Converting a Factored Form to Vertex Form: A Comprehensive Guide
Understanding how to convert a factored form quadratic equation to a vertex form is a fundamental skill in algebra. This article provides a detailed step-by-step guide and an example to illustrate the process. By the end of this guide, you will be able to easily transform any linearly factored quadratic equation into its corresponding vertex form. Let's dive in.
The Process of Converting from Factored Form to Vertex Form
Factored form and vertex form are two different ways to express quadratic equations. A quadratic equation in factored form is written as:
(ax^2 bx c a(x - r_1)(x - r_2))
Where (r_1) and (r_2) are the roots of the equation.
On the other hand, the vertex form of a quadratic equation is written as:
(y a(x - h)^2 k)
Where (h) and (k) are the coordinates of the vertex of the parabola, and (a) is the coefficient of the quadratic term.
Step-by-Step Guide
Start with a factored form quadratic equation, such as:
(y a(x - r_1)(x - r_2))
Expand the expression to standard form:
(y a[x^2 - (r_1 r_2)x r_1r_2])
Complete the square to convert the standard form into vertex form. To do this, take the coefficient of (x) (i.e., ( - (r_1 r_2) )), divide it by 2, and square it.
Let (p - frac{(r_1 r_2)}{2}), then add and subtract (p^2) inside the equation.
Group the perfect square trinomial and the constant term.
(y a[x^2 - (r_1 r_2)x p^2 - p^2 r_1r_2])
Factor the perfect square trinomial.
(y a[(x - p)^2 - p^2 r_1r_2])
Combine the constants.
(y a(x - p)^2 (r_1r_2 - ap^2))
Identify the vertex form. The equation is now in vertex form:
(y a(x - p)^2 (r_1r_2 - ap^2))
In this form, the vertex is at the point ( (p, r_1r_2 - ap^2) ).
An Example
Let's walk through an example to solidify our understanding.
Consider the factored form: (y 2(x - 3)(x 1))
Expand the factored form:
(y 2(x^2 - 2x - 3))
Complete the square:
(y 2[x^2 - 2x 1 - 1 - 3])
(y 2[(x - 1)^2 - 4])
Combine the constants:
(y 2(x - 1)^2 - 8)
In this example, the vertex is at ( (1, -8) ).
A Quick Note on Handwriting
When working through equations, especially in exams, it's important to write neatly. Poor handwriting can lead to mistakes, especially when dealing with negative signs and coefficients. If you encountered challenges with your handwriting in your example, ensure you practice writing clearly and neatly.
Conclusion
Transforming a quadratic equation from factored form to vertex form is not only a crucial skill in algebra but also helps in understanding the properties of quadratic functions. By following the steps outlined in this article, you can easily convert any quadratic equation into vertex form. If you have any questions or need further clarification, don't hesitate to reach out.