IDNLearn.com provides a collaborative environment for finding and sharing knowledge. Our platform offers detailed and accurate responses from experts, helping you navigate any topic with confidence.

Describe how the graph of the parent function [tex]y=\sqrt[3]{x}[/tex] is transformed when graphing [tex]y=\sqrt[3]{8x}-3[/tex].

The graph is translated 3 units [tex]$\square$[/tex]


Sagot :

Sure, let's analyze and describe the transformations in detail.

The parent function given is [tex]\( y = \sqrt[3]{x} \)[/tex].

The transformed function given is [tex]\( y = \sqrt[3]{8x} - 3 \)[/tex].

We need to determine the transformations applied to the parent function to obtain the transformed function.

1. Horizontal Transformation:

The term inside the cube root function, [tex]\( 8x \)[/tex], indicates a horizontal transformation. When a function is of the form [tex]\( \sqrt[3]{kx} \)[/tex], it represents a horizontal compression or stretch depending on the value of [tex]\( k \)[/tex].

- For [tex]\( y = \sqrt[3]{8x} \)[/tex]:
The factor [tex]\( 8 \)[/tex] suggests a horizontal compression by a factor of [tex]\( \frac{1}{8} \)[/tex].

2. Vertical Transformation:

The term outside of the cube root function, [tex]\( - 3 \)[/tex], indicates a vertical transformation. Specifically, it indicates a vertical translation.

- For [tex]\( y = \sqrt[3]{8x} - 3 \)[/tex]:
This implies a vertical translation downward by 3 units.

Putting it all together:

- The graph of [tex]\( y = \sqrt[3]{x} \)[/tex] is horizontally compressed by a factor of [tex]\( \frac{1}{8} \)[/tex].
- Then, the graph is translated 3 units downward.

Therefore, the answer is:

The graph is translated 3 units [tex]\(\textbf{downward}\)[/tex].