How do you graph the function #y=-1/2x^2-2x# and identify the domain and range?

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Answer 1 · 2017-01-29T22:32:03

Draw an upside parabola that goes through the origin and the point #(-4, 0)# and has a maximum at #y=2#

Step 1. Find the roots of the parabola by letting #y=0# and solve for #x#.

#-1/2 x^2-2x=0#
#x(-1/2 x - 2)=0#

When #x=0#, then #y=0#. That's at the origin #(0,0)#

Also, if we let

#-1/2 x - 2 = 0#
#-2*(-1/2 x - 2) = -2*0#
#x+4=0#
#x=-4#.

So our two roots are #(0,0)# and #(-4,0)#.

Step 2. Find the maximum #y#-value by using the formula: #x_"max"=-b/(2a)#, were #b=-2# and #a=-1/2#
#x_"max"=-(-2)/(2*(-1/2))=-2#

Plugging this value of #x_"max"=-2# into the original formula, gives the value of #y# at that #x_"max"# value. So,

#y=-1/2(-2)^2-2(-2)=-2+4=2#

That is, there is a maximum of the parabola at #(-2,2)#. Now we can draw the graph:

graph{-1/2x^2-2x [-11.91, 8.09, -5.2, 4.8]}