What is the vertex form of y=-x^2+5x-5?

May 10, 2018

$y = - {\left(x - \frac{5}{2}\right)}^{2} + \frac{5}{4}$

Explanation:

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

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

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

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

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

$\text{factor out } - 1$

$y = - \left({x}^{2} - 5 x + 5\right)$

• " add/subtract "(1/2"coefficient of the x-term")^2" to"
${x}^{2} - 5 x$

$y = - \left({x}^{2} + 2 \left(- \frac{5}{2}\right) x \textcolor{red}{+ \frac{25}{4}} \textcolor{red}{- \frac{25}{4}} + 5\right)$

$\textcolor{w h i t e}{y} = - {\left(x - \frac{5}{2}\right)}^{2} - \left(- \frac{25}{4} + 5\right)$

$\Rightarrow y = - {\left(x - \frac{5}{2}\right)}^{2} + \frac{5}{4} \leftarrow \textcolor{red}{\text{in vertex form}}$