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While simultaneously solving a quadratic equation and a linear equation, I noticed that their intercepts are actually equal to the x intercepts of another quadratic function. I find it hard to grasp that subtracting a linear equation from a quadratic equation results in a symmetric curve(parabola).

I understand that the result of the subtraction will give me a polynomial of degree 2 which forms a parabola, but why?

ashir
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  • I think that in order to answer this question, we would need to know what exactly makes something "quadratic" intuitively. What do you expect to happen when one adds a linear function to a quadratic one that might "ruin" the property of being quadratic? – Ben Grossmann Oct 29 '20 at 17:04
  • Both linear, and quadratic functions are changing according to different 'rules'. It just feels weird that their subtraction can produce something symmetric. I know about the vertex form(just completed quadratics in college) – ashir Oct 29 '20 at 17:08
  • Also, do you know what the vertex form of a quadratic is? – Ben Grossmann Oct 29 '20 at 17:08
  • Thanks for explaining. I think the thing to nail down then is what exactly is the "rule" that a quadratic function follows? Once you have that nailed down, you should find that linear functions actually follow the same rule that quadratic functions do, and adding any two such function is going to make it so that this rule will still be followed. – Ben Grossmann Oct 29 '20 at 17:12
  • One rule that characterizes a quadratic function is the property of linear differences (or constant second-order differences), which is what the answer given below describes – Ben Grossmann Oct 29 '20 at 17:13
  • THANK YOU!!! This seems to be a whole new way of thinking about functions. Can I get references for further reading(its hard to find stuff that's my level). – ashir Oct 29 '20 at 17:17
  • If you like that, then you'll love calculus. In fact, the rule that is more typically used in this kind of way is that of the derivative of a quadratic function. Instead of looking at the difference between $f(x)$ and $f(x + 1)$, we look at the slope of the function $f$ at the point $x$. It turns out as just as these differences change linearly, so does the slope change linearly. – Ben Grossmann Oct 29 '20 at 17:21
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    As for a reference: this kind of analysis usually appears in the study of difference equations, which is unfortunately rarely presented at the high school level. This wiki page gives some of the broader context. The specific usage on quadratic functions is described here in a way that I think should be "at your level". – Ben Grossmann Oct 29 '20 at 17:29

2 Answers2

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One way of looking at it is through finite differences. A quadratic function has the property that successive differences follow a linear progression. For example, with $f(n) = n^2 + n + 1$:

n  f(n)  difference
-------------------
0    1  
          2
1    3  
          4
2    7
          6
3   13
          8
   ...

A linear function has the property that successive differences are constant. Thus, adding a linear function to a quadratic function will shift the differences of the latter without disturbing their linearity.

The argument in the previous paragraph relies on an understanding that "constant + linear = linear", so in fact you can see that you can use the same reasoning inductively for adding polynomials of any degree.

Théophile
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A higher level understanding of this could come from the study of affine geometry. In affine geometry all conics are either ellipses, hyperbolas, or parabolas, and are not changed in character by any affine transformation. Your notion of

subtracting a linear equation from a quadratic equation

is a special case of an affine transformation and and hence transforms a parabola to another parabola.

Somos
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