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I am curious if there is a way to get the vertex of a parabola without calculus?

I know the calculus approach has you take the derivative set to $0$ (for $0$ slope) so if $f(x) = ax^2 + bx + c$ then $f'(x) = 2x + b$ so $0 = 2x + b$ implies $x = -\frac{b}{2a}$ with $y$ coordinate $f(-\frac{b}{2a})$.

But how would we get the vertex without calculus?

José Carlos Santos
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user525966
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1 Answers1

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You can observe that$$ax^2+bx+c=a\left(x^2+\frac bax+\frac ca\right)=a\left(\left(x+\frac b{2a}\right)^2+\frac ca-\frac{b^2}{4a^2}\right).$$So, the minimum or maximum is obtained when $x=-\frac b{2a}$.

José Carlos Santos
  • 427,504