Determining the Equation of a Parabola Given Specific Points

Introduction:
Understanding the equation of a parabola is fundamental in mathematics, especially when dealing with its intersections and intercepts. This article will walk you through the process of finding the equation of a parabola that cuts the x-axis at (x 2) and (x 6), and has a y-intercept of 12.

Defining the Equation of the Parabola

Given a parabola that intersects the x-axis at (x 2) and (x 6), we can express this using its roots. The standard form for such a parabola can be written as:

[ y a(x - 2)(x - 6) ]

Expanding this expression, we get:

[ y a(x^2 - 8x 12) ]

Or equivalently:

[ y ax^2 - 8ax 12a ]

This form shows the relationship between the coefficients and the roots of the parabola. However, to determine the value of (a), we need to use the information about the y-intercept.

Using the y-intercept to Find the Value of (a)

The y-intercept occurs when (x 0). Substituting (x 0) into the equation, we find:

[ y a(0^2 - 8 cdot 0 12) 12a ]

It is given that the y-intercept is 12. Therefore:

[ 12 12a ]

From this, we solve for (a):

[ a 1 ]

Substituting (a 1) back into the expanded form of the equation:

[ y 1 cdot x^2 - 8 cdot 1 cdot x 12 cdot 1 ]

Simplifying, we get the equation of the parabola:

[ y x^2 - 8x 12 ]

Conclusion

Thus, the defining equation of the parabola that cuts the x-axis at (x 2) and (x 6) with a y-intercept of 12 is:

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This solution demonstrates the use of the given roots and y-intercept to construct the equation of a parabola, providing a clear methodology for similar problems.