To find the values of \( f(13x) \) based on the parent function points and the horizontal stretch factor, let's first understand how the transformation works.
The function \( f(x) \) is horizontally stretched by a factor of 3, which means that the new function can be expressed as: \[ y = f\left(\frac{x}{3}\right) \]
In this case, we need to determine points for the function \( y = f(13x) \).
For each original point \( (a, b) \) from the parent function, after the transformation:
- The x-coordinates change by the factor of 13, as \( 13x = a \) means \( x = \frac{a}{13} \).
- The y-coordinates stay the same.
Using the original points:
- For the point (0, 0): \[ x = 0 \Rightarrow f(13 \cdot 0) = f(0) = 0 \]
- For the point (1, 1): \[ x = \frac{1}{13} \Rightarrow f(13 \cdot \frac{1}{13}) = f(1) = 1 \]
- For the point (2, 4): \[ x = \frac{2}{13} \Rightarrow f(13 \cdot \frac{2}{13}) = f(2) = 4 \]
- For the point (3, 9): \[ x = \frac{3}{13} \Rightarrow f(13 \cdot \frac{3}{13}) = f(3) = 9 \]
Now, let's fill in the table:
| x | f(13x) | |----|--------| | 0 | 0 | | 3 | 1 | | 4 | 4 | | 9 | 9 |
Now, for the remaining value, we need to determine when \( 13x = 4 \): \[ x = \frac{4}{13} \]
Thus, at \( x = \frac{4}{13} \), \( f(4) = 4 \).
Finally, our completed table looks like this:
| x | f(13x) | |---------------|--------| | 0 | 0 | | \(\frac{1}{13}\) | 1 | | \(\frac{2}{13}\) | 4 | | \(\frac{3}{13}\) | 9 |
So, the values for \( x = 3 \) and \( x = 4 \) are \( 1 \) and \( 4 \), respectively. Therefore, the answer is:
- For \( x = 4 \), \( f(13x) = 4 \).
- For \( x = 9 \), \( f(13x) = 9 \).
Thus, you can fill in the table under \( x \) values provided.
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