# What is the domain and range of y+2 = (x-3)^2?

Jun 14, 2018

Domain: $x \in \mathbb{R}$
Range: $y \in \left[- 2 , \infty\right)$

#### Explanation:

The function you provided is almost in vertex form of a quadratic function, which helps greatly when answering your question. Vertex form in a quadratic is when the function is written in the following form:

$y = a {\left(x - h\right)}^{2} + k$

To write your function in vertex form, I'll simply solve for $y$ by subtracting 2 from both sides:

$y = {\left(x - 3\right)}^{2} - 2$

The two parameters you want in this are $a$ and $k$, since those will actually tell you the range. Since any value of $x$ can be used in this function, the domain is:

$x \in \mathbb{R}$

Now we need the range. As stated before, it comes from the values of $a$ and $k$. If $a$ is negative, the range goes to$- \infty$. If $a$ is positive, the range goes to $\infty$. In this case, $a$ is positive, so we know the range goes to $\infty$. The lowest value will be the $k$ value, which in this case is -2. Hence, the range of your function is:

$y \in \left[- 2 , \infty\right)$