Let's break down the transformation of the function [tex]\( f(x) = x^3 \)[/tex] to derive the function [tex]\( g(x) = -\frac{1}{2}(x-3)^3 \)[/tex].
We'll apply these transformations to the reference points (-1,-1), (0,0), and (1,1).
### Identifying the Transformations:
1. Horizontal Translation: The term inside the cube function [tex]\( (x - 3) \)[/tex] represents a translation to the right by 3 units.
2. Reflection and Vertical Scaling: The multiplier [tex]\( -\frac{1}{2} \)[/tex] indicates two separate effects:
- Reflection over the x-axis: The negative sign ([tex]\(-\)[/tex]) flips the graph upside down.
- Vertical Scaling: The factor [tex]\( \frac{1}{2} \)[/tex] compresses the graph vertically by a factor of 2.
### Applying Transformations to Reference Points:
1. Reference Point (-1, -1):
- Horizontal Translation: Move right by 3 units.
[tex]\[
(-1 + 3, -1) = (2, -1)
\][/tex]
- Reflection and Vertical Scaling: Reflect over the x-axis and scale vertically by [tex]\(\frac{1}{2}\)[/tex].
[tex]\[
\left(2, -\frac{1}{2} \times -1\right) = (2, 0.5)
\][/tex]
2. Reference Point (0, 0):
- Horizontal Translation: Move right by 3 units.
[tex]\[
(0 + 3, 0) = (3, 0)
\][/tex]
- Reflection and Vertical Scaling: Reflect over the x-axis and scale vertically by [tex]\(\frac{1}{2}\)[/tex].
[tex]\[
\left(3, -\frac{1}{2} \times 0\right) = (3, 0)
\][/tex]
3. Reference Point (1, 1):
- Horizontal Translation: Move right by 3 units.
[tex]\[
(1 + 3, 1) = (4, 1)
\][/tex]
- Reflection and Vertical Scaling: Reflect over the x-axis and scale vertically by [tex]\(\frac{1}{2}\)[/tex].
[tex]\[
\left(4, -\frac{1}{2} \times 1\right) = (4, -0.5)
\][/tex]
### Summary of Transformations:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Reference Points} & \text{First Transformation} & \text{Second Transformation} \\
\hline
(-1, -1) & (2, -1) & (2, 0.5) \\
\hline
(0, 0) & (3, 0) & (3, 0) \\
\hline
(1, 1) & (4, 1) & (4, -0.5) \\
\hline
\end{array}
\][/tex]
Thus, the transformed points on the graph of [tex]\( g(x) \)[/tex] are:
[tex]\[
(2, 0.5), (3, 0), (4, -0.5)
\][/tex]
By graphing these points on the same coordinate plane, you can visualize how the graph of [tex]\( f(x) = x^3 \)[/tex] has been transformed to produce the graph of [tex]\( g(x) = -\frac{1}{2}(x-3)^3 \)[/tex].
