https://file.aiquickdraw.com/mathbot/file/17257521444662nkaq6dc.jpg

The image shows six different transformations applied to the graphs of functions. Below is how each transformation affects the equation of a general function \( f(x) \):

1. **Dilation of factor \( \frac{1}{2} \) from the \( y \)-axis**:
   - This type of transformation compresses the graph horizontally. It is represented by modifying the input variable \( x \) in the function.
   - New equation: \( f(2x) \).

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

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

4. **Dilation of factor \( 4 \) from the \( x \)-axis**:
   - This is a vertical stretch by a factor of \( 4 \), meaning the output of the function is multiplied by \( 4 \).
   - New equation: \( 4 f(x) \).

5. **Reflection in the \( x \)-axis**:
   - This transformation reflects the graph over the \( x \)-axis, meaning the output of the function is negated.
   - New equation: \( -f(x) \).

6. **Reflection in the \( y \)-axis**:
   - This transformation reflects the graph over the \( y \)-axis, meaning the input variable \( x \) is negated.
   - New equation: \( f(-x) \).

### 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 \( y \)-axis have on the domain of a function?
3. How would you combine a reflection in the \( x \)-axis with a dilation of factor \( 3 \) from the \( x \)-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 \( y \)-axis versus from the \( x \)-axis?

#### Tip:
For horizontal transformations (like dilations from the \( y \)-axis or reflections in the \( y \)-axis), changes happen inside the function’s argument \( f(x) \), while vertical transformations affect the whole function output.

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

Explore the transformations of functions through dilations and reflections. Learn how to write equations for graphs undergoing transformations like horizontal dilation from the y-axis, vertical dilation from the x-axis, and reflections over the x and y axes. Suitable for students in Grades 10-12.

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