How do you find the vertex and the intercepts for #f(x)= -x^2-3x-6 #?

1 Answer
Oct 4, 2017

#"see explanation"#

Explanation:

#"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 is"#

#•color(white)(x)x_(color(red)"vertex")=-b/(2a)#

#f(x)=-x^2-3x-6" is in standard form"#

#"with "a=-1,b=-3,c=-6#

#rArrx_(color(red)"vertex")=-(-3)/(-2)=-3/2#

#"substitute this value into f(x) for y-coordinate"#

#rArry_(color(red)"vertex")=-(-3/2)^2-3(-3/2)-6=-15/4#

#rArrcolor(magenta)"vertex"=(-3/2,-15/4)#

#color(blue)"Intercepts"#

#• " let x = 0, in equation for y-intercept"#

#• " let y = 0, in equation for x-intercepts"#

#x=0toy=-6larrcolor(red)" y-intercept"#

#y=0to-x^2-3x-6=0#

#"checking the value of the "color(blue)"discriminant"#

#Delta=b^2-4ac=(-3)^2-(4xx-1xx-6)=-15#

#"since "Delta<0" then no real solutions"#

#rArrf(x)" does not intersect with the x-axis"#
graph{-x^2-3x-6 [-10, 10, -5, 5]}