 32\frac{3}{2} from the xx-axis:

    • This is a vertical stretch by a factor of 32\frac{3}{2}, meaning the output of the function is multiplied by 32\frac{3}{2}.
    • New equation: 32f(x)\frac{3}{2} f(x).
  4. Dilation of factor 44 from the xx-axis:

    • This is a vertical stretch by a factor of 44, meaning the output of the function is multiplied by 44.
    • New equation: 4f(x)4 f(x).
  5. Reflection in the xx-axis:

    • This transformation reflects the graph over the xx-axis, meaning the output of the function is negated.
    • New equation: f(x)-f(x).
  6. Reflection in the yy-axis:

    • This transformation reflects the graph over the yy-axis, meaning the input variable xx is negated.
    • New equation: f(x)f(-x).

Let me know if you'd like further clarification or details on any of these transformations!

Here are five related questions for further exploration:

  1. How does a vertical dilation differ from a horizontal dilation in terms of the function's equation?
  2. What effect does a reflection in the yy-axis have on the domain of a function?
  3. How would you combine a reflection in the xx-axis with a dilation of factor 33 from the xx-axis?
  4. Can dilations and reflections change the range of a function? Why or why not?
  5. How can we graphically interpret a dilation of a function from the yy-axis versus from the xx-axis?

Tip:

For horizontal transformations (like dilations from the yy-axis or reflections in the yy-axis), changes happen inside the function’s argument f(x)f(x), 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