Lagrange interpolating polynomials help?

Question

Determine the linear Lagrange interpolating polynomial that passes through the points (2,4) and (5,1). Please explain all steps so I can understand how you got your answer.

Answer 1

If you only have 2 points, the Lagrange interpolating polynomial will give the equation of a straight line.  Recall that the degree of the Lagrange interpolating polynomial is 1 lower than the number of points.  It is difficult to write the equations in this window, but here goes my attempt.

 

P(x) = Σ_j=1ⁿ P_j(x)

 

P_j(x) = y_jΠ_k=1,_k≠jⁿ (x - x_k)/(x_j - x_k)

 

x₁ = 2

x₂ = 5

y₁ = 4

y₂ = 1

 

P₁(x) = y₁(x - x₂)/(x₁ - x₂) = 4(x - 5)/(2 - 5) = -(4/3)(x - 5)

 

P₂(x) = y₂(x - x₁)/(x₂ - x₁) = 1(x - 2)/(5 - 2) = (1/3)(x - 2)

 

P(x) = P₁(x) + P₂(x) = [-(4/3)(x - 5)] + [(1/3)(x - 2)] = -x + 6

 

Note that in this problem with only two points, y = P(x) is just the equation of a straight line that goes through the two given points.  

 

 

 

 

Answer 2

x0 = 2 y0 = 4
x1 = 5 y1 = 1


y0( x-x1)/(x0-x1) + y1(x-x0)/(x1-x0) =

4(x-5)/(-3) + (x-2)/(3) = -4/3(x+5) + 1/3(x-2) = -4/3X + 20/3 + 1/3X - 2/3 = -X + 6

To verify, just find the slope of the line: (4-1)/(2-5) = 3/-3 = -1

Line equation is y = Mx+B so then B = y-Mx. Using the second data point, X=5 when y=1,
so B = 1 - -1*5 = 1 + 5 = 6. This agrees with the Lagrange interpolation polynomial