How to Find the Equation of a Parabola from Three Points?

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Discussion Overview

The discussion revolves around finding the equation of a parabola given three points on its graph, specifically in the form Y = -ax² + bx + c. Participants explore methods for deriving the coefficients a, b, and c using the provided points.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant expresses uncertainty about how to derive the equation from three points, mentioning tangent lines.
  • Another participant suggests that if a point (x0, y0) lies on the parabola, it must satisfy the equation Y(x) = -ax² + bx + c, leading to a system of equations for the coefficients.
  • It is noted that having three points allows for the creation of three equations with three unknowns (a, b, c).
  • One participant mentions calculating the slope between two points but is unsure how to find the rest of the equation, indicating a lack of understanding of how to derive coefficients or the y-intercept.
  • Another participant challenges the idea of using the slope between two points on the parabola, stating that the slope varies along the curve and suggesting the use of the general equation instead.
  • There is a reiteration that plugging the coordinates of the three points into the general equation will yield three linear equations to solve for a, b, and c.

Areas of Agreement / Disagreement

Participants generally agree on the method of substituting the points into the parabola's equation to derive the coefficients. However, there is disagreement regarding the relevance of calculating the slope between two points on the parabola.

Contextual Notes

Some participants express confusion about the relationship between the slope of a line connecting two points and the slope of the parabola itself, indicating a potential misunderstanding of the properties of parabolic curves.

donlin
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I have three points on a Y= -ax^2 + bx + c graph (negative parabola). I don't know how to find the equation with this information. Tangent lines? Help.
 
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Hi donlin. Welcome to PF!
Let's say one of the points is (x0, y0). What's -ax02+bx0+c equal to?
 
Your equation is a function of one variable, Y(x)=-ax^2+bx+c. If a point (x0, y0) lies on the parabola, it must satisfy the equation f(x0) = y0. Since you have three points, after 'plugging' every one of them into the equation, you'll have three equations with three unknowns a, b and c, which are the coefficients you need.
 
I think I have the slope, by connecting point A to B and then rise over run, but I don't know how to get the rest of the equation. Basically, I have a point A and slope, but I don't have the coefficients or y-intercept.
 
neutrino said:
Hi donlin. Welcome to PF!
Let's say one of the points is (x0, y0). What's -ax02+bx0+c equal to?

I think it's equal to y0.
 
donlin said:
I think I have the slope, by connecting point A to B and then rise over run, but I don't know how to get the rest of the equation. Basically, I have a point A and slope, but I don't have the coefficients or y-intercept.
You can't get the slope at one point of a parabola by connecting two points on it. It's different at every point. Use the method that radou posted.
 
donlin said:
I think I have the slope, by connecting point A to B and then rise over run, but I don't know how to get the rest of the equation. Basically, I have a point A and slope, but I don't have the coefficients or y-intercept.
A curve does not have a slope! What you have found is the slope of the line through A and B which is irrelevant. If your question is how to find the equation of a parabola passing through three points, just put the x and y coordinates of each point into your general equation, Y= -ax^2 + bx + c, gives you three linear equations for A, B, and C.
 

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