What are the vertex, focus and directrix of # y=x^2-3x+4 #?

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Answer 1 · 2017-04-09T19:20:30

#"vertex="(1.5,1.75)#
#"focus="(1.5,2)#
#"directrix: y=1.5#

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#y=a(x-h)^2+k " the vertex form of parabola"#

#"vertex="(h,k)#

#"focus="(h,k+1/(4a))#

#y=x^2-3x+4 "your parabola equation"#

#y=x^2-3xcolor(red)(+9/4-9/4)+4#

#y=(x-3/2)^2-9/4+4#

#y=(x-3/2)^2+7/4#

#"vertex"=(h,k)=(3/2,7/4)#

#"vertex="(1.5,1.75)#

#"focus="(h,k+1/(4a))#

#"focus="(1.5,7/4+1/(4*1))=(1.5,8/4)#

#"focus="(1.5,2)#

#"Find directrix :"#

#"take a point(x,y) on parabola"#

#"let " x=0#

#y=0^2-3*0+4#

#y=4#

#C=(0,4)#

#"find distance to focus"#

#j=sqrt((1.5-0)^2+(2-4)^2)#

#j=sqrt(2.25+4)#

#j=sqrt(6.25)#

#j=2.5#

#directrix=4-2.5=1.5#

#y=1.5#