Graph Completion Challenge: Determine Increasing and Decreasing Function Trends

To determine if it is possible to create a function that is both increasing and decreasing with the two given points, we first identify that we need these points to serve as part of a curve that captures some increasing and decreasing interval logic.

Given the graphical representation of the two points, let's say Point A is approximately at $(x_1, y_1)$ and Point B at $(x_2, y_2)$ where $x_1 < x_2$ and $y_1$ and $y_2$ are different.

The challenge is to connect these points so a section of the curve shows an increase in $y$ value, followed by a section that shows a decrease, or vice versa. This means that:

• From Point A to Point B, without other constraints, a simple line might just connect increasing or flat, or decreasing, as a linear function.
• However, if we can have a part of the function increase to a point $C$, and then decrease to Point B, or vice versa, a composite or non-linear function allows variability of slopes.

For example, with polynomial functions or sinusoidal pieces, the behavior can vary such that while it rises and then falls, it can pass the given points.

Under continuity and allowing intervals, this scenario is possible. Between these two given points, you can define intervals $I_1$ and $I_2$ such that part of the function $f(x)$ is increasing over $I_1$ and decreasing over $I_2$.

Therefore, assuming correct choice of path and function forms, it is possible to create such a function.

In conclusion, the capability of implementing both increasing and decreasing sections through strategic function choice and segmentation confirms: Possible.