Find the equation to the curve that joins the following points.
(1,1),(0,1),(2,5),(-1,-5)
assuming you want a polynomial of lowest degree, that would be a cubic
y=ax^3+bx^2+cx+d
(1,1): a+b+c+d=1
(0,1): d=1
(2,5): 8a+4b+2c+d=5
(-1,-5): -a+b-c+d=-5
solve the system to get (and avoiding fractions)
3y = 5x^3-9x^2+4x+3
To find the equation of a curve that joins a set of points, we can use the concept of interpolation. Interpolation is a method to estimate values between known data points. In this case, we will use a polynomial interpolation method called Lagrange Interpolation.
Lagrange Interpolation can be used to find the equation of a curve that passes through a given set of points. The equation for Lagrange Interpolation is:
y = Σ [L(i)(x) * y(i)]
where y is the dependent variable (the y-coordinate of the point), x is the independent variable (the x-coordinate of the point), L(i)(x) is the Lagrange polynomial, and Σ represents the sum over all data points.
To find the equation for the curve passing through the given points (1,1), (0,1), (2,5), and (-1,-5), we can follow these steps:
Step 1: Determine the Lagrange polynomials for each point.
The Lagrange polynomial L(i)(x) for the ith point is defined as:
L(i)(x) = Π [(x - x(j)) / (x(i) - x(j))]
where j is the index for all points except i, x(i) and x(j) are the x-coordinates of the ith and jth points, respectively.
For the given set of points, we have:
L(1)(x) = [(x - 0)(x - 2)(x + 1)] / [(1 - 0)(1 - 2)(1 + 1)]
L(2)(x) = [(x - 1)(x - 2)(x + 1)] / [(0 - 1)(0 - 2)(0 + 1)]
L(3)(x) = [(x - 1)(x - 0)(x + 1)] / [(2 - 1)(2 - 0)(2 + 1)]
L(4)(x) = [(x - 1)(x - 0)(x - 2)] / [(-1 - 1)(-1 - 0)(-1 - 2)]
Step 2: Plug the Lagrange polynomials into the formula and simplify.
The equation for the curve passing through the given set of points is:
y = L(1)(x) * y(1) + L(2)(x) * y(2) + L(3)(x) * y(3) + L(4)(x) * y(4)
Substituting the L(i)(x) and y(i) values from the given points, we have:
y = [(x - 0)(x - 2)(x + 1)] / [(1 - 0)(1 - 2)(1 + 1)] * 1 +
[(x - 1)(x - 2)(x + 1)] / [(0 - 1)(0 - 2)(0 + 1)] * 1 +
[(x - 1)(x - 0)(x + 1)] / [(2 - 1)(2 - 0)(2 + 1)] * 5 +
[(x - 1)(x - 0)(x - 2)] / [(-1 - 1)(-1 - 0)(-1 - 2)] * -5
Simplifying the equation further will give you the final equation of the curve passing through the given points.