Let's take a detailed look at how we transform the graph of [tex]\( y = -\sqrt[3]{x-4} \)[/tex] to get [tex]\( y = -\sqrt[3]{2x} - 4 \)[/tex].
### Step-by-Step Transformation:
1. Original Function:
[tex]\[
y = -\sqrt[3]{x - 4}
\][/tex]
2. Transform 1: Compress Horizontally by a Factor of 2
Compressing horizontally by a factor of 2 involves replacing [tex]\( x \)[/tex] with [tex]\( 2x \)[/tex]. This step modifies the horizontal scale of the graph.
- In [tex]\( y = -\sqrt[3]{x - 4} \)[/tex]:
- We replace [tex]\( x \)[/tex] with [tex]\( 2x \)[/tex],
- This results in:
[tex]\[
y = -\sqrt[3]{2x - 4}
\][/tex]
3. Transform 2: Move Vertically Down by 4 Units
Moving the graph down involves subtracting 4 from the entire function.
- In [tex]\( y = -\sqrt[3]{2x - 4} \)[/tex]:
- We subtract 4,
- This results in:
[tex]\[
y = -\sqrt[3]{2x} - 4
\][/tex]
### Resulting Transformation Description:
With these transformations, we can conclude that:
- The graph is compressed horizontally by a factor of 2 (since we replaced [tex]\( x \)[/tex] with [tex]\( 2x \)[/tex]).
- Then it is moved down by 4 units (since we subtracted 4 from the function).
Therefore, the correct transformation description is:
"The graph is compressed horizontally by a factor of 2 and then moved down 4 units."
