# How do you find the domain and range of f(x)=x^2-4x+7 ?

Dec 27, 2017

$x \in \mathbb{R}$
$y \in \mathbb{R} , y \ge 3$

#### Explanation:

$f \left(x\right) \text{ is defined for all } x \in \mathbb{R}$

$\Rightarrow \text{domain is } x \in \mathbb{R}$

$\text{to find the range we require to find the vertex}$

$\text{the equation of a parabola in "color(blue)"vertex form}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a {\left(x - h\right)}^{2} + k} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where "(h,k)" are the coordinates of the vertex and a }$
$\text{is a multiplier}$

• " if "a>0" then vertex is a minimum "uuu

• " if "a<0" then vertex is a maximum"nnn

$\text{to obtain vertex form use "color(blue)"completing the square}$

• " coefficient of the "x^2" term must be 1 which it is"

• " add/subtract "(1/2"coefficient of x-term")^2" to"
${x}^{2} - 4 x$

$\Rightarrow y = {x}^{2} + 2 \left(- 2\right) x \textcolor{red}{+ 4} \textcolor{red}{- 4} + 7$

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

$\Rightarrow \text{ vertex "=(2,3)" and } a > 0$

$\Rightarrow \text{range is } y \in \mathbb{R} , y \ge 3$
graph{x^2-4x+7 [-10, 10, -5, 5]}