Question

How do you find the domain and range of `y = x^2 + 4x − 21`?

Answer 1

Domain:

`x^2+4x-21` is a polynomial. I don't need to worry about dividing by zero or taking even roots or negative numbers or other weirdness. The domain is all real numbers. (The interval `(-oo, oo)`

Range

This may depend on ow much you know. I would graph (or at least think about the graph) of the parabola whose equation id `y=x^2+4x-21`

For `y=ax^2+bx+c`,
The vertex is at `x=(-b)/(2a)` In this case the vertex is at `x=-4/(2(1))=-2`

For this value of `x=-2`, we have `y=(-2)^2+4(-2)-21=4-8-21=-25`

`a>0` tells me that the parabola opens upward, so the `y` values (the range) are all values greater than or equal to -25. The range is `[-25, oo)`

Here's the graph (you may need to zoom in to see some points. (Use your mouse wheel)

graph{x^2+4x-21 [-75.07, 73, -58.35, 15.8]} :