jmc

A $y$-intercept is a point on the graph that lies on the $y$-axis, so $x=0$. Hence, the number of $y$-intercepts corresponds to the number of real solutions of the quadratic equation $2y^2-3y+7=0$. The discriminant of this quadratic equation is $(-3{)}^2-4\cdot 2\cdot 7=-47$, which is negative, so the quadratic has no real roots. Therefore, the number of $y$-intercepts is $\boxed{0}$.