The equation you've provided is:

\[
y = \sqrt{x} - 6
\]

This describes a transformation of the square root function \( y = \sqrt{x} \). Here's how:

1. **Parent Function**: The base function is \( y = \sqrt{x} \), which starts at the point \( (0, 0) \) and gradually increases as \( x \) increases.
2. **Vertical Shift**: The entire graph of \( y = \sqrt{x} \) is shifted 6 units downwards because of the subtraction of 6.

### Key Points:
- **Domain**: Since \( \sqrt{x} \) is only defined for non-negative \( x \), the domain is \( x \geq 0 \).
- **Range**: Since the square root of any number is non-negative and the graph is shifted down by 6, the range is \( y \geq -6 \).

If you want to sketch the graph, it would start at the point \( (0, -6) \) and gradually rise as \( x \) increases.

Would you like more details or have any questions?

### Related Questions:
1. How do we find the inverse of \( y = \sqrt{x} - 6 \)?
2. What are the intercepts of the graph of \( y = \sqrt{x} - 6 \)?
3. How does the function behave as \( x \to \infty \)?
4. Can we express the equation in terms of a piecewise function?
5. How can we apply transformations to the graph \( y = \sqrt{x} - 6 \)?

**Tip**: The square root function is non-linear, so shifts impact the graph differently than for linear functions like \( y = x \).

Explore the transformation of the square root function y = √x - 6 with a detailed explanation of the vertical shift, domain, and range. Understand how the graph shifts 6 units downwards and how this affects the function. Suitable for students in Grades 9-10.

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