To demonstrate that the function \( y = \frac{1}{2}f(x) \) results in a vertical compression of the parent function \( y = f(x) = x^4 \) when \( k = \frac{1}{2} \), we will complete the second table (for \( y = \frac{1}{2}x^4 \)) based on the values from the first table.
Parent Function Table (y = x^4)
| x | y = x^4 | |-----|-----------| | 0 | 0 | | 1 | 1 | | 2 | 16 |
Compressed Function Table (y = 1/2 x^4)
To calculate the values for the second table, we will take the values from the first table and multiply them by \(\frac{1}{2}\):
- For \( x = 0 \): \( y = \frac{1}{2} \cdot 0^4 = \frac{1}{2} \cdot 0 = 0 \)
- For \( x = 1 \): \( y = \frac{1}{2} \cdot 1^4 = \frac{1}{2} \cdot 1 = \frac{1}{2} \)
- For \( x = 2 \): \( y = \frac{1}{2} \cdot 2^4 = \frac{1}{2} \cdot 16 = 8 \)
Complete Compressed Function Table (y = 1/2 x^4)
| x | y = 1/2 x^4 | |-----|---------------| | 0 | 0 | | 1 | 0.5 | | 2 | 8 |
Summary:
- At \( x=0 \), \( y=0 \) remains the same.
- At \( x=1 \), \( y=1 \) is vertically compressed to \( y=0.5 \).
- At \( x=2 \), \( y=16 \) is vertically compressed to \( y=8 \).
This confirms that the function \( y = \frac{1}{2}x^4 \) is a vertically compressed version of the parent function \( y = x^4 \).
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