If you have accurately drawn your axes, you will be able to fold the paper along the vertical axis and using a light table or a window, you can trace the first quadrant onto the opposite side of the axis. Then, with the two upper quadrants drawn, you can fold the paper along the horizontal axis and trace the bottom two quadrants from their counterparts in the upper half. Unfold your paper, and you will see a perfect ellipse sitting in front of you, and you will understand how to draw any ellipse of any size. Many people mistake ovals for ellipses. Ovals are made up of two circles connected on their sides by parallel lines.
Ovals are what they use for race tracks.
Ellipses have no straight sides. Their perimeters are constantly curved according to a mathematical formula. The earth travels an elliptical orbit around the sun, one of the interesting natural facts that we can enjoy. The cause of this is gravitational, of course. Eggs have a shape of their own, even more wonderful than the ellipse.