Part A: Complete the table to vertically stretch the parent functionf(x)=x3

Let's complete both parts of the problem step by step.

Part A: Vertically Stretch the Function \( f(x) = x^3 \) by a Factor of 3

When we vertically stretch a function by a factor of \( k \), we multiply the function by \( k \). In this case, \( k = 3 \).

So the modified function will be: \[ y = kf(x) = 3f(x) = 3x^3 \]

Now, we will calculate the values of \( y \) for \( x = 0, 1, 2 \):

1. For \( x = 0 \): \[ y = 3(0^3) = 3(0) = 0 \]

2. For \( x = 1 \): \[ y = 3(1^3) = 3(1) = 3 \]

3. For \( x = 2 \): \[ y = 3(2^3) = 3(8) = 24 \]

Completed Table for Part A:

| x | \( y = 3f(x) = 3x^3 \) | |---|-------------------------| | 0 | 0 | | 1 | 3 | | 2 | 24 |

Part B: Vertically Compress the Function \( f(x) = x^4 \) by a Factor of \( \frac{1}{2} \)

When we vertically compress a function by a factor of \( k \), we also multiply the function by \( k \). In this case, \( k = \frac{1}{2} \).

So the modified function will be: \[ y = kf(x) = \frac{1}{2}f(x) = \frac{1}{2}x^4 \]

Now, we will calculate the values of \( y \) for \( x = 0, 1, 2 \):

1. For \( x = 0 \): \[ y = \frac{1}{2}(0^4) = \frac{1}{2}(0) = 0 \]

2. For \( x = 1 \): \[ y = \frac{1}{2}(1^4) = \frac{1}{2}(1) = 0.5 \]

3. For \( x = 2 \): \[ y = \frac{1}{2}(2^4) = \frac{1}{2}(16) = 8 \]

Completed Table for Part B:

| x | \( y = \frac{1}{2}f(x) = \frac{1}{2}x^4 \) | |---|---------------------------------------------| | 0 | 0 | | 1 | 0.5 | | 2 | 8 |

These tables successfully represent the vertically stretched and compressed functions. If you have any further questions or need additional explanations, feel free to ask!