Part 4/5:
For ( x \geq 0 ), ( x^3 ) remains positive, hence ( |x^3| = x^3 ). However, for ( x < 0 ), ( x^3 ) becomes negative, leading us to rewrite it as ( -x^3 ):
[
y =
\begin{cases}
-x^3 & \text{if } x < 0 \
x^3 & \text{if } x \geq 0
\end{cases}
]
The resultant graph for ( y = |x^3| ) mirrors the negative side of the cubic curve upward, giving it a modified appearance akin to the original cubic graph's features.
Conclusion
In summary, the absolute value function serves as a crucial concept in mathematics, allowing us to understand the magnitude of numbers without regard to their sign. From the basic definition to piecewise function representations, mastering absolute values is essential for further mathematical explorations.