To show that the function \( y = \frac{1}{2} f(x) \) results in a vertical compression of the parent function \( y = x^4 \), we will complete the tables for both functions.
Parent Function Table: \( y = x^4 \)
| x | y = x^4 | |-------|-------------| | 0 | 0 | (0^4 = 0) | 1 | 1 | (1^4 = 1) | 2 | 16 | (2^4 = 16) | 3 | 81 | (3^4 = 81)
Vertically Compressed Function Table: \( y = \frac{1}{2} x^4 \)
To find the values of \( y = \frac{1}{2} x^4 \), we can substitute our \( x \) values into the equation.
| x | y = \frac{1}{2} x^4 | |-------|--------------------------| | 0 | 0 | (1/2 * 0^4 = 0) | 1 | 0.5 | (1/2 * 1^4 = 1/2) | 2 | 8 | (1/2 * 2^4 = 1/2 * 16 = 8) | 3 | 40.5 | (1/2 * 3^4 = 1/2 * 81 = 40.5)
Summary
- For the parent function \( y = x^4 \):
- At \( x = 1 \), \( y = 1 \)
- At \( x = 2 \), \( y = 16 \)
- For the vertically compressed function \( y = \frac{1}{2} x^4 \):
- At \( x = 1 \), \( y = 0.5 \)
- At \( x = 2 \), \( y = 8 \)
Thus, we can see from the tables that for \( k = \frac{1}{2} \), the function \( y = \frac{1}{2} x^4 \) indeed results in a vertical compression of the parent function \( y = x^4 \). The \( y \) values from the compressed function are half of those from the parent function for the same \( x \) values, illustrating the effect of vertical compression.
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