Problem 40 - College Algebra

Question

Find the quadratic function \(f(x)=a x^{2}+b x+c\) for which \(f(-1)=5, f(1)=3,\) and \(f(2)=5\)

Step-by-Step Solution

Answer

The solution is \(f(x)= 1/2x^2 - 1x + 2\).

Step 1: Formulate equations

First, we substitute the given points into the quadratic function to form three equations.\nFor \(f(-1)=5\), we get \(a(-1)^2 + b(-1) + c = 5\), which simplifies to \(a - b + c = 5\).\nFor \(f(1)=3\), we get \(a(1)^2 + b(1) + c = 3\), which simplifies to \(a + b + c = 3 \).\nFor \(f(2)=5\), we get \(a(2)^2 + b(2) + c = 5\), which simplifies to \(4a + 2b + c = 5\).

Step 2: Solve the system of equations

Use any of the methods for solving systems of linear equations (like substitution, elimination, matrix method etc.) to find the values of a, b and c. Here we can subtract the second equation (i) from the third equation (ii) to get an equation that only contains the variable 'a'. This simplifies to \(3a + b = 2\). Now we have a pair of simultaneous equations: \(a - b + c = 5\) and \(3a + b = 2\) that can be easily solved by addition or subtraction.

Step 3: Find the values of 'a', 'b' and 'c'

Adding the two equations together, we get \(4a + 2c = 7\), which simplifies to \(2a + c = \frac{7}{2}\) . Then, substitute the value of 'a' back into the first equation to find 'c', and finally into the last equation to find 'b'.

Key Concepts

System of Equations

In mathematics, a system of equations is a collection of two or more equations involving the same set of variables. Solving a system of equations entails finding values for the variables that satisfy all the equations at the same time.

• For the given problem, we have three equations because we've been given three points on the quadratic function.
• These equations represent various conditions that the coefficients of the quadratic function must satisfy, using different values of the variable.

To solve the system of equations, you can use methods such as:

• Substitution: Solving one equation for one variable and then plugging this into the other equations.
• Elimination: Adding or subtracting equations from each other to eliminate one variable.
• Matrix methods if the system is more complex or for systems with more equations and variables.

Solving a system can bring clarity to the relationships between variables and help identify the specific solutions that piece together the overarching equation of the function.

Function Notation

Understanding function notation is crucial in dealing with equations, particularly when working with different kinds of functions like quadratic functions. Function notation provides a way to represent functions and easily substitute values to find specific outputs.

In our example, the quadratic function is expressed as:\[ f(x) = ax^2 + bx + c \]Here, \(f(x)\) indicates a function of \(x\). The notation \(f(-1)\) or \(f(1)\) means that you substitute the given value for \(x\) into the function to calculate the corresponding value of the function, \(f(x)\).

• Function notation is a powerful tool for describing how one quantity depends on another.
• It allows for precise communication of mathematical ideas and makes it clear what input goes with which output.
• It is especially useful when working with complex functions as it keeps the expressions manageable and organized.

This notation is widely used because it simplifies the representation and manipulation of functions.

Solution of Equations

Finding the solution of equations is about determining the values of the variables that satisfy all the given conditions. This process often involves several steps of algebraic manipulation. In our quadratic function example:

• We took three points and substituted them, resulting in three separate equations.
• The process then involved solving these equations simultaneously to find the coefficients \(a\), \(b\), and \(c\).

Each solution step brings us closer to understanding the unique function that meets all the given conditions.
When we add or subtract equations to isolate one variable, or substitute one variable into another equation, we find the values that will keep the equality true for the quadratic function. Once found:

• The solutions provide the specific parameters of the quadratic equation.
• This means our function satisfies the conditions posed by those three given points, effectively tailoring the equation to fit through exactly those specified points in the coordinate plane.

This approach ensures that all conditions are met and the solution is robust and reliable.