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The quadratic equation has three general forms:

  1. $ax^2+bx+c$
  2. $a(x-r_1)(x-r_2)$
  3. $a(x-h)^2+k$

  • $r_1$ and $r_2$ are the zeroes of the quadratic.

  • $h$ is the horizontal position of the vertex, $k$ is the vertical position of the vertex.

Are there any such geometric interpretations the coefficients, $a$, $b$, and $c$?

Frank Vel
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    $c$ is the $y$-intercept. $a$ is the shape parameter. $b$ as it stands doesn't have much of an intuitive interpretation, but $b/2a$ is the axis of symmetry. Changes in $b$ translate the parabola. – Doug M Jun 29 '16 at 19:08
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    $b$ is the slope of the tangent at the $y$-intercept. – Vincenzo Tibullo Jun 29 '16 at 19:29
  • Enzotib. That's a really good point. Never thought of that. I always think of the sign of a as determining if the parabola points up and down. The size of a determines how wide or narrow the parabola is, but the is no really units of measure. Theoretically as f(x) = a^2 +bx + c then f''(x) = a, we can consider a to be an "acceleration constant". – fleablood Jun 29 '16 at 20:22

1 Answers1

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The figure shows how these quantities are related.

enter image description here

Mick
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  • This doesn't show how $a$ and $b$ are interpreted, only $\frac{b}{a}$. And technically $\alpha \leq \beta$. – Frank Vel Jun 30 '16 at 17:26
  • @FrankVel 1) WLOG, we can assume $\alpha \lt \beta$ and the theory applies equally well for the reversed case. 2) Assuming $a \gt 0$ (means the curve is concaved upward) is just one of the many sub-cases to be discussed . I think the post has served the illustration already and you need to explore other cases yourself. – Mick Jul 01 '16 at 14:50