Once you know the $x$-value at the y-intercept, plug that value into $f(x)=\tfrac12(x-3)(x+1)$ to find the corresponding $y$-value.

After substituting, simplify the expressions in parentheses first, then multiply, and finally apply the factor of $\tfrac12$. If needed, convert your fraction to a decimal to compare with the answer choices.

In Desmos, type f(x) = 0.5*(x-3)*(x+1) into an expression line. The parabola for $f(x)=\tfrac12(x-3)(x+1)$ will appear.

Look at where the graph crosses the y-axis (the vertical axis where $x=0$). Click that point on the graph; Desmos will show its coordinates $(0,y)$. The $y$-value shown is the y-intercept.

Click the gear icon next to the function and choose "Convert to table" or add a table and use f(x) as the rule. In the $x$-column, enter 0. The corresponding $y$-value in the table is $f(0)$, which is the y-coordinate of the y-intercept.

The y-intercept is the point where the graph crosses the y-axis. On the y-axis, the $x$-coordinate is always $0$, so the y-intercept has the form $(0,y)$. That means we need to find $f(0)$ and then write the point as $(0,f(0))$.

Use the given function:

Substitute $x=0$:

Now simplify inside the parentheses: $0-3=-3$ and $0+1=1$, so

As a decimal, $-\tfrac32 = -1.5$.

The y-intercept has $x=0$ and $y=f(0)$. We found $f(0)=-1.5$, so the y-intercept is $(0,-1.5)$. This matches answer choice B) $(0,-1.5)$.