How do you find the vertex of the parabola: y=0.5x^2+2x?

Aug 10, 2015

The vertex is at $\left(- 2 , - 2\right)$

Explanation:

The general vertex form of a quadratic is
$\textcolor{w h i t e}{\text{XXXX}}$$y = m {\left(x - a\right)}^{2} + b$
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$with its vertex at $\left(a , b\right)$

To convert $y = 0.5 {x}^{2} + 2 x$ into vertex form:

Extract $m$
$\textcolor{w h i t e}{\text{XXXX}}$$y = \left(0.5\right) \left({x}^{2} + 4 x\right)$

Complete the square
$\textcolor{w h i t e}{\text{XXXX}}$$y = \left(0.5\right) \left(x + 4 x + 4\right) - \left(0.5\right) \left(4\right)$

Re-write as a squared binomial and simplify the constant
$\textcolor{w h i t e}{\text{XXXX}}$$y = \left(0.5\right) {\left(x + 2\right)}^{2} - 2$

Write in explicit vertex form
$\textcolor{w h i t e}{\text{XXXX}}$$y = \left(0.5\right) {\left(x - \left(- 2\right)\right)}^{2} + \left(- 2\right)$
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$with vertex at $\left(- 2 , - 2\right)$