the point (-3, 8) lies on the graph of y = f(x). Using mapping notation, state the coordinates of of this point after the following transformation: y + 1 = f(1/4x - 1)

The given transformation is \( y + 1 = f\left( \frac{1}{4}x - 1 \right) \).

### Step-by-Step Breakdown:

1. **Horizontal Transformation:**
   - \( \frac{1}{4}x - 1 \) represents two transformations:
     - **Horizontal stretch** by a factor of 4. To account for this, multiply the \( x \)-coordinate by 4.
     - **Horizontal shift** to the right by 1. To account for this, add 1 to the \( x \)-coordinate.

   Applying this to the point \( (-3, 8) \):
   \[
   x' = 4 \times (-3 + 1) = 4 \times (-2) = -8
   \]

2. **Vertical Transformation:**
   - \( y + 1 = f(\text{...}) \) indicates a vertical shift **down** by 1. So, subtract 1 from the \( y \)-coordinate.

   Applying this to the point \( (-3, 8) \):
   \[
   y' = 8 - 1 = 7
   \]

### Final Coordinates:
After the transformation, the new coordinates of the point are \( (-8, 7) \).

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

Here are 5 related questions to explore:
1. What happens to the point if the transformation includes a reflection across the y-axis?
2. How would the transformation affect the slope of a line passing through the original point?
3. What is the effect of applying only the horizontal transformation to the point?
4. How do you reverse a vertical shift in a transformation equation?
5. What happens if we change the stretch factor from 4 to another number?

**Tip:** Always handle horizontal transformations (stretch/compression and shifts) before applying vertical changes!

The point (-3, 8) lies on the graph of y = f(x). Using mapping notation, state the coordinates of this point after the following transformation: y + 1 = f(1/4x - 1).

Explore how to find the new coordinates of the point (-3, 8) after applying the transformation y + 1 = f(1/4x - 1). This solution involves horizontal and vertical transformations of functions and is suitable for students in Grades 9-11.

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