Family of quadratic functions?

In summary, the conversation discusses finding a family of quadratic functions in the form y=ax^2+bx+c that pass through specific points. The first point given is (1,1) and the second is (2,0). The conversation goes on to discuss how to find the specific functions using systems of equations and substitution. Eventually, it is determined that the family of functions can be defined as ax^2 + x - 1 - a for nonzero constants a. There is some confusion about the exact form of the function, but it is concluded that it should be ax^2 + x - 1 - a.
  • #1
mathman100
15
0
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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  • #2
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?
 
  • #3
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!
 
  • #4
woops, points= (1,0) and (-1,-2)
 
  • #5
does anyone know?
 
  • #6
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?
 
  • #7
i don't know, now I'm lost:uhh: 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?
 
  • #8
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]). :tongue2:
 
  • #9
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!
 

1. What is a family of quadratic functions?

A family of quadratic functions is a set of functions that share the same general form of y = ax^2 + bx + c, where a, b, and c are constants. These functions have a parabolic shape and can differ in the values of their coefficients, resulting in different curves.

2. How can I identify a quadratic function in a set of data?

To identify a quadratic function, you can plot the data points on a graph and look for a parabolic shape. Alternatively, you can calculate the second differences between the data points, and if they are constant, the function is likely quadratic. You can also use the quadratic formula to find the roots of the function and determine its shape.

3. What are the key features of a quadratic function?

The key features of a quadratic function include its vertex, which is the highest or lowest point on the parabola, its axis of symmetry, which divides the parabola into two symmetrical halves, and its roots, which are the x-intercepts of the function. The leading coefficient also determines whether the parabola opens upwards or downwards.

4. How can I use a family of quadratic functions in real-life situations?

Quadratic functions are commonly used in physics and engineering to model the motion of objects, such as projectiles or cars. They can also be used in economics to analyze profit and cost functions, and in biology to describe growth and decay processes. Additionally, they can be used to solve optimization problems and predict future outcomes.

5. Can a quadratic function have more than one x-intercept?

Yes, a quadratic function can have zero, one, or two x-intercepts. The number of x-intercepts depends on the location of the vertex and the value of the discriminant, which is the part of the quadratic formula under the square root. If the discriminant is positive, the function will have two real roots, if it is zero, there will be one real root, and if it is negative, there will be no real roots.

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