"given the equation of a parabola in standard form"
•color(white)(x)y=ax^2+bx+c color(white)(x);a!=0
"then the x-coordinate of the vertex which is also"
"the axis of symmetry is"
•color(white)(x)x_(color(red)"vertex")=-b/(2a)
y=x^2-x+19" is in standard form"
"with "a=1,b=-1" and "c=19
rArrx_(color(red)"vertex")=-(-1)/2=1/2
"substitute this value into the equation for y"
rArry_(color(red)"vertex")=(1/2)^2-1/2+19=75/4
rArrcolor(magenta)"vertex "=(1/2,75/4)
rArry=(x-1/2)^2+75/4larrcolor(blue)"in vertex form"
"the translated form of a vertically opening parabola is"
•color(white)(x)(x-h)^2=4p(y-k)
"where "(h,k)" are the coordinates of the vertex and"
"p is the distance from the vertex to the focus/directrix"
rArr(x-1/2)^2=1(y-75/4)larrcolor(blue)"translated form"
"with "4p=1rArrp=1/4
"the focus lies on the axis of symmetry "x=1/2
"since "a>0" then parabola opens up "uuu
"hence the focus is "1/4" unit above the vertex and"
"the directrix "1/4" unit below the vertex"
rArrcolor(magenta)"focus "=(1/2,19)
"and equation of directrix is "y=37/2
