Family of quadratic functions?

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Discussion Overview

The discussion revolves around finding a family of quadratic functions of the form y=ax^2+bx+c that pass through specific points. Participants explore how to derive these functions based on given coordinates, with a focus on the implications of different points on the resulting equations.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant initially presents the problem of finding a family of quadratic functions that pass through the points (1,1) and (2,0), expressing confusion about how to derive specific functions from these points.
  • Another participant suggests forming two equations based on the points, leading to a system that can be solved for b and c in terms of a, indicating that there are no conditions on a.
  • A later post corrects the initial points to (1,0) and (-1,-2), leading to a new set of equations and a different approach to finding b and c.
  • Participants discuss the implications of the values of b and c, with one participant expressing uncertainty about how to define the family of functions based on these parameters.
  • There is a clarification regarding the form of the quadratic function, with one participant asserting that the family can be expressed as ax^2 + x - 1 - a for nonzero constants a.
  • Another participant questions the correctness of the expression, leading to a confirmation that the correct form is indeed ax^2 + x - 1 - a.

Areas of Agreement / Disagreement

Participants express differing views on the correct points and the resulting equations, leading to some confusion. There is no consensus on the specific family of functions until later posts clarify the form, but earlier disagreements about the values of b and c remain unresolved.

Contextual Notes

Participants exhibit uncertainty regarding the correct points and their implications for the quadratic equations. There are also unresolved issues about the derivation of the family of functions based on the parameters a, b, and c.

mathman100
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This question has been killing me for days, :
Give a family of quadratic functions of the form y=ax^2+bx+c, that passes through the following points:
(1,1) and (2,0)
I see how we can find the family, but how do we find the specific functions that pass through both those points? I tried making two separate equations (1 for each point, like(1,1) I would sub in for x and y) but that didn't give me anything useful. Even if I use elimination for the 2 equations, I don't know where it will get me...what should i do?
 
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mathman100 said:
I tried making two separate equations (1 for each point, like(1,1) I would sub in for x and y) but that didn't give me anything useful.

Of course it does. You get the two equations

[tex]1 = a + b + c[/tex]

and

[tex]0 = 4a + 2b + c,[/tex]

which can be solved to get [itex]b = -1 - 3a[/itex] and [itex]c = 2(a+1)[/itex] with no conditions on [itex]a[/itex]. So what's your family of functions?
 
Really? Well first of all I put down the wrong 2 points (aorry, my fault) they should be (1,) and (-1,-2). I got 0=a+b+c and -2=a-b+c. I solved to get b=-1 and i think c=-1-a. I don't know how to continue!
 
woops, points= (1,0) and (-1,-2)
 
does anyone know?
 
well, if b=-1 and c=-1-a in ax^2 + bx + c then what do the polynomials that go through those points look like?
 
i don't know, now I'm lost:rolleyes: do you mean like a parabola? a linear line with a slope? how do i define the family? why can't i just sub in the known values of b and c:
ax^2-x-1-a?
 
you can! that is exactly the "family" it's looking for: all the quadratics with the form [itex]a^2 + x - 1 - a[/itex] for nonzero constants [itex]a[/itex] (you actually made an error earlier. You should have found [itex]b=1[/itex], not [itex]b=-1[/itex]). :-p
 
Data said:
you can! that is exactly the "family" it's looking for: all the quadratics with the form [itex]a^2 + x - 1 - a[/itex] for nonzero constants [itex]a[/itex]
should it be [tex]a^2 + x - 1 - a[/tex]? or [tex]ax^2 + x - 1 - a[/tex]?
 
  • #10
[itex]ax^2 + x - 1 -a.[/itex] I'm not very good at typing!
 

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