# What are the vertex, focus and directrix of # y=-15+12x-2x^2 #?

##### 1 Answer

Aug 8, 2018

#### Explanation:

#"since the equation has an "x^2" term, this is a"#

#"vertically opening parabola"#

#"the equation of a vertically opening parabola is"#

#•color(white)(x)(x-h)^2=4a(y-k)#

#"where "(h,k)" are the coordinates of the vertex and a"#

#"is the distance from the vertex to the focus and directrix"#

#"if "a>0" then opens upwards"#

#"if "a< 0" then opens downwards"#

#"to obtain this form "color(blue)"complete the square"#

#y=-2(x^2-6x+15/2)#

#color(white)(y)=-2(x^2+2(-3)x+9-9+15/2)#

#color(white)(y)=-2(x-3)^2+3#

#(x-3)^2=-1/2(y-3)#

#4a=-1/2rArra=-1/8" parabola opens down"#

#"vertex "=(3,3)#

#"focus "=(h,a+k)=(3,23/8)#

#"directrix is "y=-a+k=1/8+3=25/8#

graph{(y+2x^2-12x+15)(y-0.001x-25/8)=0 [-10, 10, -5, 5]}