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.

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