Question

complete the tables to show that y=kf(x) results in the parent function being vertically compressed when k=12 .(1 point) x y=x4 0 0 1 1 2 16 x y=12x4 0 0 1 2

Answer 1

To complete the table and show that \( y = kf(x) \) results in the parent function being vertically compressed when \( k = \frac{1}{12} \), let's first analyze the given functions:

1. The parent function \( y = f(x) = x^4 \)
2. The function with the vertical compression \( y = 12f(x) = 12x^4 \)

### Parent Function Table ( \( y = x^4 \) )
| x | y = x^4 |
|-----|-----------|
| 0 | 0 |
| 1 | 1 |
| 2 | 16 |

### Compressed Function Table ( \( y = 12x^4 \) )
To find the values for \( y = 12x^4 \), we will multiply the corresponding \( y \) values by 12.

1. For \( x = 0 \):
\[
y = 12(0^4) = 12 \times 0 = 0
\]

2. For \( x = 1 \):
\[
y = 12(1^4) = 12 \times 1 = 12
\]

3. For \( x = 2 \):
\[
y = 12(2^4) = 12 \times 16 = 192
\]

### Completed Compressed Function Table ( \( y = 12x^4 \) )
| x | y = 12x^4 |
|-----|------------|
| 0 | 0 |
| 1 | 12 |
| 2 | 192 |

Now we can summarize the vertical compression when \( k = \frac{1}{12} \).

The original parent function \( y = x^4 \) values have been scaled by 12 in the vertically compressed function \( y = 12x^4 \):

- At \( x = 0 \), the output is the same (0).
- At \( x = 1 \), the output increased from 1 to 12.
- At \( x = 2 \), the output increased from 16 to 192.

This shows that when \( k = 12 \), the outputs grow much more quickly than the original function \( y = x^4 \), confirming a vertical stretching rather than compression in this case. To verify compression, we'd look at \( k = \frac{1}{12} \).

For \( k = \frac{1}{12} \):

### For Compressed Function Table ( \( y = \frac{1}{12}x^4 \) )

1. For \( x = 0 \):
\[
y = \frac{1}{12}(0^4) = 0
\]

2. For \( x = 1 \):
\[
y = \frac{1}{12}(1^4) = \frac{1}{12} \approx 0.083
\]

3. For \( x = 2 \):
\[
y = \frac{1}{12}(2^4) = \frac{16}{12} = \frac{4}{3} \approx 1.333
\]

### Completed Compressed Function Table ( \( y = \frac{1}{12}x^4 \) )
| x | y = \(\frac{1}{12}x^4\) |
|-----|---------------------|
| 0 | 0 |
| 1 | 0.083 |
| 2 | 1.333 |

This illustrates that the vertical compression caused by \( k = \frac{1}{12} \) reduces the values compared to the parent function.

If you have any further questions or need additional information, feel free to ask!