Question

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

Answer 1

`"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]}