Finding a Parabola using three points, no equations

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SUMMARY

This discussion focuses on finding the equations of a parabola given three points without any initial equations. The points provided are (-1, 0), (3, 2), and (4, 1). To derive the standard form, represented as y = ax² + bx + c, and the vertex form, y = a(x - x₀) + y₀, users must substitute the coordinates of the points into these equations to create a system of equations. Solving these equations yields the coefficients a, b, c for the standard form and a, x₀, y₀ for the vertex form.

PREREQUISITES
  • Understanding of parabolic equations, specifically standard form and vertex form.
  • Ability to solve systems of equations.
  • Familiarity with coordinate geometry.
  • Basic algebra skills for manipulating equations.
NEXT STEPS
  • Learn how to derive the coefficients of a parabola from given points using matrix methods.
  • Study the properties of parabolas, including axis of symmetry and vertex location.
  • Explore the implications of non-vertical axes of symmetry in parabolic equations.
  • Practice solving systems of equations with three variables using substitution and elimination methods.
USEFUL FOR

Students, educators, and anyone interested in algebraic geometry or quadratic functions will benefit from this discussion, particularly those looking to enhance their skills in deriving equations from points on a parabola.

DarlingAly
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Can someone help me with this problem? I already turned it in, knowing it was wrong, and now I want to know how to do it.

they gave me three points and no equations and expected me to find the standard form and vertex form of the parabola. i don't remember what the points were, so let's just say they were (-1, 0), (3,2), (4,1)

How would you find the two equations?
 
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First, it helps to know what the "standard form" and "vertex form" for a parabola are! The "standard form" for a parabola (with vertical axis of symmetry) is y= ax2+ bx+ c and the "vertex form" is y= a(x- x0)+ y0. For the first, you need to find a, b, and c. For the second, you need to find a, x0, and y0.

In either case use the points you are given to replace x and y in the equation: saying that (-1, 0) is a point on the parabola means that x= -1, y= 0 satify the equation:
0= a(-1)2+ b(-1)+ c. For (3, 2), 2= a(3)2+ b(3)+ c, and for (4, 1), 1= a(4)2+ b(4)+ c. That gives you 3 equations to solve for a, b, and c.

You can do exactly the same thing with the vertex form to find a, x0, and y0.

(If the axis of symmetry is not vertical, 3 points are not enough to determine the equation.)
 
Yeah I knew the standard form and the vertex form. I also knew the equations. It's just that I kept getting fractions while everyone else got the vertex to be a (3, -5) with the numbers i can't remember.. haha

I just wanted to know if anyone did it different.
Thanks!
 

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