Question 973502
{{{16y^2=49x^2+784}}}
The original equation can be transformed into an equivalent form
that will make all you need to know very clear.
{{{16y^2-49x^2=784}}}
It so happens that {{{16*49=784}}} ,
one of those lucky coincidences that happen very often in math homework.
{{{(16y^2-49x^2)/784=784/784}}}
{{{16y^2/784-49x^2/784=1}}}
{{{y^2/49-x^2/16=1}}}
{{{y^2/7^2-x^2/4^2=1}}}
It is clear from the last equation above that for {{{x=0}}} , {{{system(y=-7,"or",y=7)}}} .
You can also see that since {{{x^2/4^2>+0}}} ,
it must be true that {{{y^2/7^2>=1}}}<--->{{{y^2>=7^2}}} for all the points in the graph.
That means that the horizontal band with {{{-7<y<7}}} is a forbidden zone that the graph does not enter.
It also means that the points (0,-7), and (0,7) found above are the vertices, and that this is a hyperbola.
{{{y^2/7^2-x^2/4^2=1}}}<--->{{{y^2/7^2=x^2/4^2+1}}}<--->{{{y^2=7^2x^2/4^2+7^2}}}<--->{{{y^2/x^2=7^2/4^2+7^2/x^2}}} also makes it clear that as {{{x^2}}} increases,
the curve approaches the lines where
{{{y^2/x^2=7^2/4^2}}}<--->{{{system(y/x=7/4,"and",y/x=-7/4)}}}<--->{{{system(y=(7/4)x,"and",y=(-7/4)x)}}} .
Those two straight lines are the asymptotes.
So far we know that the curve looks like this
{{{graph(300,300,-10,10,-20,20,
7x/4,-7x/4,7sqrt(x^2/16+1),-7sqrt(x^2/16+1),7,-7)}}} ,
a smile and a frown tangent on the oustside to the horizontal lines {{{system(y=-7,"and",y=7)}}} ,
all symmetrical with respect to the x-axis and the origin, (0,0). 
All that is left to do, is to draw that rectangular box bounded by the lines
{{{system(y=-7,"and",y=7)}}} , and realize that the half sides, and half diagonals of that box area the {{{a}}} , {{{b}}} and {{{c}}} that they showed you in formulas in math class,
{{{a}}} , and {{{b}}} being the numbers that appear squared as denominators in  ,
and {{{c}}} being the distance from the center, (0,0), to the foci, at (0,-c), and (0,c).
{{{drawing(300,300,-10,10,-10,10,
rectangle(-4,-7,4,7),rectangle(4,0,3.5,0.5),
locate(2,3.7,c),locate(2.1,1.4,4),locate(3.5,3.5,7),
graph(300,300,-10,10,-10,10,
7x/4,-7x/4,7sqrt(x^2/16+1),-7sqrt(x^2/16+1))
)}}} In the right triangle, {{{c^2=7^2+4^2}}}<-->{{{c^2=65}}}-->{{{c=sqrt(65)}}}