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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!

First vertically compressing the graph y = f(x) by a factor of 1/3 then shifting down 2 units gives the same result as shifting the graph of y = f(x) down 2 units then vertically compressing by a factor of 1/3.

Explore how vertically compressing and shifting the graph of y = f(x) produces different results depending on the order of transformations. Learn the key differences between two function transformations, suitable for students in Grades 10-12.

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