How do you find the equation of a curved line?

Question

I have these points on the graph:

(-3, 0) (-2, 1) (1, 2)

How do I find the equation?
Any help is appreciated.

Answer 1

Let (x,y) be the given points,

where y = ƒ(x) is a polynmial.



Then :



( x₀ , y₀ ) ≡ ( -3, 0 ),



( x₁ , y₁ ) ≡ ( -2, 1 ),



( x₂ , y₂ ) ≡ ( 1, 2 ).

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Then, by Lagrange's Formula for Interpolation,



ƒ(x) =



[ ( x - x₁ )( x - x₂ ) / (( x₀ - x₁ )( x₀ - x₂ )) ] · ƒ( x₀ ) +



[ ( x - x₀ )( x - x₂ ) / (( x₁ - x₀ )( x₁ - x₂ )) ] · ƒ( x₁ ) +



[ ( x - x₀ )( x - x₁ ) / (( x₂ - x₀ )( x₂ - x₁ )) ] · ƒ( x₂ )

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Substitute the values of x₀ , x₁ and x₂ with corresponding

ƒ-values.

That will give you the required equation in x and y.

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Happy To Help !

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Answer 2

There are an infinite number of equations which contain those points. If you limit the choice to only second order equations, then you still need at least five conditions to uniquely define the curve. You only have three.



If you restrict the curve to a parabolic equation in standard position you have only two equations that can be fit to those points:



y = ax² + bx + c

and

x = ay² + by + c



Chose one of those, a parabola with a vertical axis or one with a horizontal axis and you can obtain a unique curve that meets those five conditions, 3 points, parabola, and axis direction.. Once you've made that choice, substitute the values of x and y into the equation to get three independent equations with the unknown coefficients, a, b, and c. Solve that system for the coefficients and you will have a parabolic equation through those three points.



Or you could fit a circle through those points:

(x-g)² + (y-h)² = R²

using the same technique. Substitute the x and y values from the points into the equation to get three equations in g, h, and R. Solve that system of equations and you will have the definition of a circle that contains the three points. Note that a circle is simply an ellipse with equal length axes so you have the five conditions, 3 points, ellipse, and equal length axes.



All 2nd order curves require 5 conditions for a unique definition of the conic curve.

Answer 3

Use a graphing calculator or online utility. For instance, at wolframalpha, enter "fit (-3, 0) (-2, 1) (1, 2)" and you will be provided a couple of possible equations that closely model the given points.