Question

What is the domain and range of the quadratic equation y = -x2 - 14x - 52?

Answer 1

Quadratics (function in x) always have Domain of all real numbers and Range from y-value of vertex up for parabolas opening up and from y-value of vertex down for parabolas opening down.

Because the leading coefficient is negative, the parabola opens down.

You can find the y-value of the vertex in two ways: 1) complete the square to put the quadratic into vertex form, or 2) find the axis of symmetry and use that to find y. (the 2nd way is easier)

Axis of Symmetry: x = -b / 2a = - (-14) / 2(-1) = 14 / -2 = -7
y = -x^2 - 14x - 52 → y = -(-7)^2 - 14(-7) - 52 → y = -49 + 98 - 52 → y = -3

So the Domain is All Real Numbers and the Range is y <= 3.
______________________________________…
y-intercept is found by making x = 0 → y = 0^2 +2(0) - 3 → y = -3

x-intercepts are found by making y = 0 → 0 = x^2 + 2x - 3 → 0 = (x + 3)(x - 1) → x = -3, 1