The statement asks you to compare two transformations of the function $y = f(x)$:

1. First transformation: Vertically compress the graph of $y = f(x)$ by a factor of $\frac{1}{3}$, then shift the graph down by 2 units.

   This would result in the equation: $y = \frac{1}{3} f(x) - 2$

2. Second transformation: Shift the graph $y = f(x)$ down by 2 units first, then vertically compress it by a factor of $\frac{1}{3}$.

   This would result in the equation: $y = \frac{1}{3} (f(x) - 2) = \frac{1}{3} f(x) - \frac{2}{3}$

Comparing the results of these transformations:

• The first transformation results in $y = \frac{1}{3} f(x) - 2$.
• The second transformation results in $y = \frac{1}{3} f(x) - \frac{2}{3}$.

These two transformations do not give the same result, as the constants $-2$ and $-\frac{2}{3}$ are different. Therefore, the statement is false.

Would you like further details or have any questions?

Here are 5 related questions to expand on this concept:

1. How does shifting a graph horizontally affect the function?
2. What is the difference between vertical compression and vertical stretching?
3. How does the order of transformations affect the final graph of a function?
4. Can you explain how reflecting a graph across the x-axis works?
5. What is the effect of horizontally compressing a graph by a factor?

Tip: Always remember that the order of transformations, especially when involving shifts and compressions or stretches, can affect the outcome!