Question
From the equation f(x)= ^3 square root x + 8 and f(4x)= ^3 square root 4x +8. Identify whether the graph shows a compression or a stretch
Answers
GPT-4o mini
To analyze whether the function \( f(x) = \sqrt[3]{x} + 8 \) and \( f(4x) = \sqrt[3]{4x} + 8 \) results in a compression or a stretch of the graph, we can compare the transformations involved.
1. **Understanding the function**: The function \( f(x) = \sqrt[3]{x} + 8 \) is a cubic root transformed vertically by 8 units.
2. **Analyzing \( f(4x) \)**: The function \( f(4x) = \sqrt[3]{4x} + 8 \) involves substituting \( 4x \) into the original function where \( x \) is used.
3. **Compression or Stretch**:
- A function of the form \( f(kx) \) where \( k > 1 \) compresses the graph horizontally (bringing it closer to the y-axis) because the input is multiplied by a factor greater than 1, which effectively speeds up the growth of the function.
- Conversely, if \( 0 < k < 1 \), the graph stretches horizontally.
In this case, since \( f(4x) \) involves \( k = 4 \) (which is greater than 1), it will compress the graph of \( f(x) \) horizontally.
**Conclusion**: The transformation from \( f(x) \) to \( f(4x) \) results in a horizontal compression of the graph by a factor of 4.
1. **Understanding the function**: The function \( f(x) = \sqrt[3]{x} + 8 \) is a cubic root transformed vertically by 8 units.
2. **Analyzing \( f(4x) \)**: The function \( f(4x) = \sqrt[3]{4x} + 8 \) involves substituting \( 4x \) into the original function where \( x \) is used.
3. **Compression or Stretch**:
- A function of the form \( f(kx) \) where \( k > 1 \) compresses the graph horizontally (bringing it closer to the y-axis) because the input is multiplied by a factor greater than 1, which effectively speeds up the growth of the function.
- Conversely, if \( 0 < k < 1 \), the graph stretches horizontally.
In this case, since \( f(4x) \) involves \( k = 4 \) (which is greater than 1), it will compress the graph of \( f(x) \) horizontally.
**Conclusion**: The transformation from \( f(x) \) to \( f(4x) \) results in a horizontal compression of the graph by a factor of 4.