Question

The parent function y=f(x)
is horizontally stretched by a factor of 3. Points on the parent function include (0,0)
, (1,1)
, (2,4)
, and (3,9)
. Complete the table for y=f(13x)
.(1 point)
x f(13x)
0 0
3 1
4
9

Answer 1

To complete the table for \( y = f(13x) \), we need to understand that horizontally stretching a function by a factor of 3 means that for any point \( (a, b) \) on the original function \( y = f(x) \), the corresponding point on the stretched function would be \( (a/3, b) \).

Given the points on the parent function \( y = f(x) \):
- \( (0, 0) \)
- \( (1, 1) \)
- \( (2, 4) \)
- \( (3, 9) \)

Now let's determine the corresponding x-values for \( y = f(13x) \). This means we will analyze the function by substituting values into \( 13x \).

1. For \( x = 0 \):
\[
f(13 \cdot 0) = f(0) = 0
\]

2. For \( x = 3 \):
\[
f(13 \cdot 3) = f(39)
\]
Since none of the original parent function points provide \( f(39) \), we need to apply the scaling. The previous function must have produced an x-value of 1 (from original function) with a point of \( (1, 1) \). Since \( f(13x) \) takes on values \( f(1) = 1 \), it follows that:
\[
f(39) = 1
\]

3. For \( x = 4 \):
\[
f(13 \cdot 4) = f(52)
\]
Again, we do not have \( f(52) \) from the original points, but if we consider \( f(4) = 16 \) (from parabolic transformation), then \( f(52) \) approximates to our stretched scale.

4. For \( x = 9 \):
\[
f(13 \cdot 9) = f(117)
\]
Since \( f(9) = 81 \), from the original point \( (3, 9) \), we find that \( f(117) = 81 \).

Here’s a summary of the results in the table for \( y = f(13x) \):

| x | f(13x) |
|----|--------|
| 0 | 0 |
| 3 | 1 |
| 4 | \( ? \) |
| 9 | 81 |

The final values might rely on understanding the transformed range; generally, these points would show a continuous pattern.

Thus,
- For \( x = 4 \), without exact original function \( f \), values could still generalize over stretches. Assign explicitly based on function characteristic or through closest function estimation. However, general properties established \( f(4) \) across shifting outputs between the original function rescaling could provide a rough \( f(13 \cdot 4) = f(52) \) trendwise as closing faster estimates.

Final Completion for \( x = 4 \):
If \( 4 \to 16 \):

We can speculate \( \therefore = 16 \). Thus, \( f(13x) = 16 \).

Final table can be completed:

| x | f(13x) |
|----|--------|
| 0 | 0 |
| 3 | 1 |
| 4 | 16 |
| 9 | 81 |

Please adjust the particulars where exact transformation of function \( f \) would apply or yield other estimates on properties from original points.