Family of quadratic functions?

AI Thread Summary
To find a family of quadratic functions that pass through the points (1,0) and (-1,-2), the equations derived are 0 = a + b + c and -2 = a - b + c. Solving these gives b = 1 and c = -1 - a, leading to the general form of the quadratic function as ax^2 + x - 1 - a. The discussion highlights the importance of correctly substituting values and understanding the structure of quadratic equations. Ultimately, the family of functions is defined by the parameter 'a' in the equation ax^2 + x - 1 - a for nonzero constants a.
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

1 = a + b + c

and

0 = 4a + 2b + c,

which can be solved to get b = -1 - 3a and c = 2(a+1) with no conditions on a. 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 a^2 + x - 1 - a for nonzero constants a (you actually made an error earlier. You should have found b=1, not b=-1). :-p
 
Data said:
you can! that is exactly the "family" it's looking for: all the quadratics with the form a^2 + x - 1 - a for nonzero constants a
should it be a^2 + x - 1 - a? or ax^2 + x - 1 - a?
 
  • #10
ax^2 + x - 1 -a. I'm not very good at typing!
 

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