Question

a parent function and transformed function are shown:
y = \sqrt3{x}
y = - (0.4) \sqrt3{x - 2}
which of the following describes the graph of the transformed function compared with the parent function? choose three correct answers.
reflected over the x-axis compressed by a factor of 0.4 translated 2 units up
translated 2 units right stretched by a factor of 0.4 translated 2 units left
translated 2 units down

Explanation:

Step1: Recall Transformations

For a function \( y = a\sqrt[3]{x - h}+k \), \( a \) affects vertical stretch/compression and reflection, \( h \) horizontal translation, \( k \) vertical translation.
Parent: \( y=\sqrt[3]{x} \), Transformed: \( y = - 0.4\sqrt[3]{x - 2} \)

Step2: Analyze Reflection

The negative sign in front (\( a=-0.4 \)) means reflection over \( x \)-axis. So "reflected over the \( x \)-axis" is correct.

Step3: Analyze Stretch/Compression

\( |a| = 0.4<1 \), so vertical compression by factor 0.4. "compressed by a factor of 0.4" is correct.

Step4: Analyze Horizontal Translation

\( h = 2 \) (from \( x - 2 \)), so horizontal translation 2 units right. "translated 2 units right" is correct.

Answer:

• reflected over the \( x \)-axis
• compressed by a factor of 0.4
• translated 2 units right