Calculate the Area Function: Shaded Triangle in a Square with Side Length a

To solve the problem, we will calculate the area of the triangle within a square of side length $a$.

Since the triangle is formed by a diagonal of the square, its base and height are both equal to the side length $a$ of the square. Thus, base = $a$ and height = $a$.

Using the formula for the area of a triangle, we have:

$\text{Area of the triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times a = \frac{1}{2} \times a^2$

This simplifies to $\frac{a^2}{2}$.

Therefore, the function that expresses the area of the shaded triangle is $\frac{a^2}{2}$, matching the choice with the answer: $y=\frac{a^2}{2}$.

The solution to the problem is $y=\frac{a^2}{2}$.