9. Determine the transformations being applied to the graph of the function 𝑦 = 𝑓(𝑥). If you are struggling with this, try letting 
𝑓(𝑥) = √𝑥 and rewrite each expression appropriately.
(a) 3𝑓(𝑥) (b) −𝑓(𝑥) (c) 𝑓(−𝑥) (d) 𝑓(𝑥) + 2 (e) 𝑓(𝑥 − 2)

To analyze the transformations being applied to the function \( y = f(x) \), we will interpret each modified version of the function in terms of its effect on the graph. For simplicity, let \( f(x) = \sqrt{x} \), as suggested.

1. **\( 3f(x) \)**:
   - This is a **vertical stretch** by a factor of 3.
   - The graph of \( f(x) = \sqrt{x} \) will be stretched vertically, making it 3 times taller than the original graph.

2. **\( -f(x) \)**:
   - This is a **reflection** across the x-axis.
   - The graph of \( f(x) = \sqrt{x} \) will be flipped upside down.

3. **\( f(-x) \)**:
   - This is a **reflection** across the y-axis.
   - The graph of \( f(x) = \sqrt{x} \) will be reflected over the y-axis, flipping it horizontally.

4. **\( f(x) + 2 \)**:
   - This is a **vertical translation** (or shift) upwards by 2 units.
   - The graph of \( f(x) = \sqrt{x} \) will move upward by 2 units.

5. **\( f(x - 2) \)**:
   - This is a **horizontal translation** (or shift) to the right by 2 units.
   - The graph of \( f(x) = \sqrt{x} \) will move 2 units to the right.

### Summary of transformations:
- \( 3f(x) \): Vertical stretch by 3.
- \( -f(x) \): Reflection across the x-axis.
- \( f(-x) \): Reflection across the y-axis.
- \( f(x) + 2 \): Vertical shift up by 2.
- \( f(x - 2) \): Horizontal shift right by 2.

Would you like further details or have any questions?

Here are 5 questions to deepen understanding:
1. How would \( \frac{1}{2}f(x) \) affect the graph of \( f(x) = \sqrt{x} \)?
2. What transformation happens if we add \( -2 \) instead of \( +2 \) to \( f(x) \)?
3. If we use \( f(x+3) \), how does this transformation affect the graph of \( f(x) = \sqrt{x} \)?
4. What happens if we combine multiple transformations, such as \( -3f(x - 1) \)?
5. How would the graph change if we used \( f(x) - 5 \)?

### Tip:
When analyzing transformations, focus on whether the change occurs *inside* or *outside* the function \( f(x) \)—this helps in identifying whether the shift is horizontal or vertical.

Determine the transformations being applied to the graph of the function y = f(x). If you are struggling with this, try letting f(x) = √x and rewrite each expression appropriately. (a) 3f(x) (b) −f(x) (c) f(−x) (d) f(x) + 2 (e) f(x − 2)

Explore the transformations of the function y = f(x), including vertical stretches, reflections, and translations. This guide explains how each transformation affects the graph, using f(x) = √x as an example. Suitable for students in Grades 9-12 studying algebraic graph transformations.

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