To understand how the graph of [tex]\( y = -\sqrt[3]{x-4} \)[/tex] is transformed to produce the graph of [tex]\( y = -\sqrt[3]{2x} - 4 \)[/tex], we need to analyze the transformations step-by-step.
Let's start with the given function:
### Original Function:
[tex]\[ y = -\sqrt[3]{x-4} \][/tex]
### Target Function:
[tex]\[ y = -\sqrt[3]{2 x} - 4 \][/tex]
### Step 1: Horizontal Compression
The term [tex]\( 2x \)[/tex] inside the cubic root indicates a horizontal compression. In general, [tex]\( y = f(ax) \)[/tex] represents a horizontal compression by a factor of [tex]\( \frac{1}{a} \)[/tex]. For our case, [tex]\( a = 2 \)[/tex], so the graph is compressed horizontally by a factor of [tex]\( \frac{1}{2} \)[/tex] or simply 2.
### Step 2: Vertical Shift
The term [tex]\( -4 \)[/tex] outside the cubic root indicates a vertical shift downward. In general, [tex]\( y = f(x) + c \)[/tex] shifts the graph of [tex]\( f \)[/tex] vertically by [tex]\( c \)[/tex] units. Since [tex]\( c \)[/tex] here is [tex]\( -4 \)[/tex], the graph is shifted downward by 4 units.
Putting these transformations together:
1. Horizontal Compression by a factor of 2:
- The original graph [tex]\( y = -\sqrt[3]{x-4} \)[/tex] is horizontally compressed by a factor of 2, transforming it to [tex]\( y = -\sqrt[3]{2(x-4)} = -\sqrt[3]{2x-8} \)[/tex].
2. Vertical Shift Downward by 4 units:
- The function [tex]\( y = -\sqrt[3]{2x} \)[/tex] is then moved down by 4 units, resulting in [tex]\( y = -\sqrt[3]{2x} - 4 \)[/tex].
Therefore, the correct transformation description is:
- The graph is compressed horizontally by a factor of 2 and then moved down 4 units.
Hence, the correct answer is:
- The graph is compressed horizontally by a factor of 2 and then moved down 4 units.
