Function of a concave curve given pts and slopes

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    Concave Curve Function
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SUMMARY

To derive a function of a concave downward sloping curve given specific points and slopes, one can utilize a third-degree polynomial of the form f(x) = a0 + a1x + a2x^2 + a3x^3. By substituting the known points and their respective slopes into the polynomial, a system of linear equations is formed. Solving this system allows for the determination of the coefficients a0, a1, a2, and a3, thus providing the desired function that describes the curve.

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  • Understanding of polynomial functions and their properties
  • Knowledge of calculus, specifically derivatives and slopes
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el_llavero
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It's been a while since my last calc course and I'm not sure if I even encountered how to do this but is there a way to derive a function of a concave downward sloping curve given two or more points and their respective slopes? I'm obviously aware how the curve looks but I would like to know how to derive the function if at all posible. i.e. (2,10) slope -8 and (6,4) slope -1/2, this curve has a negative decreasing slope and is concave towards the (0,0) origin. The curve's appearance is obvious from the information given but is there a way to derive a general function for this curve?
 
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Take a third degree polynomial f(x) = a0 + a1x + a2x^2 + a3x^3, plug your four conditions in, and solve the linear system of equations for the coefficients a0, a1, a2 and a3.
 

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