In Exercises 19–22, find the quadratic function y = ax²+bx+c whose graph passes through the given points. (−1, 6), (1, 4), (2, 9)

Verified step by step guidance

1. Write the general form of the quadratic function as $y=a{x}^2+bx+c$, 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 point (−1, 6), substitute $x=\; -1$ and $y=\; 6$ to get $6\; =\; a(-1)^2\; +\; b(-1)\; +\; c$. Repeat this for points (1, 4) and (2, 9).
3. Simplify each equation to form a system of three linear equations in terms of $a$, $b$, and $c$:
   1) $6\; =\; a\; -\; b\; +\; c$
   2) $4\; =\; a\; +\; b\; +\; c$
   3) $9\; =\; 4a\; +\; 2b\; +\; c$.
4. Solve the system of equations using substitution or elimination methods to find the values of $a$, $b$, and $c$.
5. Write the quadratic function by substituting the found values of $a$, $b$, and $c$ back into the general form $y\; =\; ax^2\; +\; bx\; +\; c$.

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, which can open upwards or downwards depending on the sign of a.

System of Equations

To find the coefficients a, b, and c, you substitute the given points into the quadratic equation, creating a system of linear equations. Solving this system simultaneously yields the values of a, b, and c that satisfy all points.

Substitution and Solving Linear Systems

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