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

Jan 29, 2017

Domain : $x \in \left(- \infty , \infty\right)$
Range : $y \in \left(- \infty , - 3\right]$

#### Explanation:

Let y = a polynomial of degree n

$= {a}_{0} {x}^{+} {a}_{1} {x}^{n - 1} + \ldots {a}_{n}$

$= {x}^{n} \left({a}_{0} + {a}_{1} / x + \ldots {a}_{n} / {x}^{n}\right)$

As $x \to \pm \infty , y \to \left(s i g n \left({a}_{0}\right)\right) \infty$, when n is even, and

$y \to \left(s i g n \left({a}_{0}\right)\right) \left(- \infty\right)$, 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 $x \in \left(- \infty , \infty\right)$ and the range is

$y \in \left(- \infty , \max y\right] = \left(- \infty , - 3\right]$.

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)