This is for week 2 assignment, correct? Please always make sure to mention which assignment you are referring to. Also, make sure you let us know what you have tried so far so we can give you better hints.

Here are some hints to get you started:

4: The equation of a parabola with vertex $(h,k)$ looks like $y - k = a(x - h)^2$ for some $a \neq 0$. This $a$ is the leading coefficient of the quadratic.

7: The graphs will intersect at a point $(x_0,y_0)$ iff $(x_0,y_0)$ satisfies BOTH equations. Use this to find the intersection points (in terms of $L$), and then find which values of $L$ make the $y$-coordinate of the point non-negative.

9: Recall $a$ is a root of a polynomial (in $x$) iff $(x-a)$ is a factor.