Question

which of the following describes the graph of $y = \sqrt3{27x - 54} + 5$ compared with the parent cube root function?
horizontal translation:

vertical translation:

stretch/compression:

reflection:

Explanation:

Step1: Simplify the function

First, simplify \( y = \sqrt[3]{27x - 54}+5 \). Factor out 27 from the cube - root: \( y=\sqrt[3]{27(x - 2)}+5 \). Since \( \sqrt[3]{27}=3 \), the function can be rewritten as \( y = 3\sqrt[3]{x - 2}+5 \). The parent cube - root function is \( y=\sqrt[3]{x} \).

Step2: Analyze horizontal translation

For a function of the form \( y = \sqrt[3]{x - h}+k \), the horizontal translation is determined by \( h \). In our function \( y = 3\sqrt[3]{x - 2}+5 \), comparing with the parent function \( y=\sqrt[3]{x} \) (where \( h = 0\) in the form \( y=\sqrt[3]{x - h}+k \)), here \( h = 2 \). So there is a horizontal translation of 2 units to the right.

Step3: Analyze vertical translation

For the function \( y = 3\sqrt[3]{x - 2}+5 \), comparing with the parent function \( y=\sqrt[3]{x} \) (where \( k = 0\) in the form \( y=\sqrt[3]{x - h}+k \)), here \( k = 5 \). So there is a vertical translation of 5 units up.

Step4: Analyze stretch/compression

For a function of the form \( y = a\sqrt[3]{x - h}+k \), the factor \( a \) determines the stretch or compression. In our function \( y = 3\sqrt[3]{x - 2}+5 \), \( a = 3>1 \). So there is a vertical stretch by a factor of 3 (or a horizontal compression by a factor of \( \frac{1}{3} \), but since the coefficient is in front of the cube - root, we consider vertical stretch).

Step5: Analyze reflection

For a reflection, we look at the sign of \( a \). In our function, \( a = 3>0 \), so there is no reflection (reflection would occur if \( a<0 \)).

Answer:

• Horizontal translation: 2 units to the right
• Vertical translation: 5 units up
• Stretch/compression: Vertical stretch by a factor of 3 (or horizontal compression by a factor of \( \frac{1}{3} \))
• Reflection: No reflection