y = f(x) points are (-4,4), (-2,0),(0,-4),(4,4) the new points are (0,-2),(2,-2),(4,2),(8,2) whats the equation for the transformation

To find the transformation between the original function \( f(x) \) and the new function, we need to analyze how the points have changed.

### Original Points of \( f(x) \):
- \( (-4, 4) \)
- \( (-2, 0) \)
- \( (0, -4) \)
- \( (4, 4) \)

### New Points:
- \( (0, -2) \)
- \( (2, -2) \)
- \( (4, 2) \)
- \( (8, 2) \)

#### Step 1: Analyze the transformation
- First, observe the \( x \)-coordinates. 
  - In the original points, \( x = -4 \) is mapped to \( x = 0 \), \( x = -2 \) is mapped to \( x = 2 \), \( x = 0 \) to \( x = 4 \), and \( x = 4 \) to \( x = 8 \).
  - This suggests a horizontal shift and scaling of the \( x \)-coordinates.

    By observing these shifts, we can determine that the \( x \)-coordinates have been transformed by the rule:
    \[
    x_{\text{new}} = 2x + 8
    \]
    This involves a horizontal scaling by 2 and a rightward shift by 8 units.

- Now, let's analyze the \( y \)-coordinates.
  - In the original function, the \( y \)-coordinates are \( 4, 0, -4, 4 \).
  - In the transformed function, the \( y \)-coordinates are \( -2, -2, 2, 2 \).
  
    The pattern shows that the \( y \)-coordinates have been scaled and shifted. It appears that:
    \[
    y_{\text{new}} = \frac{y}{2} + 2
    \]
    This involves a vertical compression by a factor of 2 and a vertical shift upwards by 2 units.

#### Step 2: Write the transformation equation
From the above analysis, the transformation of the function \( f(x) \) to the new function involves:
- A horizontal scaling by 2 and a rightward shift by 8 units.
- A vertical compression by a factor of 2 and a shift upward by 2 units.

Thus, the transformed function \( g(x) \) can be expressed as:
\[
g(x) = \frac{1}{2}f\left(\frac{x-8}{2}\right) + 2
\]

### Summary:
The equation for the transformation is:
\[
g(x) = \frac{1}{2}f\left(\frac{x-8}{2}\right) + 2
\]

Would you like more details, or do you have any questions?

Here are 5 related questions to explore further:
1. How would the transformation change if the vertical shift was downward instead of upward?
2. What would the transformation look like if there was no horizontal scaling?
3. How would a reflection across the y-axis affect the transformation equation?
4. How do you interpret the effect of a transformation on the graph of a function visually?
5. What happens if the original function \( f(x) \) is a specific known function like a parabola?

**Tip:** To identify transformations, always analyze the changes in both the x- and y-coordinates separately—this helps in detecting scaling, shifting, and reflecting patterns.

Learn how to determine the transformation equation for a function based on changes in coordinates. This problem involves horizontal scaling and shifting, as well as vertical compression and shifts. Suitable for students in Grades 10-12.

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