Chapter 10 - Parametric And Polar Curves; Conic Sections - 10.4 Conic Sections - Exercises Set 10.4 - Page 745: 6

$$\eqalign{ & \left( {\text{a}} \right){\text{Vertex: }}\left( { - 4,3} \right),{\text{Focus: }}\left( { - \frac{7}{2},3} \right){\text{directrix: }}x = - \frac{9}{2} \cr & \left( {\text{b}} \right){\text{Vertex: }}\left( { - 1,1} \right),{\text{Focus: }}\left( { - 1,\frac{{17}}{{16}}} \right){\text{directrix: }}y = \frac{{15}}{{16}} \cr} $$

$$\eqalign{ & \left( {\text{a}} \right)\,\,{y^2} - 6y - 2x + 1 = 0 \cr & {\text{Factoring}} \cr & \left( {{y^2} - 6y + 9} \right) - 2x + 1 = 9 \cr & {\left( {y - 3} \right)^2} = 2x + 8 \cr & {\left( {y - 3} \right)^2} = 2\left( {x + 4} \right) \cr & {\text{The equation has been written in the form }}{\left( {y - k} \right)^2} = 4p\left( {x - h} \right){\text{,}} \cr & \underbrace {{{\left( {y - 3} \right)}^2} = 2\left( {x + 4} \right)}_{{{\left( {y - k} \right)}^2} = 4p\left( {x - h} \right)}\,\,\, \Rightarrow \,\,\,k = 3,\,\,h = - 4,\,\,\,4p = 2,\,\,\,p = \frac{1}{2} \cr & {\text{The vertex of a parabola is }}\left( {h,k} \right) \cr & {\text{Vertex: }}\left( { - 4,3} \right) \cr & {\text{The focus of a parabola is }}\left( {p + h,k} \right) \cr & {\text{Focus: }}\left( { - \frac{7}{2},3} \right) \cr & {\text{The directrix of a parabola is }}x = - p + h \cr & {\text{directrix: }}x = - \frac{9}{2} \cr & \cr & \left( {\text{b}} \right)\,\,y = 4{x^2} + 8x + 5 \cr & {\text{Factoring}} \cr & y - 5 = 4\left( {{x^2} + 2x} \right) \cr & y - 5 + 4\left( 1 \right) = 4\left( {{x^2} + 2x + 1} \right) \cr & y - 1 = 4{\left( {x + 1} \right)^2} \cr & \frac{1}{4}\left( {y - 1} \right) = {\left( {x + 1} \right)^2} \cr & {\text{The equation has been written in the form }}{\left( {x - h} \right)^2} = 4p\left( {y - k} \right){\text{,}} \cr & \underbrace {\frac{1}{4}\left( {y - 1} \right) = {{\left( {x + 1} \right)}^2}}_{{{\left( {x - h} \right)}^2} = 4p\left( {y - k} \right)}\,\,\, \Rightarrow \,\,\,k = 1,\,\,h = - 1,\,\,\,4p = \frac{1}{4},\,\,\,p = \frac{1}{{16}} \cr & {\text{The vertex of a parabola is }}\left( {h,k} \right) \cr & {\text{Vertex: }}\left( { - 1,1} \right) \cr & {\text{The focus of a parabola is }}\left( {h,p + k} \right) \cr & {\text{Focus: }}\left( { - 1,\frac{{17}}{{16}}} \right) \cr & {\text{The directrix of a parabola is }}y = - p + k \cr & {\text{directrix: }}y = \frac{{15}}{{16}} \cr & \cr & {\text{See the graphs below}} \cr} $$