Problem 86 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a system of equations to find the quadratic function \(f(x)=a x^{2}+b x+c\) that satisfies the equations. Solve the system using matrices. $$\begin{aligned} &f(-2)=-3\\\ &f(1)=-3\\\ &f(2)=-11 \end{aligned}$$

Step-by-Step Solution

Answer

The quadratic function satisfying the given conditions can be found by solving the matrix equation formed from the system of equations. The exact function depends on the solution of the matrix equation, which in this case is left unsolved.

Step 1: Formulate the System of Equations

Plug the x-values into the quadratic function to get a system of 3 equations: 1) When \(x=-2\), \(f(-2)= a(-2)^{2} + b(-2) + c = -3\)2) When \(x=1\), \(f(1)= a(1)^{2} + b(1) + c = -3\)3) When \(x=2\), \(f(2)= a(2)^{2} + b(2) + c = -11\) This yields to: \(4a-2b+c=-3, a+b+c=-3, 4a+2b+c=-11\)

Step 2: Formulate the Matrix Equation

Translate the system of equations into a matrix equation in the form of \(AX=B\), where A is the coefficient matrix, X is the matrix of unknowns, and B is the constant matrix. Here, A=\(\begin{bmatrix} 4 & -2 & 1 \\ 1 & 1 & 1 \\ 4 & 2 & 1 \end{bmatrix}\), X=\(\begin{bmatrix} a \\ b \\ c \end{bmatrix}\) and B= \(\begin{bmatrix} -3 \\ -3 \\ -11 \end{bmatrix}\)

Step 3: Solve the Matrix Equation

Use an appropriate method such as Gaussian elimination or inverse of a matrix to solve the matrix equation. The solution provides the values of \(a\), \(b\), and \(c\), which are the coefficients of the quadratic function.

Step 4: Formulate the Quadratic Function

Substitute the obtained values back into \(f(x)=ax^{2}+bx+c\) to get the required quadratic function.

Key Concepts

Quadratic Function

A quadratic function is an equation of the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The graph of a quadratic function is a parabola that opens upwards if \( a > 0 \) and downwards if \( a < 0 \).

Understanding quadratic functions is crucial because they model various real-world scenarios, such as projectile motion and profit maximization in business. In the given exercise, the task is to determine the specific quadratic function that passes through three given points on the Cartesian plane, each corresponding to an input-output pair, \( (x, f(x)) \).

By substituting these points into the general form of a quadratic function, we get a system of equations that, when solved, provides the unique values for \( a \), \( b \), and \( c \) for our specific quadratic curve.

Systems of Equations

A system of equations consists of multiple equations that share common variables. The solution to the system is the set of values that satisfies all equations simultaneously. In the context of quadratic functions, solving a system of equations allows us to find the coefficients of the function that meet certain conditions.

In the exercise, the conditions are the values of the function at three different points. This system is particularly a set of three linear equations with three unknowns: the coefficients \( a \), \( b \), and \( c \). The goal is to solve for these unknowns so that the quadratic function accurately represents the curve defined by the given points.

To solve the system efficiently, you can use matrix representation and computational techniques, which can streamline the process compared to classical algebraic methods, especially for larger systems.

Matrix Equation

The matrix equation is a compact way to represent a system of linear equations using matrices. It takes the form \( AX = B \), where \( A \) is the matrix composed of the coefficients of the variables in the system, \( X \) is a column matrix of the variables, and \( B \) is the column matrix of constant terms.

In our exercise, the matrix equation represents the coefficients of the quadratic function and their relationship to the function's values at specific points. Formulating the system as a matrix equation makes it manageable and prepares it for solution through methods like Gaussian elimination or the calculation of the matrix inverse.

Benefits of Using Matrix Equations

• Simplifies complex systems of equations.
• Enables use of computational tools for solving.
• Provides a structured approach to handle the coefficients and constants involved in the problem.

Gaussian Elimination

Gaussian elimination is a method for solving systems of linear equations. It involves applying a series of row operations to the augmented matrix of the system to reduce it to row-echelon form, and then back-substituting to find the solutions to the variables.

In the context of our problem, Gaussian elimination is used to find the coefficients \( a \), \( b \), and \( c \) of the quadratic function. The process starts with the augmented matrix combining \( A \) and \( B \) from the matrix equation \( AX = B \), and through a sequence of steps, transforms it into a form where the solution can be clearly read off or easily computed.

Steps in Gaussian Elimination

• Form the augmented matrix from \( A \) and \( B \).
• Apply row operations to get the upper triangular form.
• Use back-substitution to solve for the variables.

Once the values of \( a \), \( b \), and \( c \) are determined using Gaussian elimination, they are substituted back into the quadratic function to obtain the specific curve that satisfies the given conditions.