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.
translated 2 units up
compressed by a factor of 0.4
stretched by a factor of 0.4
translated 2 units right
translated 2 units left
reflected over the $x$-axis

Explanation:

To determine the transformations from the parent function \( y = \sqrt[3]{x} \) to the transformed function \( y = - (0.4) \sqrt[3]{x - 2} \), we analyze each part of the transformed function:

Step 1: Horizontal Translation

For a cube root function of the form \( y = \sqrt[3]{x - h} \), the graph is translated \( h \) units to the right if \( h > 0 \). In the transformed function, we have \( \sqrt[3]{x - 2} \), so \( h = 2 \). This means the graph is translated 2 units to the right.

Step 2: Vertical Stretch/Compression

For a function of the form \( y = a \sqrt[3]{x} \), if \( 0 < |a| < 1 \), the graph is compressed vertically by a factor of \( |a| \). Here, \( a = 0.4 \) (and \( 0 < 0.4 < 1 \)), so the graph is compressed by a factor of 0.4.

Step 3: Reflection

For a function of the form \( y = -a \sqrt[3]{x} \), the negative sign indicates a reflection over the \( x \)-axis. In the transformed function, we have a negative sign in front of the \( 0.4 \sqrt[3]{x - 2} \) term, so the graph is reflected over the \( x \)-axis.

Now let's analyze each option:

• "translated 2 units up": Incorrect. The translation is horizontal (right), not vertical (up).
• "compressed by a factor of 0.4": Correct (from Step 2).
• "stretched by a factor of 0.4": Incorrect. Since \( 0.4 < 1 \), it's a compression, not a stretch.
• "translated 2 units right": Correct (from Step 1).
• "translated 2 units left": Incorrect. The translation is to the right (because of \( x - 2 \)), not left.
• "reflected over the \( x \)-axis": Correct (from Step 3).

Answer:

The three correct answers are:

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