In the country I'm living we learn how to draw the graph of quadratic equations such as $y=x^2$ or $y=x^2+5x+3$ before knowing calculus (in fact we don't even learn calculus until we begin the undergraduate studies) using the fact they are parabolas.

For me this is just a memorization. Are there some standard recognized methods to show students quadratic equations are parabolas?

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    $\begingroup$ Frankly, "the graph of a quadratic equation" is the definition of a parabola. The old "locus of points" and "intersection of a cone with a plane" definitions are hopelessly old-fashioned. $\endgroup$
    – Jim Belk
    Commented Feb 26, 2017 at 18:12
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    $\begingroup$ Hopelessly oldfashioned? What's the world coming to? Parabolic mirrors, Lithotripsy, headlights, Kepler's laws. Archimedes, Apollonius, Pascal, Dandelin, Poncelet. Oh my God . . . $\endgroup$ Commented Feb 27, 2017 at 19:10
  • $\begingroup$ How, in your context, is "parabola" defined? $\endgroup$
    – mweiss
    Commented Mar 3, 2017 at 19:04

3 Answers 3


Start with the definition of a parabola as a locus of points:

A parabola is a set of points, such that for any point $P$ of the set the distance $|PF|$ to a fixed point $F$, the focus, is equal to the distance $|Pl|$ to a fixed line $l$, the directrix: $\{P\mid|PF|=|Pl|\}$

Then locate the parabola in a rectangular coordinate system. Start with the simplest case of a parabola with its focus $F$ at coordinates $(0,f)$ (with $f>0$) and its directrix at $y=-f$. For a point $P$ (with coordinates $(x,y)$) on the parabola, we get from $|PF|^2=|Pl|^2$ the equation $x^2+(y-f)^2=(y+f)^2$. This results in the quadratic equation $y=\frac{1}{4f}x^2$.

Give more complicated examples until you get to the general case.

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    $\begingroup$ What I presented was how to show that parabolas are quadratic equations, but I think that that should be enough to see that quadratic equations are parabolas. $\endgroup$
    – JRN
    Commented Feb 25, 2017 at 14:22
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    $\begingroup$ +1 for starting with a reasonable definition of "parabola". FYI, this is how my high school text dealt with equations for circles, ellipses, parabolas, and hyperbolas, and how nearly every (if not absolutely every) precalculus, college algebra, and the older analytic geometry texts I've seen in 40+ years did this, which has me wondering whether the wording of the original question accurately conveys what user26832 wanted to know. $\endgroup$ Commented Feb 27, 2017 at 16:22

This is one of the most celebrated applications of completing the square. $$y=ax^2+bx+c \Longrightarrow y=a\left(x+\frac{b}{2a}\right)^2+\left(c-\frac{b^2}{4a}\right)$$ Since a square (in your non-complex context) must be nonnegative, and will have a minimum when it is zero, the vertex of the parabola will be at $$(x,y)=(-b/2a,c-b^2/4a)$$ If this equation doesn't look like the stretched parabola $ax^2$ shifted by this vertex, what would?

Also, you can find the places it hits the $x$-axis by solving for $y=0$ using the quadratic formula or factoring. The $y$-intercept would be $c$, of course. Hopefully with these three pieces of information you could plot as much as you want.

By the way, this isn't really a matheducators.SX type answer, but I hope it leads you to thinking about the many problems typically assaulted with calculus that may not need it, proving that dogs don't need calculus.

  • $\begingroup$ The problem is how do you know this is a parabola. Once we know it's a parabola is easy to plot the graph. $\endgroup$
    – user26832
    Commented Feb 25, 2017 at 14:03
  • $\begingroup$ How do we know $ax^2$ is a parabola? That is a little deeper - do you mean you want to prove that it is the locus of points such that ... ? Looks like someone else has covered that case. But I find that this is more sophisticated than students could handle at that point; the "locus" definition is not so easy. $\endgroup$
    – kcrisman
    Commented Feb 25, 2017 at 15:04
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    $\begingroup$ Use the idea of "graph transformations" -- when you replace x with (x-a) you shift the graph a units to the right, when you add or subtract outside the squaring you shift the graph up or down, and multiplying by a constant scales vertically. $\endgroup$
    – Opal E
    Commented Feb 27, 2017 at 3:42

I'd say this is a highly visual problem. I'd agree with Mr. Noche in that the best way to approach this is from a geometry perspective. In addition. I'd use something like Desmos, to show students how the graph of a line changes when we "multiply by x" (how, for example, the equation $y=ax^{2}+b$ seems to "open and bend" as a gets progressively closer to 0 and the equation gets closer to $y=b.$

Like Mr. Noche suggests, start slowly, adding complications as students get more comfortable the equation's structure. Here's how I would approach this thing:

  • Refresh students on linear variation.
  • Numerically compare linear variation to quadratic variation.
  • Have them plot $y=x$ (by hand or in Desmos)
  • Have them plot $y=x^{2}.$ Discuss similarities and differences.
  • Ask something like: "What do you think will happen if I add some number $b$ to this equation?" After some discussion, show them what happens.
  • Ask them something like "What do you think will happen if I make this number $b$ bigger/smaller?"

  • Ask them something like: "What would happen if I multiply this $x^{2}$ by some number $a$?

  • Build progressively, until students are playing around with the general quadratic.


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