"for the quadratic equation in standard form"for the quadratic equation in standard form
•color(white)(x)y=ax^2+bx+c color(white)(x);a!=0∙xy=ax2+bx+cx;a≠0
x_(color(red)"vertex")=-b/(2a)xvertex=−b2a
f(x)=-x^2+6x+6" is in standard form"f(x)=−x2+6x+6 is in standard form
"with "a=-1,b=6,c=6with a=−1,b=6,c=6
rArrx_(color(red)"vertex")=-6/(-2)=3⇒xvertex=−6−2=3
"substitute this value into the equation for y"substitute this value into the equation for y
y=f(3)=-(3)^2+6(3)+6=15y=f(3)=−(3)2+6(3)+6=15
rArrcolor(magenta)"vertex "=(3,15)⇒vertex =(3,15)
color(blue)"for intercepts"for intercepts
• " let x = 0, in equation for y-intercept"∙ let x = 0, in equation for y-intercept
• " let y = 0, in equation for x-intercepts"∙ let y = 0, in equation for x-intercepts
x=0toy=6larrcolor(red)" y-intercept"x=0→y=6← y-intercept
y=0to-x^2+6x+6=0y=0→−x2+6x+6=0
"solve for x using the "color(blue)"quadratic formula"solve for x using the quadratic formula
x=(-6+-sqrt(36+24))/(-2)x=−6±√36+24−2
color(white)(x)=(-6+-sqrt60)/(-2)x=−6±√60−2
color(white)(x)=(-6+-2sqrt15)/(-2)=3+-sqrt15x=−6±2√15−2=3±√15
rArrx~~ -0.87,x~~ 6.87larrcolor(red)" x-intercepts"⇒x≈−0.87,x≈6.87← x-intercepts
