How do you use the important points to sketch the graph of # f(x)= -x^2-3x-6#?

1 Answer
May 13, 2017

#"see explanation"#

Explanation:

#" the main points in sketching the graph are"#

#• " the x and y intercepts"#

#• " shape, minimum / maximum"#

#• " coordinates of the vertex"#

#color(blue)"to obtain the intercepts"#

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

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

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

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

#"check the discriminant " Delta=b^2-4ac#

#"here " a=-1,b=-3" and " c=-6#

#b^2-4ac=(-3)^2-(4xx1xx-6)=-15<0#

#Delta<0rArr" no real roots, no x-intercepts"#

#color(blue)"maximum / minimum"#

#• " If " a>0" then minimum " uuu#

#• "If " a<0" then maximum " nnn#

#"here " a=-1<0rArrnnn#

#color(blue)"coordinates of vertex"#

#x_(color(red)"vertex")=-b/(2a)=-(-3)/(-2)=-3/2#

#"substitute this value into function for y"#

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

#rArrcolor(magenta)"vertex" =(-3/2,-15/4)#
graph{-x^2-3x-6 [-20, 20, -10, 10]}