SOLUTION: sketch the portion of graph in the first quadrant and use symmetry to complete the rest of the graph 4x^2+16y^2=16

It's symmetrical with respect to the x-axis 
because if we replace x by -x and simplify, we
get the same equation we started with:







It's symmetrical with respect to the y-axis also 
because if we replace y by -y and simplify, we
get the same equation we started with:






 
It's symmetrical with respect to the origin 
because if we replace x by -x and y by -y
at the same time and simplify, we also
get the same equation we started with:







Next we find some first quadrant points.

Choose 

Substitute in










This gives  and 

So a point in the first quadrant is (0,1)

(Actually it's a point on the BORDER of the first
quadrant, on the y-axis)

Choose 

Substitute in









This gives  and 

So a point in the first quadrant is (0,),

which is about (1, .9)

Choose 

Substitute in









This gives  

So another point in the first quadrant is (2,0)

(Actually it's another point on the BORDER of the first
quadrant, this time on the x-axis)

Plot those three first quadrant points:



Next by the symmetry, we can reflect these three points
in both the x-axis and the y-axis, (like a mirror!) and 
we have this:



Now we can sketch the curve:




Edwin