Determining the Coefficients of a Parabolic Equation: A Comprehensive Guide

In this article, we will explore the process of determining the coefficients of a parabolic equation that passes through the given points. This is a fundamental concept in algebra and has numerous applications in various fields of science and engineering. Moreover, understanding these methods is critical for effective search engine optimization (SEO), as it can help in creating content that is more likely to be indexed and found by search engines.

Understanding the Parabolic Equation

A parabolic equation is a type of quadratic equation, typically represented as (y ax^2 bx c). The coefficients (a), (b), and (c) determine the specific shape, orientation, and position of the parabola. Our goal is to find the values of (a), (b), and (c) that make the parabola pass through a given set of points. In this case, we have the points ((-1, 12)), ((0, 5)), and ((2, -3)).

Solving the System of Equations

Given the points ((-1, 12)), ((0, 5)), and ((2, -3)), we can use these points to create a system of equations to solve for (a), (b), and (c).

Setting Up the Equations

For the point ((-1, 12)), we have:

(a(-1)^2 b(-1) c 12), which simplifies to (a - b c 12).

For the point ((0, 5)), we have:

(a(0)^2 b(0) c 5), which simplifies to (c 5).

For the point ((2, -3)), we have:

(a(2)^2 b(2) c -3), which simplifies to (4a 2b c -3).

Substituting and Solving

Using the equation (c 5), we can substitute (c) into the other two equations:

Substitute (c 5):

(a - b c 12) becomes (a - b 5 12), which simplifies to (a - b 7). (4a 2b c -3) becomes (4a 2b 5 -3), which simplifies to (4a 2b -8), or (2a b -4).

We now have a system of two equations:

(a - b 7) (2a b -4)

Solving the system of equations, we can express (b) in terms of (a) from the first equation:

(b a - 7)

Substituting (b a - 7) into the second equation:

(2a (a - 7) -4)

Combining like terms, we get:

(3a - 7 -4)

(3a 3)

(a 1)

Substituting (a 1) back into (b a - 7):

(b 1 - 7 -6)

Since we already found (c), (c 5).

Conclusion: The Values of (a), (b), and (c)

The values of (a), (b), and (c) that satisfy the given conditions are:

(a 1) (b -6) (c 5)

Therefore, the parabolic equation passing through the points ((-1, 12)), ((0, 5)), and ((2, -3)) is:

[y x^2 - 6x 5]

Finally, the value of (a b c) is:

[1 (-6) 5 0]

Hence, the value of (a b c) is 0.