Graphing Ellipses

In this section, we learn how to graph ellipses. An ellipse is the set of all points in a plane where the sum of the distances from two fixed points is constant. The two fixed points are known as foci. The first scenario we will encounter involves an ellipse that is centered at the origin. In this case, we see two equations:
1) Horizontal Ellipse: $$\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1$$ 2) Vertical Ellipse: $$\frac{x^2}{b^2}+ \frac{y^2}{a^2}=1$$ To graph an equation of this format, we plot the x-intercepts and the y-intercepts. We can find the x-values by taking the positive and negative square roots of the denominator under x-squared. We can find the y-values by taking the positive and negative square roots of the denominator under y-squared. We then draw a smooth curve through the four points. A more challenging scenario occurs when we see an ellipse that is shifted horizontally or vertically. As an example, suppose we see the equation: $$\frac{(x-3)^2}{25}+ \frac{(y+2)^2}{9}=1$$ Here the center has shifted to (3, -2) and so all of our points have shifted 3 units right and 2 units down.