# How do you find the domain and range of  -x^2+7?

Jul 20, 2017

Domain: $\mathbb{R}$

Range: $\left(- \infty , 7\right]$

#### Explanation:

The domain is the set of all possible $x$-values of that function. It is the “input” values that spew out the “output” or $y$- values of the function which make up the range. The domain of a quadratic function is usually all the real numbers ($\mathbb{R}$). The range will vary as in this function but we will look at that below:

Let's start by taking a look at the graph $y = {x}^{2}$ which, you know, is a parabola.

The domain of $y = {x}^{2}$ is $\mathbb{R}$ or $\left(- \infty , \infty\right)$ while the range is $\left[0 , \infty\right)$

graph{x^2 [-10.54, 9.46, -0.56, 9.44]}

Domain: $\mathbb{R}$
Range: $\left[0 , \infty\right)$

Now let's look at the graph of $y = - {x}^{2}$. The domain remains the same but the range is now $\left(- \infty , 0\right]$.

graph{-x^2 [-10.17, 9.83, -9.48, 0.52]}

Domain: $\mathbb{R}$
Range: $\left(- \infty , 0\right]$

Lastly, let's look at the graph for the function given: $y = - {x}^{2} + 7$ which is the function $y = {x}^{2}$ reflected over the $x$-axis and shifted $7$ units upward:

graph{-x^2+7 [-9.88, 10.12, -2.72, 7.28]}

Domain: $\mathbb{R}$
Range: $\left(- \infty , 7\right]$

The domain is still unchanged but the range is $\left(- \infty , 7\right]$