Question

which of the following describes the graph of \\(y = \sqrt3{8x - 64} - 5\\) compared to the parent cube root function?

stretched by a factor of 2 and translated 64 units right and 5 units down
stretched by a factor of 8 and translated 8 units right and 5 units down
stretched by a factor of 2 and translated 8 units right and 5 units down
stretched by a factor of 8 and translated 64 units right and 5 units down

Explanation:

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<reasoning>

Factor the radicand

Using the Radical Simplification knowledge point
\[

$$\begin{aligned} y &= \sqrt[3]{8x - 64} - 5 \\ y &= \sqrt[3]{8(x - 8)} - 5 \end{aligned}$$

\]

Simplify the radical expression

Using the Radical Simplification knowledge point
\[

$$\begin{aligned} y &= \sqrt[3]{8} \cdot \sqrt[3]{x - 8} - 5 \\ y &= 2\sqrt[3]{x - 8} - 5 \end{aligned}$$

\]

Analyze the transformations

Using the Cube Root Transformations knowledge point
\[

$$\begin{aligned} &\text{Vertical stretch factor: } 2 \\ &\text{Horizontal translation: } 8 \text{ units right} \\ &\text{Vertical translation: } 5 \text{ units down} \end{aligned}$$

\]

Match with the given options

Comparing the derived transformations with the choices:

• Stretched by a factor of 2, translated 8 units right, and 5 units down.

This matches the third option.
</reasoning>

<answer>
<mcq-option>stretched by a factor of 2 and translated 64 units right and 5 units down</mcq-option>
<mcq-option>stretched by a factor of 8 and translated 8 units right and 5 units down</mcq-option>
<mcq-correct>stretched by a factor of 2 and translated 8 units right and 5 units down</mcq-correct>
<mcq-option>stretched by a factor of 8 and translated 64 units right and 5 units down</mcq-option>
</answer>

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Answer:

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<reasoning>

Factor the radicand

Using the Radical Simplification knowledge point
\[

$$\begin{aligned} y &= \sqrt[3]{8x - 64} - 5 \\ y &= \sqrt[3]{8(x - 8)} - 5 \end{aligned}$$

\]

Simplify the radical expression

Using the Radical Simplification knowledge point
\[

$$\begin{aligned} y &= \sqrt[3]{8} \cdot \sqrt[3]{x - 8} - 5 \\ y &= 2\sqrt[3]{x - 8} - 5 \end{aligned}$$

\]

Analyze the transformations

Using the Cube Root Transformations knowledge point
\[

$$\begin{aligned} &\text{Vertical stretch factor: } 2 \\ &\text{Horizontal translation: } 8 \text{ units right} \\ &\text{Vertical translation: } 5 \text{ units down} \end{aligned}$$

\]

Match with the given options

Comparing the derived transformations with the choices:

• Stretched by a factor of 2, translated 8 units right, and 5 units down.

This matches the third option.
</reasoning>

<answer>
<mcq-option>stretched by a factor of 2 and translated 64 units right and 5 units down</mcq-option>
<mcq-option>stretched by a factor of 8 and translated 8 units right and 5 units down</mcq-option>
<mcq-correct>stretched by a factor of 2 and translated 8 units right and 5 units down</mcq-correct>
<mcq-option>stretched by a factor of 8 and translated 64 units right and 5 units down</mcq-option>
</answer>

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