How do you write f(x) = x^2+7x+10 in vertex form?

Aug 6, 2017

$y = {\left(x + \frac{7}{2}\right)}^{2} - \frac{9}{4}$

Explanation:

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

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a {\left(x - h\right)}^{2} + k} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where ( h , k ) are the coordinates of the vertex and a is a constant.

$\text{for a parabola in standard form } y = a {x}^{2} + b x + c$

${x}_{\textcolor{red}{\text{vertex}}} = - \frac{b}{2 a}$

$y = {x}^{2} + 7 x + 10 \text{ is in standard form}$

$\text{with } a = 1 , b = 7 , c = 10$

$\Rightarrow {x}_{\textcolor{red}{\text{vertex}}} = - \frac{7}{2}$

$\text{substitute this value into f(x) for y-coordinate}$

${y}_{\textcolor{red}{\text{vertex}}} = {\left(- \frac{7}{2}\right)}^{2} + \left(7 \times - \frac{7}{2}\right) + 10 = - \frac{9}{4}$

$\Rightarrow \textcolor{m a \ge n t a}{\text{vertex }} = \left(- \frac{7}{2} , - \frac{9}{4}\right)$

$\Rightarrow y = {\left(x + \frac{7}{2}\right)}^{2} - \frac{9}{4} \leftarrow \textcolor{red}{\text{ in vertex form}}$