How to find the vertex of a parabola

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Solution

Dear Student,

Mention below is the derivation for finding the vertex of a parabola:
$We have, y = {ax}^{2} + bx + c = a (x^{2} + \frac{b}{a} x + \frac{c}{a}) = a (x^{2} + 2 \times \frac{b}{2 a} \times x + {(\frac{b}{2 a})}^{2} - {(\frac{b}{2 a})}^{2} + \frac{c}{a}) \Rightarrow y = a ({(x + \frac{b}{2 a})}^{2} + \frac{4 ac - b^{2}}{4 a^{2}}) \Rightarrow \{y - (\frac{4 ac - b^{2}}{4 a})\} = a {(x + \frac{b}{2 a})}^{2} As, the standard form of the parabola is Y = 4 {AX}^{2}, whose vertex is V (0, 0) \Rightarrow Y = y - (\frac{4 ac - b^{2}}{4 a}), X = x + \frac{b}{2 a} and A = \frac{a}{4} So, the vertex of the given function is given by : X = 0 and Y = 0 \Rightarrow y = (\frac{4 ac - b^{2}}{4 a}), x = - \frac{b}{2 a} So, the vertex of y = {ax}^{2} + bx + c is V (\frac{4 ac - b^{2}}{4 a}, \frac{- b}{2 a}) .$

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