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We are given two sets of dataWe are given two sets of  one for \( f(x) \) and one for \( g(x) \), where \( g(x) \) is a transformed version of \( f(x) \). The goal is to find the equation for \( g(x) \) in the form \( g(x) = a \cdot f(x-h) + k \).

### Step-by-step approach:

1. **Identify the function transformation components**:
   The equation \( g(x) = a \cdot f(x-h) + k \) indicates:
   - \( a \) is a vertical stretch/compression factor.
   - \( h \) is a horizontal shift.
   - \( k \) is a vertical shift.

2. **Compare points from \( f(x) \) and \( g(x) \)**:
   - For \( x = -2 \), \( f(-2) = 4 \) and \( g(-2) = -4 \).
   - For \( x = -1 \), \( f(-1) = 2 \) and \( g(-1) = -2 \).
   - For \( x = 0 \), \( f(0) = 1 \) and \( g(0) = -1 \).
   - For \( x = 1 \), \( f(1) = 2 \) and \( g(1) = -2 \).
   - For \( x = 2 \), \( f(2) = 4 \) and \( g(2) = -4 \).

3. **Check for shifts**:
   - There is no evidence of horizontal shift, so \( h = 0 \).
   
4. **Determine the stretch/compression factor \( a \)**:
   - Compare the values of \( f(x) \) and \( g(x) \). At each corresponding \( x \), \( g(x) \) is consistently the negative of \( f(x) \).
   - This suggests that \( a = -1 \).

5. **Check for vertical shifts**:
   - There is no consistent offset between \( f(x) \) and \( g(x) \), implying \( k = 0 \).

### Final equation for \( g(x) \):
\[
g(x) = -f(x)
\]

Would you like any additional clarification or further details on any steps?

Here are some related questions to explore:

1. How do vertical shifts affect the graph of a function?
2. What is the impact of horizontal shifts on function graphs?
3. How would the equation change if there was a horizontal shift?
4. Can you provide an example where \( a \), \( h \), and \( k \) are all non-zero?
5. How do vertical stretches and compressions affect a graph?

**Tip:** When transforming functions, always analyze changes in both the x- and y-values to determine the nature of the shift or stretch.

Use the table of values for f(x) and the transformed function g(x) to write the equation for g(x) in function notation (g(x) = a(f(x-h))+k).

Explore how function transformations work using an example involving linear functions. Learn how to apply the transformation formula g(x) = a * f(x-h) + k to derive new function values from an existing function table. Suitable for high school students.

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