Problem 96 - Precalculus with Limits

Question

In Exercises 95-98, use a system of equations to find the quadratic function \(f(x) = ax^2 + bx + c\) that satisfies the equations. Solve the system using matrices. \(f(1) = 2\), \(f(2) = 9\), \(f(3) = 20\)

Step-by-Step Solution

Answer

The quadratic function that fits the given points is \(f(x) = 2x^2 + 3x - 1\).

Step 1: Set Up the Equations

First, set up three equations based on the given points. Using the general form of the quadratic function \(f(x) = ax^2 + bx + c\), we can plug in the given points to get: \(a*1^2 + b*1 + c = 2\), \(a*2^2 + b*2 + c = 9\), and \(a*3^2 + b*3 + c = 20\). These simplify to: \(a + b + c = 2\), \(4a + 2b + c = 9\), and \(9a + 3b + c = 20\).

Step 2: Set Up the Matrix

Writing the system of equations in matrix form yields the matrix equation \(Ax = B\), where \(A\) is the matrix of the coefficients of \(a, b,\) and \(c\), \(x\) is the column matrix \([a, b, c]^T\), and \(B\) is the column matrix of the right hand side of the equations. In this case, we obtain: \(\begin{bmatrix} 1 & 1 & 1 \\ 4 & 2 & 1 \\ 9 & 3 & 1 \end{bmatrix}\begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} 2 \\ 9 \\ 20 \end{bmatrix}\)

Step 3: Solve the Matrix Equation

Using a software or a calculator, we solve the matrix equation to find \(a\), \(b\), and \(c\). By performing Gaussian elimination or by using the formula \(x = A^{-1}B\), we get: \([a, b, c]^T = [2, 3, -1]^T\).

Key Concepts

System of Equations

A system of equations is a set of multiple equations, each with multiple variables, which are solved simultaneously to find a common solution for the variables. In the context of finding a quadratic function, we often come across systems where the equations represent the function evaluated at certain points.

For example, to find the quadratic function in the form of \(f(x) = ax^2 + bx + c\), we can set up a system using given values of \(f(x)\) at different \(x\) coordinates. In this case:

• \(f(1) = 2\)
• \(f(2) = 9\)
• \(f(3) = 20\)

This system can then be solved using various methods, including matrix algebra, to find the coefficients of the quadratic function.

Matrix Equations

The concept of matrix equations is pivotal when translating systems of linear equations into a compact matrix form. Matrix notation is especially helpful because it allows for the application of linear algebra techniques to solve systems efficiently. The general form of a matrix equation is \(Ax = B\), where:

• \(A\) is a matrix of coefficients.
• \(x\) is a column matrix of variables.
• \(B\) is a column matrix of constant terms.

In the example of finding a quadratic function, matrix \(A\) would consist of the coefficients from the system of equations derived from the function’s evaluation at specific points, \(x\) would represent the unknowns \(a, b, c\), and \(B\) would have the results of \(f(x)\) given in the problem.

Gaussian Elimination

The method of Gaussian elimination is a step-by-step process used to solve systems of linear equations. It involves three key operations: swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting a multiple of one row from another. These operations aim to transform the matrix into row-echelon form and, eventually, reduced row-echelon form.

Once in reduced row-echelon form, the solution to the system can be easily read off or found through back-substitution. This technique is very efficient and is often executed by computer programs and calculators to deliver solutions to systems of equations, like the ones we encounter when determining the coefficients of a quadratic function.

Quadratic Function Formula

The quadratic function formula is the standard form of a quadratic function and is expressed as \(f(x) = ax^2 + bx + c\). In this formula, \(a\), \(b\), and \(c\) are coefficients that determine the shape and position of a parabola on a graph.

Identifying these coefficients requires solving the system of equations that arises from known points on the parabola. Once solved, we can define a unique quadratic function that passes through these points, and the coefficients will reveal critical properties, such as the direction of the parabola opening, its width, and the location of its vertex.