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.