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How is the graph of [tex]\( y = -\sqrt[3]{x-4} \)[/tex] transformed to produce the graph of [tex]\( y = -\sqrt[3]{2x} - 4 \)[/tex]?

A. The graph is stretched horizontally by a factor of 2 and then moved right 4 units.
B. The graph is compressed horizontally by a factor of 2 and then moved down 4 units.
C. The graph is compressed horizontally by a factor of 2, moved left 4 units, and moved down 4 units.
D. The graph is stretched horizontally by a factor of 2, moved left 4 units, and moved down 4 units.

Sagot :

GinnyAnswer
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."
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