Finding a parabola given two x intercepts

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Homework Help Overview

The discussion revolves around finding the equation of a parabola given two x-intercepts, specifically (5,0) and (-1,0). Participants explore whether it is possible to determine a unique parabola from this information alone.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the implications of having only two x-intercepts and question whether this is sufficient to define a specific parabola. Some mention the need for additional points to establish a unique equation.

Discussion Status

There is a recognition that while a general form of the parabola can be established, a specific equation cannot be determined with only two x-intercepts. Participants acknowledge the existence of multiple parabolas that can share the same roots but differ in steepness and concavity.

Contextual Notes

Some participants note that the line of symmetry can provide insight into the vertex's x-coordinate, but they also highlight the limitation of having only two points to define the parabola fully.

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If you are given (5,0) and (-1,0) as the two x intercepts of a parabola is it possible to find the equation of the parabola? I have tried using the vertex formula for the x co-ordinate which is x = 2 the line of symmetry and plugging in either of these co-ordinates into y = ax^2 + bx + c but have too many unknowns to solve for. Is this unsolvable with only this information?
 
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This is possible only if you are looking for a general parabola; NOT for a specific one. In fact, you need THREE points to establish a specific parabola.
 
Only being given the roots of the parabola isn't sufficient enough to compile a specific parabola from it. There are an infinite number of parabolas having those 2 roots, all having different "steepness" and concavity.

e.g.

y=(x-5)(x+1)
y=2(5-x)(3x+3)

etc.
 
Thanks. I suppose if I tried different values for a and b I would come up with different parabolas. Wasn't sure if there was some way with the line of symmetry to determine the y value of the vertex but I realize that there are different options with only two points. Thanks again.
 

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