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The task is to fill out the values for \( y = f(\frac{1}{2}x) \) and \( y = f(2x) \) using the values provided for \( f(x) \).

### For \( y = f(\frac{1}{2}x) \)
1. When \( x = -9 \), \( \frac{1}{2}x = -4.5 \) – Since -4.5 is not in the list, we write NONE.
2. When \( x = -6 \), \( \frac{1}{2}x = -3 \) – For \( f(-3) = 3 \).
3. When \( x = -3 \), \( \frac{1}{2}x = -1.5 \) – Since -1.5 is not in the list, we write NONE.
4. When \( x = 0 \), \( \frac{1}{2}x = 0 \) – For \( f(0) = 2 \).
5. When \( x = 3 \), \( \frac{1}{2}x = 1.5 \) – Since 1.5 is not in the list, we write NONE.
6. When \( x = 6 \), \( \frac{1}{2}x = 3 \) – For \( f(3) = -1 \).
7. When \( x = 9 \), \( \frac{1}{2}x = 4.5 \) – Since 4.5 is not in the list, we write NONE.

### For \( y = f(2x) \)
1. When \( x = -9 \), \( 2x = -18 \) – Since -18 is not in the list, we write NONE.
2. When \( x = -6 \), \( 2x = -12 \) – Since -12 is not in the list, we write NONE.
3. When \( x = -3 \), \( 2x = -6 \) – For \( f(-6) = -6 \).
4. When \( x = 0 \), \( 2x = 0 \) – For \( f(0) = 2 \).
5. When \( x = 3 \), \( 2x = 6 \) – For \( f(6) = 0 \).
6. When \( x = 6 \), \( 2x = 12 \) – Since 12 is not in the list, we write NONE.
7. When \( x = 9 \), \( 2x = 18 \) – Since 18 is not in the list, we write NONE.

Filling in the table, it should look like this:

| \( x \) | \( f(x) \) | \( f(\frac{1}{2}x) \) | \( f(2x) \) |
|---------|------------|-----------------------|-------------|
| -9      | -3         | NONE                  | NONE        |
| -6      | -6         | 3                     | NONE        |
| -3      | 3          | NONE                  | -6          |
| 0       | 2          | 2                     | 2           |
| 3       | -1         | NONE                  | 0           |
| 6       | 0          | -1                    | NONE        |
| 9       | -4         | NONE                  | NONE        |

Would you like more details or do you have any other questions? Here are some related topics you might find interesting:

1. What is the concept of a transformation of functions?
2. How does scaling the input \( x \) by a factor affect the function \( f(x) \)?
3. Examples of real-world scenarios where transformation of functions is applied.
4. The mathematical theory behind linear transformations.
5. What other transformations can be applied to functions besides scaling?

**Tip:** Always make sure the values used for transformation \( x \) values are present in the original \( f(x) \) table to avoid introducing errors.

Fill out the values for y = f(1/2x) and y = f(2x) using the given function table. x: [-9, -6, -3, 0, 3, 6, 9] with f(x): [-3, -6, 3, 2, -1, 0, -4].

Learn how to transform the function f(x) by scaling inputs using y = f(1/2x) and y = f(2x). This example problem uses the function f(x) with specific values of x and covers key algebraic transformations, suitable for students in Grades 9-12.

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