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We are given two functions:

- The parent function: \( f(x) = x^2 \)
- The transformed function: \( k(x) = 0.5(x - 5)^2 - 2 \)

We need to identify the transformations applied to the parent function \( f(x) \) to obtain the new function \( k(x) \).

### Step-by-step transformation analysis:

1. **Horizontal Translation**: The term \( (x - 5) \) inside the squared term indicates a **translation 5 units to the right**. The general form for a horizontal translation is \( (x - h) \), where \( h = 5 \) translates the graph to the right.

2. **Vertical Translation**: The constant term outside the squared term, \( -2 \), indicates a **translation 2 units down**. This moves the graph vertically downward by 2 units.

3. **Vertical Dilation**: The coefficient \( 0.5 \) in front of the squared term compresses the graph vertically by a factor of 0.5. This is a **vertical dilation by a factor of 0.5**. A vertical dilation compresses or stretches the graph, and since \( 0.5 < 1 \), the graph is compressed (squished vertically).

### Correct Answers:
- Dilated by 0.5
- Translated 5 right
- Translated 2 down

Would you like further clarification or more details on any specific point?

Here are 5 related questions for further practice:
1. How would the transformation change if the dilation factor was \(2\) instead of \(0.5\)?
2. What is the effect of translating a graph horizontally by a negative number?
3. How does the graph of \( f(x) = x^2 \) change if the translation was \(3\) units left instead of right?
4. What happens to the graph if the vertical translation was \(+4\) instead of \(-2\)?
5. How would the transformation differ if there was a reflection across the x-axis?

**Tip**: Always check the signs in the equation carefully, as they determine the direction of translations!

Choose all correct answers that describe the transformation from the parent graph to the new graph: f(x) = x^2; k(x) = 0.5(x - 5)^2 - 2.

Explore the transformations applied to the parent function f(x) = x^2 to obtain the new function k(x) = 0.5(x - 5)^2 - 2. This analysis includes vertical dilation, horizontal and vertical shifts. Suitable for high school students studying graph transformations.

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