#"this is a quadratic function which can be sketched in "#
#"the usual way"#
#y=x^2-x+1#
#"with "a=1,b=-1,c=1#
#"since "a>0" then minimum turning point "uuu#
#x_(color(red)"vertex")=-b/(2a)#
#rArrx_(color(red)"vertex")=-(-1)/2=1/2#
#rArry_(color(red)"vertex")=(1/2)^2-1/2+1=3/4#
#rArr"minimum turning point at "(1/2,3/4)#
#"the discriminant shows"#
#Delta=b^2-4ac=1-4=-3#
#"that the quadratic does not intersect the x-axis"#
graph{x^2-x+1 [-10, 10, -5, 5]}
#color(blue)"turning point using calculus"#
#y=x^2-x+1#
#rArrdy/dx=2x-1#
#"for turning point equate "dy/dx=0#
#rArr2x-1=0rArrx=1/2#
#rArr"turning point at "(1/2,3/4)#
#"to test if max/min use the "color(blue)"second derivative test"#
#• " if "(d^2y)/(dx^2)" at "x=1/2>0" then minimum"#
#• " if "(d^2y)/(dx^2)" at "x=1/2<0" then maximum"#
#dy/dx=2x-1#
#rArr(d^2y)/(dx^2)=2>0#
#rArr"minimum turning point at "(1/2,3/4)#
