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Why do some quadratic graphs have a flat bottom? This is concerning sketching the graph on a physical graph. An example of it would be the equation:y=x^2 -5x+4

user324713
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community May 26 '22 at 09:33
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    What do you mean by "flat bottom"? – 5xum May 26 '22 at 09:35
  • If you plot points, don't join them with straight lines? Maybe – Paul May 26 '22 at 09:39
  • The "flatness" is not striking for degree 2. It is more evident for higher degrees, for example with $y=x^4, y=x^6...$ – Jean Marie May 26 '22 at 09:40
  • @5xum What I mean by flat bottom is that when x is 2 or 3 in the equation given above, y is -2 for both the values of x. – user324713 May 26 '22 at 09:40
  • @Paul Yes, but in our high school textbook, it is said to do so when a curve should turn smoothly. – user324713 May 26 '22 at 09:43
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    @user324713 Sure, but $y$ is smaller for $x$ between $2$ and $3$ – 5xum May 26 '22 at 09:44
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    @5xum Yes, but the x can only be an integer in the question. Another thing, why did it reach the same point? – user324713 May 26 '22 at 09:50
  • @user324713 For the same reason that $x^2$ reaches the same point at $x=1$ and $x=-1$. – 5xum May 26 '22 at 09:51
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    @user324713 The curve only "turns smoothly" if you draw it for all values of $x$. If you only use integer values of $x$, then it doesn't really make any sense to talk about "smoothness", because you don't even have a curve. You just have a discrete set of points. – 5xum May 26 '22 at 09:56
  • @5xum But I think it isn't the reason for this equation. The roots of x here, are 1 and 4. – user324713 May 26 '22 at 10:02
  • @user324713 It is exactly the reason. Every quadratic equation takes all values (except for one) twice. – 5xum May 26 '22 at 10:03
  • Welcome to Math.SE! <> Mathematically, all quadratic graphs $y = ax^2+bx+c$ with $a\neq0$ are parabolas. Because you're restricting to integer values of $x$ (and connecting successive dots with line segments?), you'll see a "flat bottom" if $0 < a$ and the vertex of the parabola (the point on the line of vertical symmetry) lies halfway between two integers. If the vertex lies at an integer, there will be a "corner" at the vertex, between two segments of equal absolute slope; at other locations of the vertex there will be a varyingly-sloped line segment "spanning the vertex". – Andrew D. Hwang May 26 '22 at 11:43

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To illustrate the answer you've been given in the comments (using MATLAB code)

x = 0:0.001:5;
plot(x,x.^2-5*x+4);
x = 0:1:5;
hold on;plot(x,x.^2-5*x+4)

enter image description here

Johan Löfberg
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  • I think you are right is situating the issue of the OP in the plotting step (I think it is important to give the name). – Jean Marie May 26 '22 at 12:49