To complete the tables for the transformations of the given parent functions, we'll apply the specified factors:
Part A: Vertically Stretching \( f(x) = x^3 \) by a Factor of 3
The transformation for a vertical stretch by a factor \( k \) can be expressed as:
\[ y = k f(x) = 3 f(x) = 3 x^3 \]
Now we will plug in the values of \( x \) into the equation to find \( y \).
Table Calculation
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For \( x = 0 \): \[ y = 3 (0^3) = 3 \times 0 = 0 \]
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For \( x = 1 \): \[ y = 3 (1^3) = 3 \times 1 = 3 \]
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For \( x = 2 \): \[ y = 3 (2^3) = 3 \times 8 = 24 \]
Completed Table for Part A:
| x | y = kf(x) = 3x³ | |---|------------------| | 0 | 0 | | 1 | 3 | | 2 | 24 |
Part B: Vertically Compressing \( f(x) = x^4 \) by a Factor of \( \frac{1}{2} \)
The transformation for a vertical compression by a factor \( k \) can be expressed as:
\[ y = k f(x) = \frac{1}{2} f(x) = \frac{1}{2} x^4 \]
Now we will plug in the values of \( x \) into the equation to find \( y \).
Table Calculation
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For \( x = 0 \): \[ y = \frac{1}{2} (0^4) = \frac{1}{2} \times 0 = 0 \]
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For \( x = 1 \): \[ y = \frac{1}{2} (1^4) = \frac{1}{2} \times 1 = 0.5 \]
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For \( x = 2 \): \[ y = \frac{1}{2} (2^4) = \frac{1}{2} \times 16 = 8 \]
Completed Table for Part B:
| x | y = kf(x) = \( \frac{1}{2} x^4 \) | |---|----------------------------------| | 0 | 0 | | 1 | 0.5 | | 2 | 8 |
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