Chapter 10 - Analytic Geometry - 10.3 Hyperbolas - 10.3 Exercises - Page 988: 23

$$\eqalign{ & {\text{Center }}\left( { - 3,2} \right) \cr & {\text{Vertices}}:\left( { - 7,2} \right){\text{ and }}\left( {1,2} \right) \cr & {\text{Foci: }}\left( { - 8,2} \right){\text{ and }}\left( {2,2} \right) \cr & {\text{Asymptotes: }}y = \pm \frac{3}{4}\left( {x + 3} \right) + 2 \cr & {\text{domain: }}\left( { - \infty , - 7} \right] \cup \left[ {1,\infty } \right) \cr & {\text{range: }}\left( { - \infty , + \infty } \right) \cr} $$

$$\eqalign{ & \frac{{{{\left( {x + 3} \right)}^2}}}{{16}} - \frac{{{{\left( {y - 2} \right)}^2}}}{9} = 1 \cr & {\text{The equation is written in the form }}\frac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} - \frac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1 \cr & \frac{{{{\left( {x + 3} \right)}^2}}}{{16}} - \frac{{{{\left( {y - 2} \right)}^2}}}{9} = 1,{\text{ then }}a = 4,\,\,b = 3,\,\,\,h = - 3,\,\,\,k = 2 \cr & c = \sqrt {{a^2} + {b^2}} = \sqrt {16 + 9} = 5 \cr & \cr & {\text{Characteristics:}} \cr & {\text{Center: }}\left( {h,k} \right) \cr & {\text{Center: }}\left( { - 3,2} \right) \cr & {\text{Transverse axis: horizontal}} \cr & {\text{vertices: }}\left( {h \pm a,k} \right) \cr & {\text{vertices: }}\left( { - 3 \pm 4,2} \right) \cr & {\text{vertices: }}\left( { - 7,2} \right){\text{ and }}\left( {1,2} \right) \cr & {\text{foci:}}\left( {h \pm c,k} \right) \cr & {\text{foci:}}\left( { - 3 \pm 5,2} \right) \cr & {\text{foci:}}\left( { - 8,2} \right){\text{ and }}\left( {2,2} \right) \cr & {\text{asymptotes: }}y = \pm \frac{b}{a}\left( {x - h} \right) + k \cr & {\text{asymptotes: }}y = \pm \frac{3}{4}\left( {x + 3} \right) + 2 \cr & {\text{The domain is: }}\left( { - \infty ,h - a} \right] \cup \left[ {h + a,\infty } \right) \cr & {\text{domain: }}\left( { - \infty , - 7} \right] \cup \left[ {1,\infty } \right) \cr & {\text{The range is: }}\left( { - \infty , + \infty } \right) \cr} $$