By applying the features of polynomials, the quadratic function g(x) = (1/5) · x² - 2 · x + 21/5 passes through the points (x₁, y₁) = (-8, 33), (x₂, y₂) = (2, 1) and (x₃, y₃) = (8, 1).
Polynomials are algebraic expressions of the form [tex]\sum \limits_{i=0}^{n} c_{i}\cdot x^{i}[/tex], where [tex]c_{i}[/tex] is the i-th coefficient of the polynomial and n is the grade of the polynomial. In addition, we know that polynomials have n + 1 coefficients.
In the case of quadratic functions, the grade is 2 and the function have three coefficients. Hence, we need three distinct points to determine all the coefficients of a quadratic function.
(x₁, y₁) = (-8, 33)
64 · a - 8 · b + c = 33 (1)
(x₂, y₂) = (2, 1)
4 · a + 2 · b + c = 1 (2)
(x₃, y₃) = (8, 1)
64 · a + 8 · b + c = 1 (3)
Then, we find the solution of this system of linear equations:
a = 1/5, b = -2, c = 21/5
By applying the features of polynomials, the quadratic function g(x) = (1/5) · x² - 2 · x + 21/5 passes through the points (x₁, y₁) = (-8, 33), (x₂, y₂) = (2, 1) and (x₃, y₃) = (8, 1).
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