To graph the equation $Y = 3 \sqrt{8x}$, we first need to rewrite it in a more familiar form to understand its behavior.

Steps:

1. Rewrite the equation: $Y = 3 \sqrt{8x} = 3 \times \sqrt{8} \times \sqrt{x} = 6\sqrt{2} \times \sqrt{x}$ This simplifies to: $Y = 6\sqrt{2} \cdot \sqrt{x}$ Here, $\sqrt{x}$ indicates that this function is defined only for $x \geq 0$.

2. Shape of the graph:

   • Since the equation involves $\sqrt{x}$, it has a half-parabola shape (rightward curve starting from the origin).
   • As $x$ increases, $Y$ grows at a decreasing rate because square root functions grow slower than linear functions.

3. Important points:

   • At $x = 0$: $Y = 6\sqrt{2} \times \sqrt{0} = 0$.
   • At $x = 1$: $Y = 6\sqrt{2} \times \sqrt{1} = 6\sqrt{2} \approx 8.49$.
   • At $x = 4$: $Y = 6\sqrt{2} \times \sqrt{4} = 12\sqrt{2} \approx 16.97$.

4. Behavior:

   • The graph starts at the origin $(0, 0)$ and moves upwards and to the right.
   • The curve is steep at first but then flattens out as $x$ increases.

Would you like a visual representation of the graph?

Related Questions:

1. How does the behavior of square root functions differ from linear ones?
2. Can we transform this function to shift or stretch the graph?
3. What happens if we introduce negative values for $x$?
4. How can we find the domain and range of this function?
5. What is the role of the constant $6\sqrt{2}$ in shaping the graph?

Tip: When graphing square root functions, always check the domain to ensure you are only plotting valid points (i.e., where the function is defined).