Find the quadratic function y = ax^2 + bx + c whose graph passes through the points (1, 4), (3, 20), and (-2, 25).

Verified step by step guidance

1. <insert step 1> Start by substituting each point into the quadratic equation y = ax^2 + bx + c to create a system of equations.
2. <insert step 2> Substitute the point (1, 4) into the equation: 4 = a(1)^2 + b(1) + c, which simplifies to 4 = a + b + c.
3. <insert step 3> Substitute the point (3, 20) into the equation: 20 = a(3)^2 + b(3) + c, which simplifies to 20 = 9a + 3b + c.
4. <insert step 4> Substitute the point (-2, 25) into the equation: 25 = a(-2)^2 + b(-2) + c, which simplifies to 25 = 4a - 2b + c.
5. <insert step 5> Solve the system of equations formed by the three equations: a + b + c = 4, 9a + 3b + c = 20, and 4a - 2b + c = 25, to find the values of a, b, and c.

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 structure of quadratic functions is essential for solving problems related to their graphs and properties.

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 the coefficients a, b, and c, which define the unique quadratic function that fits all the specified points.

Substitution Method

The substitution method is a technique used to solve systems of equations by isolating one variable and substituting it into another equation. In the context of finding a quadratic function, this method can simplify the process of solving for the coefficients by reducing the number of variables in the equations, making it easier to find the values of a, b, and c that satisfy all conditions.