In Exercises 19–22, 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. Start by writing the general form of the quadratic function: $y=a{x}^2+bx+c$.
2. Substitute each given point into the quadratic equation to create a system of equations. For point (−1, −4), substitute $x=\; -1$ and $y=\; -4$ to get: $-4\; =\; a(-1)^2\; +\; b(-1)\; +\; c$.
3. Repeat the substitution for the other points: (1, −2) gives $-2\; =\; a(1)^2\; +\; b(1)\; +\; c$, and (2, 5) gives $5\; =\; a(2)^2\; +\; b(2)\; +\; c$.
4. Simplify each equation to form a system of three linear equations in terms of $a$, $b$, and $c$: $-4\; =\; a\; -\; b\; +\; c$, $-2\; =\; a\; +\; b\; +\; c$, and $5\; =\; 4a\; +\; 2b\; +\; c$.
5. Solve the system of equations using substitution or elimination methods to find the values of $a$, $b$, and $c$, which will give you the quadratic function.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial of degree two, generally written as y = ax^2 + 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. Understanding this form is essential to model and analyze the curve passing through given points.

System of Equations

To find the coefficients a, b, and c of the quadratic function, you substitute each given point into the equation y = ax^2 + bx + c, creating a system of linear equations. Solving this system simultaneously allows you to determine the unique values of a, b, and c that fit all points.

Substitution and Solving Linear Systems

Substitution involves replacing variables with known values to simplify equations. In this context, plugging in the x and y values from each point forms equations that can be solved using methods like substitution, elimination, or matrix operations to find the unknown coefficients of the quadratic.