To understand the transformation of the graph of the parent function [tex]\( y = \sqrt[3]{x} \)[/tex] when it becomes [tex]\( y = \sqrt[3]{8x} - 3 \)[/tex], let's break it down step-by-step.
1. Identify the components of the transformation:
- Horizontal Compression:
The given function is [tex]\( y = \sqrt[3]{8x} - 3 \)[/tex]. Notice the argument of the cubed root, [tex]\( 8x \)[/tex]. This indicates a horizontal transformation. When a function is in the form [tex]\( y = \sqrt[3]{kx} \)[/tex], where [tex]\( k \)[/tex] is a constant, the graph undergoes a horizontal compression or stretch. Specifically:
- If [tex]\( k > 1 \)[/tex], the graph is compressed horizontally by a factor of [tex]\( \frac{1}{k} \)[/tex].
- If [tex]\( 0 < k < 1 \)[/tex], the graph is stretched horizontally.
In our case, [tex]\( k = 8 \)[/tex] which means the graph is compressed horizontally by a factor of [tex]\( \frac{1}{8} \)[/tex].
2. Vertical Translation:
- The term [tex]\(-3\)[/tex] at the end of the function [tex]\( y = \sqrt[3]{8x} - 3 \)[/tex] suggests a vertical translation. In general, for a function [tex]\( y = f(x) + c \)[/tex]:
- If [tex]\( c > 0 \)[/tex], the graph shifts upward by [tex]\( c \)[/tex] units.
- If [tex]\( c < 0 \)[/tex], the graph shifts downward by [tex]\( |c| \)[/tex] units.
Here, [tex]\( c = -3 \)[/tex], so the graph shifts 3 units downward.
Combining these transformations:
- Horizontal Compression: The factor [tex]\( 8x \)[/tex] compresses the graph by a factor of [tex]\( \frac{1}{8} \)[/tex]. This means each point on the graph of [tex]\( y = \sqrt[3]{x} \)[/tex] will be closer to the y-axis.
- Vertical Translation: The term [tex]\(-3\)[/tex] translates the entire graph downward by 3 units.
Step-by-step transformation:
1. Start with the parent function [tex]\( y = \sqrt[3]{x} \)[/tex].
2. Apply the horizontal compression by a factor of [tex]\( \frac{1}{8} \)[/tex], resulting in [tex]\( y = \sqrt[3]{8x} \)[/tex].
3. Translate the graph downward by 3 units, giving [tex]\( y = \sqrt[3]{8x} - 3 \)[/tex].
So, the final transformation from the original graph of [tex]\( y = \sqrt[3]{x} \)[/tex] to [tex]\( y = \sqrt[3]{8x} - 3 \)[/tex] involves:
- A horizontal compression by a factor of [tex]\( \frac{1}{8} \)[/tex].
- A vertical translation 3 units downward.
Hence, the graph is horizontally compressed by a factor of [tex]\( \frac{1}{8} \)[/tex] and translated 3 units down.
