Certainly! Let's analyze the transformations of the parent function [tex]\( y = \sqrt[3]{x} \)[/tex] to the transformed function [tex]\( y = -(0.4)\sqrt[3]{x-2} \)[/tex].
1. Horizontal Shift:
- In the transformed function, the argument of the cube root is [tex]\( x-2 \)[/tex]. This suggests a horizontal shift.
- The term [tex]\( x-2 \)[/tex] indicates a shift to the right by 2 units.
- Transformation: Horizontal shift to the right by 2 units.
2. Vertical Compression:
- The factor multiplying the cube root is [tex]\( 0.4 \)[/tex].
- This means that the output values of the function are scaled by a factor of [tex]\( 0.4 \)[/tex], resulting in a vertical compression.
- Transformation: Vertical compression by a factor of [tex]\( 0.4 \)[/tex].
3. Reflection Over the x-axis:
- The negative sign outside the [tex]\( (0.4) \sqrt[3]{x-2} \)[/tex] indicates a reflection.
- This reflection is over the x-axis because the entire function is multiplied by -1, which inverts all the y-values.
- Transformation: Reflection over the x-axis.
So, the step-by-step transformations from the parent function [tex]\( y = \sqrt[3]{x} \)[/tex] to the transformed function [tex]\( y = -(0.4)\sqrt[3]{x-2} \)[/tex] are:
1. Horizontal shift to the right by 2 units,
2. Vertical compression by a factor of [tex]\( 0.4 \)[/tex],
3. Reflection over the x-axis.
These steps describe how the parent function is altered to become the transformed function.
