#color(blue)("General shape of the graph")#
Consider the standardised quadratic form of #y=ax^2+bx+c#
If #a# is positive the general shape of the graph is #uu#
If #a# is negative the general shape of the graph is #nn#
Lets multiply out the brackets:
#y=color(blue)((x-7))color(green)((x-1))#
Multiply everything in the right bracket by everything in the left.
#color(green)(y=color(blue)(x)(x-1)color(blue)(" "-7)(x-1)#
#y=x^2-x" "-7x+7#
#color(white)(.)#
#y=x^2-8x+7#
So for #y=ax^2+bx+c#
#a=1"; "b=-8"; "c=7#
So #a=+1# which is positive so the graph is of type: #uu#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("To determine the vertex - a sort of cheat way")#
Consider the form #y=x^2-8x+7#
Write in the form of #y=a(x^2+b/ax)+c#
Where #x_("vertex")=(-1/2)xxb/a#
#=>color(green)(x_("vertex")="axis of symmetry" = (-1/2)xx(-8)=+4)#
Substituting #x=4#
#y=x^2-8x+7" "->" "y_("vertex")=(4)^2-8(4)+7#
#y_("vertex")=16-32+7#
#color(green)(y_("vertex")=-9)#
#color(blue)("Veretex"->(x,y)=(4,-9)#
