Problem 22 - Algebra and Trigonometry

Question

Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. $$ (1,3),(3,-1),(4,0) $$

Step-by-Step Solution

Answer

The quadratic function that passes through the given points is \(y=x^{2}-5x+7\).

Step 1: Set up the system of equations

Three points \((1,3)\), \((3,-1)\), \((4,0)\) are given. A quadratic function has the form \(y=a x^{2}+b x+c\). Substitute the coordinates of the given points into the equation to form a system of three equations. The equations are: \(a+b+c=3 \) (from point (1,3)), \(9a+3b+c=-1\) (from point (3,-1)), \(16a+4b+c=0\) (from point (4,0)).

Step 2: Solve the system of equations

The task is to solve this system of equations. This can be done with the method of substitution or by using the Gaussian elimination method. Solving this system gives the solutions \(a=1\), \(b=-5\) and \(c=7\).

Step 3: Form the function

Now replace a, b and c in the equation \(y=a x^{2}+b x+c\) with the obtained solutions. This gives the function \(y=x^{2}-5x+7\).

Key Concepts

System of Equations

In mathematics, a system of equations is a collection of two or more equations with a same set of variables. A common goal in algebra is to find the values of these variables that satisfy all equations simultaneously. For example, when determining a quadratic function that fits a set of points, we use the general form of a quadratic equation, which is \(y = ax^2 + bx + c\), and plug in the coordinates of the given points to create a system.

In the case of the quadratic function solution, we had three points: \(1,3\), \(3,-1\), \(4,0\). Substituting these into the quadratic formula gives us the specific equations:

• \(a + b + c = 3\)
• \(9a + 3b + c = -1\)
• \(16a + 4b + c = 0\)

These equations form our system which, when solved, gives us the exact function that passes through the given points. The solving process could employ methods like substitution, elimination, or matrix operations to find the values of \(a\), \(b\), and \(c\).

Gaussian Elimination Method

The Gaussian elimination method, also known as row reduction, is a sequence of operations used to solve systems of linear equations. This method involves three types of row operations:

• Swapping two rows,
• Multiplying a row by a nonzero number,
• Adding or subtracting a multiple of one row from another row.

By performing these operations, the system's augmented matrix transforms into a row-echelon form, and eventually, reduced row-echelon form, which can then be solved with back substitution.

To apply Gaussian elimination to our system of equations for the quadratic function, we would create an augmented matrix that represents our equations and then use row operations to simplify. This process systematically eliminates variables one by one, making it possible to solve for each variable starting from the last equation up to the first.

Substitution Method

Another handy technique for solving systems of equations is the substitution method. With this approach, we solve one of the equations for one variable in terms of the others and then substitute this expression into the remaining equations. This is typically most effective when one of the equations can easily be solved for one variable or when there are only two variables.

Considering our quadratic function example, we could solve the first equation, \(a + b + c = 3\), for \(c\) and then substitute \(c = 3 - a - b\) into the other equations. This would give us two equations with two variables (\(a\) and \(b\)) that we can then handle using substitution, elimination, or further simplification. While substitution can be more straightforward conceptually, it can sometimes lead to more complicated algebra than other methods such as Gaussian elimination.