# What is the domain and range of f(x) = -7(x - 2)^2 - 9?

May 5, 2018

See below.

#### Explanation:

$- 7 {\left(x - 2\right)}^{2} - 9$

This is a polynomial, so its domain is all $\mathbb{R}$.

This can be expressed in set notation as:

$\left\{x \in \mathbb{R}\right\}$

To find the range:

We notice that the function is in the form:

color(red)(y=a(x-h)^2+k

Where:

$\boldsymbol{a} \textcolor{w h i t e}{88}$is the coefficient of ${x}^{2}$.

$\boldsymbol{h} \textcolor{w h i t e}{88}$ is the axis of symmetry.

$\boldsymbol{k} \textcolor{w h i t e}{88}$ is the maximum or minimum value of the function.

Because $\boldsymbol{a}$ is negative we have a parabola of the form, $\bigcap$.

This means $\boldsymbol{k}$ is a maximum value.

$k = - 9$

Next we see what happens as $x \to \pm \infty$

as $x \to \infty$ , $\textcolor{w h i t e}{8888} - 7 {\left(x - 2\right)}^{2} - 9 \to - \infty$

as $x \to - \infty$ , $\textcolor{w h i t e}{8888} - 7 {\left(x - 2\right)}^{2} - 9 \to - \infty$

So we can see that the range is:

$\left\{y \in \mathbb{R} | - \infty < y \le - 9\right\}$

The graph confirms this:

graph{-7x^2+28x-37 [-1, 3, -16.88, -1]}