Question 1055058
The first two example parts of the question take an equation form,  {{{4p(x-h)=(y-0)^2}}}, which you would be able to derive if you assumed a given vertex, directrix, focus.  See a video about the derivation or the discussion in your textbook.




The example for p=2 would give  {{{8(x-h)=(y-0)^2}}}, and since you're also given that vertex is the origin, (0,0), the equation becomes simply {{{8x=y^2}}}.  If you want this in the more typical standard form, then  x={{{highlight((1/8)y^2)}}}.




The example for the parabola to contain  point (4,2), and axis of symmetry still be x axis, means you have {{{4p(x-h)=(y-0)^2}}} or  better,  {{{4px=y^2}}}; and you use the given included point to find the value of p.
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{{{p=(y^2)/(4x)}}}
and putting in the coordinates for the point,
{{{p=(2^2)/(4*4)}}}
{{{p=1/4}}}
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and this finished equation is  {{{highlight(x=y^2)}}}.