Can someone tell me the range and domain of these functions? Thanks?

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  • Alan
    Lv 7
    3 weeks ago

    The domain is all real numbers for both functions. 

    (-infinity, + infinity)

    There are no restrictions on the domain.   

    Range: 

    Both functions are parabolas. 

    Vertex format is: 

    y = a(x-h)^2  + k 

    has vertex (h,k) 

    if a is positive, the parabola turns upward

    if a is postive,  the range is y>=k  

    if a is negative,  the parabola turns downward 

    if a is negative, the range is is y<= k   

    1st parabola is already in vertex format 

    y = 2(x+2)^2  + 3  

    vertex ( -2, 3) 

    and "a" = + 2 , the parabola goes upward from the vertex (-2,3) 

    so range is   y >=  3       

    2nd parabola 

    convert to vertex format 

    y=-2x^2 -8x +3   = -2(x^2 + 4x)   + 3   

    but (x+2)^2 = x^2 + 4x  + 4   

    so 

    x^2 + 4x  = (x+2)^2 - 4    

    now, substitute  " (x+2)^2 -4"  for x^2 +4x  

    y=   -2( (x+2)^2 -4)   + 3  

    y = -2(x+2)^2   + 8 + 3 

    y = -2(x+2)^2   + 11 

    so vertex  (-2, 11) 

    and "a" is negative , so the parabola goes 

    downward  from the vertex  (-2,11) 

    so range is  y<= 11

  • 3 weeks ago

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