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

The function \( g(x) = -\frac{1}{2}(x - 2) \) represents a linear transformation of the basic identity function \( f(x) = x \). To determine the mapping rule for this transformation, we need to break down the modifications:

1. **Horizontal translation (shift)**: The term \( (x - 2) \) inside the parentheses indicates a **horizontal shift** to the **right by 2 units**.
   
2. **Vertical stretch and reflection**: The coefficient \( -\frac{1}{2} \) outside the parentheses affects the slope. 
   - The negative sign reflects the graph across the **x-axis**.
   - The \( \frac{1}{2} \) means that the graph is **compressed vertically** by a factor of 2 (i.e., it’s less steep than the original identity function).

### Mapping rule:
For a given input \( x \), the point \( (x, y) \) on the graph of \( f(x) = x \) will be transformed to:
\[
(x, y) \rightarrow (x + 2, -\frac{1}{2} y)
\]
Thus, the function translates the graph right by 2 units, compresses it vertically by a factor of 2, and reflects it across the x-axis.

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Here are 5 related questions:
1. What is the mapping rule for \( g(x) = 3(x + 1) \)?
2. How does a horizontal stretch differ from a vertical stretch in a function?
3. How does the graph of \( f(x) = -x \) differ from \( f(x) = x \)?
4. What would be the mapping rule for \( g(x) = 2(x - 5) + 3 \)?
5. Can you explain the mapping rule for \( g(x) = -\frac{1}{3}(x + 4) \)?

**Tip**: A negative coefficient in front of a function not only compresses or stretches the graph but also flips it across the x-axis.

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

Discover how to find the mapping rule for the function g(x) = -1/2(x - 2). This example covers horizontal shifts, vertical stretching, and reflections across the x-axis, key concepts in algebra and graphing functions, ideal for students in Grades 9-11.

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