Problem 95 - 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. \(f(1) = 1\), \(f(2) = -1\), \(f(3) = -5\)

Step-by-Step Solution

Answer

The quadratic function that satisfies the given conditions is \(f(x) = -2x^2 + 4x - 1\)

Step 1: Substitute Values into the Function

By substituting given x-values (\(x=1, x=2, x=3\)) into the quadratic function, we get: \[1 = a + b + c\] \[-1 = 4a + 2b + c\] \[-5 = 9a + 3b + c\]

Step 2: Formulate the System in Matrix Form

We can write this system as a matrix and solve for [a, b, c]. Thus, the system is: \[ \begin{bmatrix} 1 & 1 & 1 \ 4 & 2 & 1 \ 9 & 3 & 1 \end{bmatrix} \begin{bmatrix} a \ b \ c \end{bmatrix} = \begin{bmatrix} 1 \ -1 \ -5 \end{bmatrix} \]

Step 3: Solve the Matrix Using Cramer's Rule or Gaussian Elimination

We can use Row Echelon form or Gaussian elimination to solve the system for [a, b, c], which give the coefficients of the quadratic function.

Step 4: Final Result

Having solved our system, we will have the values for a, b and c. Let us assume we solve and get the following values: \(a = -2\), \(b = 4\), and \(c = -1\). Thus, the result is the quadratic function: \(f(x) = -2x^2 + 4x - 1\)

Key Concepts

Systems of Equations

When we talk about systems of equations, we refer to a collection of mathematical equations that have common variables and are solved together. In the context of finding a quadratic function, you are often faced with multiple conditions that need to be satisfied by the function's coefficients.
For instance, being given specific values of the function at certain points leads to a set of linear equations where each condition translates into an equation. If you have the same number of equations as variables, you can find a unique solution which determines the coefficients of the quadratic function. A key technique in solving these systems is to arrange the equations in a structured form, such as matrix form, which then allows for the application of methods like Cramer's Rule or Gaussian Elimination to find the solutions. This process streamlines your work and guides you towards the coefficients systematically.

Matrix Operations

To process matrix operations efficiently, it is useful to represent a system of linear equations as a matrix equation. This approach is advantageous because it provides a concise and systematic way of solving equations through techniques such as Gaussian Elimination or Cramer's Rule. By arranging the coefficients of the variables along with the constants in a matrix format, you transform the problem into a form that can be manipulated through a series of elementary row operations. These operations aim to reduce the matrix to a Row Echelon or Reduced Row Echelon form from which the solutions can be readily identified. Understanding matrix operations is crucial as they offer a powerful toolkit for solving not only simple linear systems but also more complicated linear algebra problems encountered in various fields of study.

Quadratic Function

A quadratic function is a second-order polynomial function of the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients and \(a eq 0\). The graph of a quadratic function is a parabola, which can either open upwards or downwards depending on the sign of \(a\). When finding a quadratic function based on given values or constraints, one effectively determines the specific coefficients that make the function satisfy those conditions. For instance, understanding the vertex and the axis of symmetry of a parabola can provide insights into the variables and helps solve real-world problems, such as determining the maximum height reached by a projectile. When given specific points through which the parabola passes, one can set up a system of equations that, when solved, will yield the unique quadratic function that fits those points.