How to Write a Quadratic Function in Factored Form for a Parabola Opening Downward with Given Zeros

Writing a quadratic function in factored form to represent a parabola that opens downward and has specific zeros is a fundamental concept in algebra. This guide covers the process in detail and provides additional insights that may be valuable for SEO purposes.

Identifying the Zeros

The zeros of a quadratic function are the x-values where the function equals zero. In this case, the zeros are x -6 and x -2.

Writing the Factored Form of the Quadratic Function

The general form of a quadratic function in factored form is:

Step 1: Write the factored form

For the given zeros, the factored form of the quadratic function can be written as:

f(x) a(x - r_1)(x - r_2)

where r_1 and r_2 are the zeros of the function and a is a constant that determines the direction and width of the parabola.

Substituting the zeros:

f(x) a(x 6)(x 2)

Determining the Value of a

Since the parabola opens downward, the value of a must be negative. For simplicity, a common choice is -1. Thus, the quadratic function in factored form is:

f(x) -1(x 6)(x 2)

Expanding this to standard form:

f(x) -x^2 - 8x - 12

This function represents a parabola that opens downward with zeros at -6 and -2.

Finding the Family of Parabolas

Actually, there is a family of parabolas whose equation is:

Equation of Parabola

y ax^2 8x 12

where a is any positive real number. For example:

y 2x^2 8x 12y (2/3)x^2 8x 12

Vertex, Focus, and Directrix of the Parabola

The quadratic function in factored form can also be written in the form:

y C(x 4)^2 - 4C

This is a form of a downward-opening parabola, where C is a constant. The vertex of the parabola h, k is (-4, -4C) and the focus is (-4, -4C - 1/(4C)). The directrix is the line y -4C 1/(4C).

Key Points and SEO Optimization

The zeros of the quadratic function determine the factors of the function. In this case, the factors are (x 6) and (x 2).

The constant a determines the direction and width of the parabola. For a downward-opening parabola, a must be negative.

The value of a can be any negative number. To simplify, -1 is often used.

By clearly understanding and leveraging these key points, you can more effectively write and optimize quadratic functions in factored form for SEO purposes. This will help improve the visibility and relevance of your content in search engine results.