Chapter 10 - Analytic Geometry - 10.2 Ellipses - 10.2 Exercises - Page 979: 36

$$\eqalign{ & {\text{The domain of the function is }}[ - 10,10] \cr & {\text{The range of the function is }}y = [ - 1,0] \cr} $$

$$\eqalign{ & y = - \sqrt {1 - \frac{{{x^2}}}{{100}}} \cr & {\text{Square each side}} \cr & {\left( y \right)^2} = {\left( { - \sqrt {1 - \frac{{{x^2}}}{{100}}} } \right)^2} \cr & {y^2} = 1 - \frac{{{x^2}}}{{100}} \cr & {\text{Write in standard form}} \cr & \frac{{{x^2}}}{{100}} + \frac{{{y^2}}}{1} = 1 \cr & {\text{This is the equation of an ellipse }}\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1 \cr & a = 10,\,\,\,\,b = 1 \cr & {\text{with }}x{\text{ - intercepts}}\left( { \pm a,0} \right):\left( { \pm 10,0} \right) \cr & {\text{with }}y{\text{ - intercepts}}\left( {0, \pm b} \right):\left( {0, \pm 1} \right) \cr & \cr & {\text{In the original equation}},{\text{ the radical expression}}\,\,\, - \sqrt {1 - \frac{{{x^2}}}{{100}}} \cr & {\text{represents a nonnegative number, number}},{\text{ so the only }} \cr & {\text{possible values of }}y{\text{ are negative}} \cr & {\text{The domain of the function is }}[ - 10,10] \cr & {\text{The range of the function is }}y = [ - 1,0] \cr & \cr & {\text{Graph}} \cr & \cr} $$