Understanding Quadratic Functions and Algebraic Manipulation

Quadratic functions, represented in the form fx ax2 bx c, are powerful tools for modeling a wide range of real-world phenomena. However, determining the specific quadratic function that passes through given points can be a bit intricate. In this article, we will guide you through the process of finding the quadratic function that passes through the points (0, 6), (3, -6), and (-1, 14).

Setting Up Equations Using Given Points

To find the quadratic function that passes through the given points, we need to set up a system of equations based on these points. The given points are:

Let's start by setting up these equations using the general form of a quadratic function:

fx ax2 bx c

Step 1: Setting Up the Equations

From the point (0, 6):f(0) 6 c.

From the point (3, -6):32 * a 3 * b 6 -6 → 9a 3b 6 -6 → 9a 3b -12 → 3a b -4 (Equation 1).

From the point (-1, 14):(-1)2 * a (-1) * b 6 14 → a - b 6 14 → a - b 8 (Equation 2).

Step 2: Substituting the Known Values

We know from the first point that c 6, so we substitute this into the other two equations:

Equation for (3, -6):

Equation for (-1, 14):

Step 3: Solving the System of Equations

Now, we solve the system of equations to find the values of a and b:

3a b -4 → 3a (a - 8) -4 → 4a - 8 -4 → 4a 4 → a 1.

Substituting a 1 back into the equation for b:b a - 8 1 - 8 -7.

Step 4: Writing the Final Quadratic Function

Using the values of a, b, and c, we write the final quadratic function:fx ax2 bx c 1x2 - 7x 6.

Conclusion and Further Implications

The quadratic function that passes through the points (0, 6), (3, -6), and (-1, 14) is:fx x2 - 7x 6.

This function can be used to model various real-world scenarios, such as the trajectory of a projectile or the profit function in economics. Understanding how to determine the specific quadratic function that passes through given points not only helps in solving algebraic problems but also enhances problem-solving skills in other domains.

By familiarizing yourself with the steps and processes involved, you can effectively apply the knowledge of quadratic functions to a variety of scenarios. In the next section, we will explore the implications of the vertex form and the geometric interpretation of the function.

Implications: Understanding how to find a quadratic function that passes through specific points is crucial in algebra and has applications in higher mathematics, physics, and engineering. By mastering this skill, you can solve complex problems and model real-world phenomena more accurately.