Problem 97 - 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. \(f(-2) = -15\), \(f(-1) = 7\), \(f(1) = -3\)

Step-by-Step Solution

Answer

The solution will give the coefficients \(a\), \(b\), and \(c\) which represent the quadratic function that satisfies the given equations. The matrix representation and matrix operations are used to solve the system of equations.

Step 1: Formulate the System of Equations

Start by substituting the given \(x\) values (-2, -1, 1) into the quadratic function \(f(x) = ax^2 + bx + c\). This yields three different equations: \n 1. For \(x = -2\), we have \( -15 = 4a - 2b + c\) 2. For \(x = -1\), we have \( 7 = a - b + c\) 3. For \(x = 1\), we have \( -3 = a + b + c\)

Step 2: Represent the System as a Matrix

Next, represent the system of equations in matrix form. This gives us: \n \(\begin{bmatrix} 4 & -2 & 1 \ 1 & -1 & 1 \ 1 & 1 & 1 \ \end{bmatrix}\) \(\begin{bmatrix} a \ b \ c \ \end{bmatrix}\) \(=\begin{bmatrix} -15 \ 7 \ -3 \ \end{bmatrix}\)

Step 3: Solve Using Inverse of the Matrix

To find the matrix \(\begin{bmatrix} a \ b \ c \ \end{bmatrix}\), multiply the inverse of the coefficient matrix (if it exists) with the constant matrix. This gives us the values of \(a\), \(b\), \(c\) as the solution to the system.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. In this exercise, we are working with three equations as they represent the quadratic function at specific points.

• The first equation comes from substituting x = -2 into the quadratic function, resulting in \(-15 = 4a - 2b + c\).
• The second equation is formed by substituting x = -1, leading to the equation \(7 = a - b + c\).
• The third is derived from substituting x = 1, which gives us \(-3 = a + b + c\).

The goal is to solve for the unknowns \(a\), \(b\), and \(c\) that satisfy all these equations. Solving such systems can be efficiently done using matrices, especially for systems with multiple variables.

Matrices

Matrices are a powerful mathematical tool used to handle systems of equations. They are simply rectangular arrays of numbers, which can be manipulated to find solutions to complex problems. In matrix form, our system of equations is represented as:\[\begin{bmatrix} 4 & -2 & 1 \ 1 & -1 & 1 \ 1 & 1 & 1 \end{bmatrix}\begin{bmatrix} a \ b \ c \end{bmatrix}= \begin{bmatrix} -15 \ 7 \ -3 \end{bmatrix}\]The first matrix on the left is called the coefficient matrix and contains the coefficients of \(a\), \(b\), and \(c\) from the system of equations. The matrix on the right represents the constants from each equation. By using matrix operations, we can find the values for the unknowns.

Inverse Matrix

An inverse matrix is a type of matrix that, when multiplied by the original matrix, yields the identity matrix. This property is crucial for solving systems of equations with matrix algebra. The process involves the following steps:

• Check for invertibility: Not all matrices have an inverse. A matrix must be square (the same number of rows and columns) and have a non-zero determinant to have an inverse.
• Calculate the inverse: If the coefficient matrix has an inverse, it can be found using various methods, such as row reduction, adjoint method, or computational tools.
• Multiply by the inverse: To solve for the unknowns \(\begin{bmatrix} a \ b \ c \end{bmatrix}\), multiply the inverse of the coefficient matrix by the constant matrix.This computation gives us:\[\begin{bmatrix} a \b \c \end{bmatrix}= \begin{bmatrix} \text{inverse of coefficient matrix}\end{bmatrix}\begin{bmatrix} -15 \7 \-3 \end{bmatrix}\]

This operation effectively isolates the variables \(a\), \(b\), and \(c\) and provides their values.

Coefficient Matrix

The coefficient matrix is crucial in representing a system of equations in matrix form. It solely contains the coefficients of the variables from each equation, omitting the constants.

• Structure: In our example, the coefficient matrix is a 3x3 matrix:\[\begin{bmatrix} 4 & -2 & 1 \1 & -1 & 1 \1 & 1 & 1 \end{bmatrix}\]Each row represents the coefficients from one of the system's equations.
• Utility: This matrix is used to perform operations such as addition, subtraction, multiplication, and finding an inverse if applicable. The coefficients represent the variables, and when combined with matrix operations, they allow us to solve for unknowns efficiently.

Understanding how to create and manipulate the coefficient matrix is key to solving systems of equations using matrices. It's a foundational step in translating word problems or equations into a format that can be tackled systematically.