To 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], we will break down the transformations step by step.
1. Translation (Vertical Shift):
- The original function is [tex]\( y=\sqrt[3]{x} \)[/tex].
- The term [tex]\(-3\)[/tex] in [tex]\( y=\sqrt[3]{8x}-3 \)[/tex] indicates a vertical shift. Specifically, it translates the graph 3 units downwards.
So, the first transformation is:
[tex]\[
\text{Translation: 3 units downwards}
\][/tex]
2. Simplify the Expression:
- Now, we need to simplify the expression inside the radical to determine the vertical scaling.
- In the transformed function [tex]\( y=\sqrt[3]{8x}-3 \)[/tex], the term inside the radical is [tex]\( 8x \)[/tex].
3. Vertical Scaling:
- The factor [tex]\( 8 \)[/tex] inside the radical affects the [tex]\( x \)[/tex]-values in the function.
- For the cube root function [tex]\( y=\sqrt[3]{8x} \)[/tex], the coefficient 8 scales the [tex]\( x \)[/tex]-values horizontally. This can be interpreted as a horizontal compression by a factor of [tex]\( \frac{1}{8} \)[/tex].
- Hence, the parent function [tex]\( y=\sqrt[3]{x} \)[/tex] is horizontally compressed by a factor of [tex]\( \frac{1}{8} \)[/tex].
Thus, the second transformation is:
[tex]\[
\text{Horizontal compression by a factor of } \frac{1}{8}
\][/tex]
4. Function Form:
- We can now write the transformation in the standard format [tex]\( y = a \sqrt[3]{x-h} + k \)[/tex].
- Here, [tex]\( a = 8 \)[/tex], [tex]\( h = 0 \)[/tex] (no horizontal shift), and [tex]\( k = -3 \)[/tex].
From these steps, we can summarize the transformations of the graph [tex]\( y=\sqrt[3]{x} \)[/tex] when graphing [tex]\( y=\sqrt[3]{8x}-3 \)[/tex] as follows:
1. Translation (Vertical Shift): The graph is translated 3 units downwards.
2. Horizontal Compression: The graph is compressed horizontally by a factor of [tex]\( \frac{1}{8} \)[/tex].
So, the transformed graph [tex]\( y = \sqrt[3]{8x} - 3 \)[/tex] involves a vertical shift of 3 units downwards and a horizontal compression by a factor of [tex]\( \frac{1}{8} \)[/tex].
