To determine the transformations that have been performed on the graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] to obtain the graph of [tex]\( g(x) = \frac{1}{4} \sqrt[3]{x} \)[/tex], we can analyze how the equation [tex]\( g(x) = \frac{1}{4} f(x) \)[/tex] modifies the graph of [tex]\( f(x) \)[/tex].
1. Original Function:
- The original function is [tex]\( f(x) = \sqrt[3]{x} \)[/tex].
2. Transformed Function:
- The transformed function is [tex]\( g(x) = \frac{1}{4} \sqrt[3]{x} \)[/tex].
To understand the effect of multiplying by [tex]\(\frac{1}{4}\)[/tex], follow these steps:
### Step-by-Step Transformation Analysis:
- Vertical Compression:
- The function [tex]\( g(x) = \frac{1}{4} \sqrt[3]{x} \)[/tex] is derived by multiplying [tex]\( f(x) \)[/tex] by [tex]\(\frac{1}{4}\)[/tex].
- This means that every y-value of [tex]\( f(x) \)[/tex] is scaled by a factor of [tex]\(\frac{1}{4}\)[/tex].
- When the function is multiplied by a positive constant between 0 and 1, it causes a vertical compression of the graph.
- Specifically, the y-values are brought closer to the x-axis by the factor [tex]\(\frac{1}{4}\)[/tex].
### Conclusion:
- The transformation applied to the graph is compressing the graph closer to the x-axis.
Therefore, the correct answer is:
- compress the graph closer to the x-axis
