Find the quadratic function y = ax²+bx+c whose graph passes through the given points. (−1,−4), (1,−2), (2, 5)

Verified step by step guidance

1. Identify the general form of the quadratic function: \(y = ax^{2} + bx + c\), where \(a\), \(b\), and \(c\) are constants to be determined.
2. Substitute each given point into the quadratic equation to create a system of equations. For the point \((-1, -4)\), substitute \(x = -1\) and \(y = -4\) to get: \(a(-1)^{2} + b(-1) + c = -4\).
3. Similarly, substitute the point \((1, -2)\) into the equation: \(a(1)^{2} + b(1) + c = -2\).
4. Substitute the point \((2, 5)\) into the equation: \(a(2)^{2} + b(2) + c = 5\).
5. Solve the resulting system of three equations with three unknowns (\(a\), \(b\), and \(c\)) using methods such as substitution, elimination, or matrix operations to find the values of \(a\), \(b\), and \(c\).

Key Concepts

Quadratic Functions

A quadratic function is a polynomial of degree two, generally written as y = ax² + bx + c, where a, b, and c are constants and a ≠ 0. Its graph is a parabola that opens upward if a > 0 and downward if a < 0.

System of Equations

To find the coefficients a, b, and c of the quadratic function, you substitute the given points into the equation y = ax² + bx + c, creating a system of linear equations. Solving this system yields the values of a, b, and c.

Substitution and Solving Linear Systems

Substitution involves replacing variables with known values to form equations. Solving the resulting system of linear equations can be done using methods like substitution, elimination, or matrix operations to find the unknown coefficients.