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

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Step 1: Start by substituting each of the given points into the quadratic equation y = ax^2 + bx + c. For the point (1, 4), substitute x = 1 and y = 4 to get the equation: 4 = a(1)^2 + b(1) + c, which simplifies to 4 = a + b + c.

Step 2: Repeat the substitution process for the point (3, 20). Substitute x = 3 and y = 20 into the equation y = ax^2 + bx + c to get: 20 = a(3)^2 + b(3) + c, which simplifies to 20 = 9a + 3b + c.

Step 3: Substitute the third point (-2, 25) into the quadratic equation. Using x = -2 and y = 25, you get: 25 = a(-2)^2 + b(-2) + c, which simplifies to 25 = 4a - 2b + c.

Step 4: Now, you have a system of three equations: (1) a + b + c = 4, (2) 9a + 3b + c = 20, and (3) 4a - 2b + c = 25. Solve this system of equations using substitution or elimination to find the values of a, b, and c.

Step 5: Once you solve for a, b, and c, substitute these values back into the quadratic equation y = ax^2 + bx + c to write the final quadratic function.

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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 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 and substituting it into the other equations. 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 isolate and calculate the values of a, b, and c.