what is the mapping rule for g(x)= -1/ (2(x-2))

The mapping rule describes how a transformation alters the graph of a base function. For the function \( g(x) = \frac{-1}{2(x-2)} \), we can break it down into a series of transformations applied to the parent function \( f(x) = \frac{1}{x} \).

### Step-by-Step Mapping Rule:

1. **Horizontal Shift**: The term \( (x - 2) \) in the denominator means that the graph of the parent function \( f(x) = \frac{1}{x} \) is shifted **2 units to the right**.
   
2. **Vertical Stretch/Compression**: The factor \( \frac{1}{2} \) means the graph is **compressed vertically** by a factor of \( \frac{1}{2} \). This reduces the "steepness" of the graph.
   
3. **Reflection**: The negative sign \( -1 \) in front of the fraction reflects the graph **across the x-axis**.

### Mapping Rule Summary:

- The point \( (x, y) \) on the parent function \( f(x) = \frac{1}{x} \) maps to:
  \[
  (x, y) \rightarrow (x + 2, -\frac{y}{2})
  \]
  where \( x + 2 \) represents the horizontal shift by 2 units to the right, and \( -\frac{y}{2} \) represents the vertical compression and reflection.

Would you like more details or have any questions?

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**Related Questions:**
1. How does horizontal shifting affect the asymptotes of a function?
2. What are the domain and range of \( g(x) = \frac{-1}{2(x-2)} \)?
3. How does vertical compression affect the behavior near asymptotes?
4. How do reflections affect the shape and direction of a graph?
5. Can you determine the asymptotes for \( g(x) = \frac{-1}{2(x-2)} \)?

**Tip**: Always look for horizontal shifts by checking for \( (x - c) \) in the denominator, which indicates a movement left or right.

What is the mapping rule for g(x)= -1/ (2(x-2))?

Explore the mapping rule for the rational function g(x) = -1 / (2(x-2)). This solution includes a breakdown of transformations such as horizontal shifts, vertical compression, and reflection. Suitable for students in Grades 9-12, this problem helps in understanding the graph transformations of rational functions.

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