Question

How do you find the range of the equation `y = -x^2 – 6x – 13`?

Answer 1

Range of `y = [-4,-oo)`

Explanation:

`y = -x^2-6x-13`

`y` is a quadratic function, represented on the `xy-`plane as a parabola of the form: `ax^2+bx+c`

The vertex of the parabola will be at `x=( -b)/(2a)`

In our case, `b=-6, a=-1`

Hence, `x_(vertex) = (6)/(-2) =-3`

Since `a<0` then `y(x_(vertex) )` will be a maximum of `y`

`:. y_max = y(-3) = -(-3)^2+6*3-13 = -9+18-13=-4`

`:.` the greatest value of `y` is `-4`

Since `y` has no lower bounds, the range of `y` is `[-4, -oo)`

As can be seen from the graph of `y` below.

graph{-x^2-6x-13 [-23.18, 22.45, -15.1, 7.71]}