What is the range of the function #f(x) = -3x^2 + 3x - 2#?

1 Answer
May 14, 2018

Answer:

#(-oo,-5/4]#

Explanation:

#"we require to find the vertex and it's nature, that is"#
#"maximum or minimum"#

#"the equation of a parabola in "color(blue)"vertex form"# is.

#color(red)(bar(ul(|color(white)(2/2)color(black)(y=a(x-h)^2+k)color(white)(2/2)|)))#

#"where "(h,k)" are the coordinates of the vertex and a"#
#"is a multiplier"#

#"to obtain this form use "color(blue)"completing the square"#

#• " the coefficient of the "x^2" term must be 1"#

#"factor out "-3#

#y=-3(x^2-x+2/3)#

#• " add/subtract "(1/2"coefficient of the x-term")^2" to"#
#x^2-x#

#y=-3(x^2+2(-1/2)xcolor(red)(+1/4)color(red)(-1/4)+2/3)#

#color(white)(y)=-3(x-1/2)^2-3(-1/4+2/3)#

#color(white)(y)=-3(x-1/2)^2-5/4larrcolor(red)"in vertex form"#

#rArrcolor(magenta)"vertex "=(1/2,-5/4)#

#"to determine if vertex is max/min"#

#• " if "a>0" then minimum "uuu#

#• " if "a<0" then maximum "nnn#

#"here "a=-3<0" hence maximum"#

#"range "y in(-oo,-5/4]#
graph{-3x^2+3x-2 [-8.89, 8.89, -4.444, 4.445]}