Use a system of equations to solve each problem. Find an equation of the parabola y = ax² + bx + c that passes through the points (2, 3), (-1, 0), and (-2, 2).

Verified step by step guidance

1. Start by substituting each given point into the general form of the parabola equation \(y = ax^2 + bx + c\). For the point \((2, 3)\), substitute \(x = 2\) and \(y = 3\) to get the equation \$3 = a(2)^2 + b(2) + c$.
2. Next, substitute the point \((-1, 0)\) into the equation by setting \(x = -1\) and \(y = 0\), which gives \$0 = a(-1)^2 + b(-1) + c$.
3. Then, substitute the point \((-2, 2)\) by setting \(x = -2\) and \(y = 2\), resulting in \$2 = a(-2)^2 + b(-2) + c$.
4. Now, simplify each equation to form a system of three linear equations in terms of \(a\), \(b\), and \(c\). This system will look like: \$4a + 2b + c = 3\(, \)a - b + c = 0\(, and \)4a - 2b + c = 2$.
5. Solve this system of equations using either substitution, elimination, or matrix methods to find the values of \(a\), \(b\), and \(c\). These values will give you the specific equation of the parabola.

Key Concepts

System of Equations

A system of equations consists of multiple equations with multiple variables that are solved together. In this problem, each point provides an equation when substituted into y = ax^2 + bx + c, creating a system to find a, b, and c.

Quadratic Function Form

The quadratic function y = ax^2 + bx + c represents a parabola, where a, b, and c are constants. Understanding this form allows you to set up equations by plugging in x and y values from given points.

Substitution of Points into Equations

Substituting each point's coordinates into the quadratic equation generates specific equations. This step translates geometric information into algebraic form, enabling the creation of a solvable system.