Answer :
By using algebra resolution methods for systems of linear equations, we find that the equation of the form f(x) = a · x² + b · x + c that passes through the three points is equal to f(x) = (5 / 2) · x² + (3 / 2) · x.
How to determine the quadratic equation that passes through three points
In this problem we must determine the coefficients of a quadratic equation that passes through the points (x₁, y₁) = (1, 4), (x₂, y₂) = (2, 13) and (x₃, y₃) = (4, 46). First, we need to create a system of linear equations by substituting on y and x thrice:
(x₁, y₁) = (1, 4)
a · 1² + b · 1 + c = 4
a + b + c = 4
(x₂, y₂) = (2, 13)
a · 2² + b · 2 + c = 13
4 · a + 2 · b + c = 13
(x₃, y₃) = (4, 46)
a · 4² + b · 4 + c = 46
16 · a + 4 · b + c = 46
Then, we find a system of three linear equations with three variables that offers an unique solution:
a + b + c = 4 (1)
4 · a + 2 · b + c = 13 (2)
16 · a + 4 · b + c = 46 (3)
There are different methods to find the solution to this system, we proceed to use algebraic substitution:
By (1):
c = 4 - a - b
(1) in (2) and (3):
4 · a + 2 · b + (4 - a - b) = 13
3 · a + b = 9 (2b)
16 · a + 4 · b + (4 - a - b) = 46
15 · a + 3 · b = 42 (3b)
By (2b):
b = 9 - 3 · a
(2b) in (3b):
15 · a + 3 · (9 - 3 · a) = 42
15 · a + 27 - 9 · a = 42
6 · a = 15
a = 15 / 6
a = 5 / 2
By (2b):
b = 9 - 3 · (5 / 2)
b = 9 - 15 / 2
b = 18 / 2 - 15 / 2
b = 3 / 2
By (1):
c = 4 - 5 / 2 - 3 / 2
c = 4 - 8 / 2
c = 4 - 4
c = 0
The coefficients of the quadratic equation are (a, b, c) = (5 / 2, 3 / 2, 0).
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