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

Answer 1

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

Step 1: Let #y=0# and solve for #x# to find the parabola's roots.
#x(-1/2 x - 2)=0# #-1/2 x^2-2x=0#
The origin, #(0,0)#, is where #y=0# occurs when #x=0#.

Furthermore, if we allow

#x+4=0# #x=-4# #-1/2 x - 2 = 0# #-2*(-1/2 x - 2) = -2*0#.
Therefore, #(0,0)# and #(-4,0)# are our two roots.
Step 2: Utilizing the formula #x_"max"=-b/(2a)#, where #b=-2# and #a=-1/2#, find the maximum #y#-value. #x_"max"=-(-2)/(2*(-1/2))=-2#
The value of #y# at that #x_"max"# value is obtained by plugging this value of #x_"max"=-2# into the original formula.
#y=-1/2(-2)^2-2(-2) = -2+4 = 2#
In other words, the parabola has a maximum at #(-2,2)#. At this point, we can create the graph:

graph{-11.91, 8.09, -5.2, 4.8]}

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Answer 2

To graph the function ( y = -\frac{1}{2}x^2 - 2x ), follow these steps:

  1. Plot key points:
    • Choose x-values and calculate corresponding y-values.
    • Use a table of values or algebraic methods to find points.
  2. Draw the graph:
    • Plot the points on the coordinate plane.
    • Use smooth curves to connect the points.

To identify the domain and range:

  • Domain: All real numbers (-∞, ∞)
  • Range: ( (-\infty, -\frac{9}{2}] )
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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