The Equation For The Axis of Symmetry Of A Parabola

The vertex form equation of a parabola is:

\[ y=\frac{(x-h)^2}{4p} + k\]

The axis of symmetry here is at \(x=h\). The standard form equation of a parabola is:

\[ y= ax^2 + b x + c\]

Let's find the axis of symmetry here in terms of \(a\) and \(b\). We can express \(a\) and \(b\) in terms of \(h\) and \(p\):

\[ a=\frac{1}{4p} \quad b=\frac{-h}{2p} \]

This means:

\[\begin{gather} b=\frac{-2h}{4p} = -2ha \\ h = -\frac{b}{2a}\end{gather} \]

This means the axis of symmetry lies at \(x = -\frac{b}{a}\).

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