Question

For each modeling function, the residual, R₁, of data point (xi, yi) is calculated as R₁ = yi - f(x₁). For which function is the condition below true?

graphed data
I : 1, 2, 3, 4
xi: 0, 2, 4, 6
yi; 2, 1, -1, 1
ri : , , , ,
the functions provided :
A. f(x) = x - 2
B. f(x) = 3 - x
C. f(x) = 0
D. f(x) = 1​

Answer 1

To determine which function satisfies the condition for the given residual values, we can calculate the residuals for each function and compare them to the provided residual values.

Given data:
xi: 0, 2, 4, 6
yi: 2, 1, -1, 1

Calculating residuals for each function:

A. f(x) = x - 2:
R₁ = yi - f(xi)
R₁ = 2 - (0 - 2) = 2 - (-2) = 4
R₂ = 1 - (2 - 2) = 1 - 0 = 1
R₃ = -1 - (4 - 2) = -1 - 2 = -3
R₄ = 1 - (6 - 2) = 1 - 4 = -3

B. f(x) = 3 - x:
R₁ = yi - f(xi)
R₁ = 2 - (3 - 0) = 2 - 3 = -1
R₂ = 1 - (3 - 2) = 1 - 1 = 0
R₃ = -1 - (3 - 4) = -1 - (-1) = 0
R₄ = 1 - (3 - 6) = 1 - (-3) = 4

C. f(x) = 0:
R₁ = yi - f(xi)
R₁ = 2 - 0 = 2
R₂ = 1 - 0 = 1
R₃ = -1 - 0 = -1
R₄ = 1 - 0 = 1

D. f(x) = 1:
R₁ = yi - f(xi)
R₁ = 2 - 1 = 1
R₂ = 1 - 1 = 0
R₃ = -1 - 1 = -2
R₄ = 1 - 1 = 0

Comparing the calculated residuals to the provided residuals:

Given residuals: , , ,
Calculated residuals (A): 4, 1, -3, -3
Calculated residuals (B): -1, 0, 0, 4
Calculated residuals (C): 2, 1, -1, 1
Calculated residuals (D): 1, 0, -2, 0

Based on the comparison, the function that satisfies the condition for the given residuals is Function C, f(x) = 0, as the calculated residuals match the provided residuals: 2, 1, -1, 1.