In Exercises 19–22, find the quadratic function y = ax^2+bx+c whose graph passes through the given points. (−1,−4), (1,−2), (2, 5)

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form y = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. Understanding the general form is essential for identifying the coefficients that will satisfy the conditions given by specific points.

Systems of Equations

To find the specific quadratic function that passes through given points, one must set up a system of equations. Each point (x, y) provides an equation when substituted into the quadratic formula. Solving this system allows us to determine the values of a, b, and c that define the unique quadratic function for the specified points.

Substitution Method

The substitution method is a technique used to solve systems of equations by expressing one variable in terms of another. In the context of finding a quadratic function, this method can simplify the process of solving for the coefficients a, b, and c by substituting known values from the equations derived from the points into one another, ultimately leading to a solution.