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

Mar 26, 2015

Domain:

${x}^{2} + 4 x - 21$ 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 $\left(- \infty , \infty\right)$

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 $y = {x}^{2} + 4 x - 21$

For $y = a {x}^{2} + b x + c$,
The vertex is at $x = \frac{- b}{2 a}$ In this case the vertex is at $x = - \frac{4}{2 \left(1\right)} = - 2$

For this value of $x = - 2$, we have $y = {\left(- 2\right)}^{2} + 4 \left(- 2\right) - 21 = 4 - 8 - 21 = - 25$

$a > 0$ 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 $\left[- 25 , \infty\right)$

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]} :