To determine the points for the transformation \( y = f\left(\frac{1}{3}x\right) \), we need to understand how the horizontal stretch affects the original points on the parent function \( y = f(x) \).
The transformation \( y = f\left(\frac{1}{3}x\right) \) indicates that every \( x \) value from the parent function is multiplied by 3. In other words:
- If \( (a, b) \) is a point on the parent function \( y = f(x) \), then the corresponding point on the transformed function \( y = f\left(\frac{1}{3}x\right) \) will be \( (3a, b) \).
Now, let's apply this transformation to the points given from the parent function:
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For the point \( (0, 0) \):
- \( x = 0 \) results in \( (3 \cdot 0, 0) \) = \( (0, 0) \)
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For the point \( (1, 1) \):
- \( x = 1 \) results in \( (3 \cdot 1, 1) \) = \( (3, 1) \)
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For the point \( (2, 4) \):
- \( x = 2 \) results in \( (3 \cdot 2, 4) \) = \( (6, 4) \)
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For the point \( (3, 9) \):
- \( x = 3 \) results in \( (3 \cdot 3, 9) \) = \( (9, 9) \)
Now we can fill out the table. Given the points we calculated, the table will look like this:
| \( x \) | \( f\left(\frac{1}{3}x\right) \) | |---------|----------------------------------| | 0 | 0 | | 3 | 1 | | 6 | 4 | | 9 | 9 |
So, the missing values in your table would be:
- When \( x = 6 \), \( f\left(\frac{1}{3} \cdot 6\right) = 4 \)
- When \( x = 9 \), \( f\left(\frac{1}{3} \cdot 9\right) = 9 \)
The completed table is as follows:
| \( x \) | \( f\left(\frac{1}{3}x\right) \) | |---------|----------------------------------| | 0 | 0 | | 3 | 1 | | 6 | 4 | | 9 | 9 |
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