The graph of the function \( y = \sqrt{2x} \) is a **square root function**, which typically has the following properties:

### Key Features of \( y = \sqrt{2x} \):
1. **Domain**: Since we cannot take the square root of a negative number in real numbers, the function is defined for \( x \geq 0 \). Therefore, the domain is \( [0, \infty) \).
2. **Range**: The output of the square root function is always non-negative. Thus, the range is \( [0, \infty) \).
3. **Intercept**: The graph passes through the origin, as \( y(0) = \sqrt{2 \cdot 0} = 0 \).
4. **Shape**: The graph starts at the origin (0,0) and gradually increases as \( x \) increases. It is concave down, meaning the slope of the curve decreases as \( x \) increases, but remains positive.
5. **End behavior**: As \( x \to \infty \), \( y \to \infty \) but at a decreasing rate because the square root function grows more slowly than linear functions.

### Graph Characteristics:
- The function is **increasing** for all \( x \geq 0 \).
- It is a **non-linear** function, specifically a radical function.
- The graph is smooth and starts at the point \( (0, 0) \) and rises to the right.

Would you like more details or an image of the graph? Let me know if you have any questions!

### Related Questions:
1. What is the domain and range of the function \( y = \sqrt{x} \)?
2. How does the graph of \( y = \sqrt{2x} \) differ from \( y = \sqrt{x} \)?
3. What are the transformations applied to the graph of \( y = \sqrt{x} \) to obtain \( y = \sqrt{2x} \)?
4. How do square root functions compare to quadratic functions in terms of their graphs?
5. Can the graph of \( y = \sqrt{2x} \) ever produce negative values for \( y \)?

### Tip:
Remember, when working with square root functions, the argument inside the square root must always be non-negative in real number solutions.

Explore the graph of the square root function y = √2x, focusing on its domain, range, and key properties. Learn how this radical function behaves, including its intercept, end behavior, and why it is always increasing for x ≥ 0. Suitable for students in Grades 9-12.

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