The graph you've shared represents the function $p(x) = \sqrt{3x}$. To graph this function using transformations, let's break down the transformations compared to the basic square root function $y = \sqrt{x}$.

Step-by-step analysis:

1. Starting with $y = \sqrt{x}$:

   • This is the basic square root function, which has a domain of $x \geq 0$ and a range of $y \geq 0$. It starts at the origin $(0, 0)$.

2. Transformation due to $3x$:

   • The expression $\sqrt{3x}$ implies a horizontal compression of the graph by a factor of $\frac{1}{\sqrt{3}}$. This is because the factor of 3 inside the square root causes the graph to stretch or compress horizontally.
   • Essentially, every point on the graph of $y = \sqrt{x}$ will now occur at a smaller $x$-coordinate, which causes the graph to "shrink" horizontally.

Key points for the graph of $p(x) = \sqrt{3x}$:

• Domain: The domain is $x \geq 0$ (just like the basic square root function).
• Starting point: The graph starts at the origin $(0, 0)$.
• Shape: The graph increases more rapidly than $y = \sqrt{x}$ due to the compression by a factor of $\frac{1}{\sqrt{3}}$.

This transformation results in a graph that is steeper compared to $y = \sqrt{x}$, which is consistent with what you've shown in the image.

Would you like further details or have any questions about the transformation process?

Related questions to explore:

1. How does the graph of $p(x) = \sqrt{3x}$ change if the coefficient of $x$ inside the square root was 2 instead of 3?
2. How would the graph of $p(x) = \sqrt{x}$ behave if we had a negative coefficient inside the square root, such as $p(x) = \sqrt{-x}$?
3. How would the graph change if there was a vertical shift involved, like in $p(x) = \sqrt{3x} + 2$?
4. How does the function $p(x) = \sqrt{3x}$ compare to $p(x) = 3\sqrt{x}$?
5. What are the effects of other transformations like reflections and stretches on the square root function?

Tip: Always analyze the function inside the square root first (for horizontal shifts or stretches) and then any external changes like vertical shifts or reflections.