Converting y -2(x-1)2 3 to Standard Form: A Comprehensive Guide

Introduction to Quadratic Equations

In the realm of mathematics, quadratic equations are foundational in understanding various phenomena, from physics to engineering. A quadratic equation generally follows the form y ax2 bx c and can represent a parabola in the coordinate plane. This piece will guide you through the process of converting a quadratic equation from its vertex form to its standard form. We will focus on the specific equation y -2(x-1)2 3 and demonstrate the steps to transform it into the standard form y ax2 bx c. Let’s dive in!

The Structure of Quadratic Equations

Understanding the structure of quadratic equations starts with recognizing the differences between various forms. The vertex form of a quadratic equation, represented as y a(x-h)2 k, provides a direct way to identify the vertex of the parabola at (h, k). Here, ‘a’ affects the width and direction of the parabola, ‘h’ is the x-coordinate of the vertex, and ‘k’ is the y-coordinate. The standard form, on the other hand, is y ax2 bx c, where the coefficients ‘a’, ‘b’, and ‘c’ help us to understand the behavior of the parabola more comprehensively.

Converting from Vertex Form to Standard Form

Given the equation in vertex form, y -2(x-1)2 3, our goal is to convert it to the standard form. Let’s break down the process step by step.

Step 1: Expand (x-1)2

First, we need to expand the expression (x-1)2. This follows the binomial theorem, which states that (a-b)2 a2 - 2ab b2. Applying this to (x-1)2:

(x-1)2 x2 - 2x(1) 12 x2 - 2x 1

Step 2: Distribute the -2

After expanding, we distribute the -2 across the terms of the expanded form:

-2(x2 - 2x 1) -2x2 4x - 2

Step 3: Add the Constant Term

Next, we add the constant term from the original vertex form equation (which is 3) to the result from the previous step:

-2x2 4x - 2 3

After performing this addition, we get the final form of the equation:

y -2x2 4x 1

Understanding the Result

With the equation in standard form, y -2x2 4x 1, we can now easily identify the coefficients that define the parabola. Here, ‘a’ is -2, ‘b’ is 4, and ‘c’ is 1. This form allows us to analyze the parabola’s direction, vertex, and intercepts more effectively.

Conclusion

In conclusion, converting a quadratic equation from vertex form to standard form involves a series of algebraic steps. Starting from the vertex form y -2(x-1)2 3, we expanded (x-1)2, distributed the -2, and added the constant term to arrive at the standard form y -2x2 4x 1. Understanding these steps is crucial for mastering quadratic equations and applying them in various mathematical contexts. If you have any further questions or need more practice, don’t hesitate to explore additional resources or seek guidance from a mathematics tutor.

Further Learning

For those interested in delving deeper into quadratic equations and their applications, consider exploring the following resources: Advanced Algebra Textbook Algebra 2 Course on Online Platforms YouTube Tutorial Series on Quadratic Equations By expanding your knowledge and practicing with various examples, you can enhance your understanding of quadratic equations and their practical implications.