If a 2x8 joist has a 3 inch x 3 inch notch at the bottom where it rests on a beam:

When a joist is notched, particularly at its bottom edge, the structural integrity of the joist can be compromised. Notching removes material that contributes to the joist's ability to resist bending and shear forces. This kind of modification is generally discouraged in structural applications because it can lead to stress concentrations in the remaining material of the joist. If a crack develops from the notch, it is likely to extend horizontally as the joist experiences loads, due to the increased stress in that area.

Therefore, the statement regarding the potential for a horizontal crack extending from the notch is valid in recognizing that the notching can lead to structural failure or weakening of the joist. Notching at the bottom is particularly detrimental because it reduces the tension capacity of the joist, which can lead to serious issues like cracking, especially under load.

Understanding the implications of notching is critical for ensuring the safety and durability of structural elements in construction. The other possible choices do not directly relate to the consequences of the notch in the same way, focusing less on structural integrity and more on potential alterations or conditions that don't necessarily stem from the notch itself.