Use the *r* notation to write the rule for transforming from f(x) to h(x)

To write a rule for transforming a function \( f(x) \) to another function \( h(x) \) using the notation \( r \), we need to define how \( h(x) \) is related to \( f(x) \). The \( r \)-notation typically represents transformations such as translations, reflections, stretches, or compressions. 

Let's consider a few types of common transformations using \( r \)-notation:

1. **Translation**: 
   - Horizontal: \( h(x) = f(x - r) \) represents a translation of \( f(x) \) to the right by \( r \) units (if \( r > 0 \)) or to the left by \( |r| \) units (if \( r < 0 \)).
   - Vertical: \( h(x) = f(x) + r \) represents a translation of \( f(x) \) upwards by \( r \) units (if \( r > 0 \)) or downwards by \( |r| \) units (if \( r < 0 \)).

2. **Reflection**: 
   - Across the y-axis: \( h(x) = f(-x) \) reflects \( f(x) \) across the y-axis.
   - Across the x-axis: \( h(x) = -f(x) \) reflects \( f(x) \) across the x-axis.

3. **Stretch/Compression**: 
   - Horizontal Stretch/Compression: \( h(x) = f(rx) \). If \( |r| > 1 \), this represents a horizontal compression; if \( 0 < |r| < 1 \), it represents a horizontal stretch.
   - Vertical Stretch/Compression: \( h(x) = r \cdot f(x) \). If \( |r| > 1 \), this represents a vertical stretch; if \( 0 < |r| < 1 \), it represents a vertical compression.

### Example:
If you wanted to represent a function \( h(x) \) that is a vertical stretch of \( f(x) \) by a factor of 2, you would write:
\[ h(x) = 2f(x) \]
where \( r = 2 \).

If you can specify the particular type of transformation you're interested in, I can provide a more tailored rule using \( r \)-notation.

Would you like to specify the type of transformation, or do you have another question related to function transformations?

### Follow-Up Questions:

1. What is a horizontal translation of \( f(x) \) by 3 units to the left?
2. How do you reflect \( f(x) \) across the y-axis using \( r \)-notation?
3. What happens when you vertically compress \( f(x) \) by a factor of 0.5?
4. How would you write a rule for stretching \( f(x) \) horizontally by a factor of 2?
5. How can multiple transformations be combined using \( r \)-notation?

**Tip**: When combining multiple transformations, apply horizontal changes before vertical ones to maintain clarity.

Understand how to use r-notation to write rules for transforming functions. Learn about translations, reflections, stretches, and compressions in function transformations. Explore examples and practical applications. Suitable for high school students and those interested in mathematical transformations.

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