# What is the domain and range of y=-x^2+4x-1?

May 25, 2017

Domain: $x \in \mathbb{R}$
Range: $y \in \left(- \infty , 3\right]$

#### Explanation:

This is a polynomial, so the domain (all possible $x$ values for which $y$ is defined) is all real numbers, or $\mathbb{R}$.

To find the range, we need to find the vertex.

To find the vertex, we need to find the axis of symmetry.

The axis of symmetry is $x = - \frac{b}{2 a} = - \frac{4}{2 \cdot \left(- 1\right)} = 2$

Now, to find the vertex, we plug in $2$ for $x$ and find $y$.

$y = - {\left(2\right)}^{2} + 4 \left(2\right) - 1$

$y = - 4 + 8 - 1$

$y = 3$

The vertex is either the maximum or minimum value, depending on whether the parabola faces up or down.

For this parabola, $a = - 1$, so the parabola faces down.

Therefore, $y = 3$ is the maximum value.

So the range is $y \in \left(- \infty , 3\right]$