Question 1039314
Only doing one of those for you.


C.  V(4,-4), y+5=0



This might be vertex and directrix.  Solving for y gives y=-5 and no restriction on x for the directrix.  This can also be stated as a variable point  (x,-5), the directrix.


The focus is on the other side of the vertex.  From -4 to -5 is a distance of 1.  The focus is therefore 1 unit away from the vertex and this focus must be (4,-3).


Use the definition of a parabola.  Find the equation for Distance from (x,y) to (x,-5) is equal to Distance from (x,y) to (4,-3).


{{{sqrt((x-x)^2+(y-(-5))^2)=sqrt((x-4)^2+(y-(-3))^2)}}}
Simplify this into whatever form or forms you need.
Find a good demonstration example for how this works, here:  <a href="https://www.youtube.com/watch?v=Wworlx39KfQ">deriving equation of parabola for given focus and directrix - video example</a>
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The finished general form for this example  in your list is  {{{highlight(y=(1/4)x^2-2x)}}}.


Its graph appears like {{{graph(300,300,-1,9,-5,5,x^2/4-2x)}}}.



The same channel with that video also has something to explain your last two example equations.