What is the domain and range of the quadratic equation #y = –x^2 – 14x – 52#?

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Answer 1 · 2017-01-29T05:03:31

Domain : #x in (-oo, oo)#
Range : #y in (-oo, -3]#

Let y = a polynomial of degree n

#=a_0x^+a_1x^(n-1)+...a_n#

#=x^n(a_0+a_1/x+...a_n/x^n)#

As #x to +-oo, y to (sign (a_0))oo#, when n is even, and

#y to (sign (a_0))( -oo)#, when n is odd.

Here, n = 2 and #sign (a_0#) is #-#.

y = -x^2-14x-52)=-(x+7)^2-3<=-3, giving #max y = -3#.

The domain is #x in (-oo, oo)# and the range is

#y in (-oo, max y]=(-oo, -3]#.

See graph. graph{(-x^2-14x-52-y)(y+3)((x+7)^2+(y+3)^2-.01)=0 [-20, 0, -10, 0]}

Graph shows the parabola and its highest point, the vertex V(-7, -3)