Y 1 2x 2 3 In Standard Form

Writing the Equation y = 1/2x² + 2x + 3 in Standard Form: A Comprehensive Guide

The equation y = 1/2x² + 2x + 3 represents a parabola. Understanding how to rewrite this equation in standard form is crucial for various mathematical applications, including graphing, finding the vertex, and determining key features of the parabola. This comprehensive guide will walk you through the process step-by-step, exploring the underlying concepts and providing practical examples.

Understanding Standard Form of a Quadratic Equation

The standard form of a quadratic equation is expressed as:

ax² + bx + c = 0

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. Notice that in this form, the equation is set equal to zero. Our given equation, y = 1/2x² + 2x + 3, is in a different form; it's solved for 'y'. While not technically "standard form," converting it to a form where it's equal to zero is a necessary precursor to many analytical techniques.

Converting the Equation to Standard Form

To convert y = 1/2x² + 2x + 3 to standard form, we need to rearrange the terms to equal zero:

1. Subtract 'y' from both sides: This moves the 'y' term to the right side of the equation, resulting in:

   0 = 1/2x² + 2x + 3 - y

2. Rearrange the terms (optional but recommended): While technically in standard form, it's generally more aesthetically pleasing and conventional to arrange the terms in descending order of powers of x:

   0 = 1/2x² + 2x - y + 3

This equation is now in a form that allows us to apply various techniques such as the quadratic formula or completing the square. While it isn't the exact form ax² + bx + c = 0 because of the 'y' term, it's the closest equivalent for this particular problem.

Applications of Standard Form

Converting the equation to this adjusted standard form unlocks several analytical capabilities:

1. Finding the Vertex

The vertex of a parabola, which represents its lowest or highest point, is crucial for graphing and understanding the parabola's behavior. While the original form is useful for understanding the y-intercept (the point where the parabola intersects the y-axis, which is (0,3) in this case), the adjusted standard form facilitates the calculation of the vertex more easily using the formula:

x = -b / 2a

In our adjusted standard form, a = 1/2 and b = 2. Therefore:

x = -2 / (2 * (1/2)) = -2

To find the y-coordinate of the vertex, substitute x = -2 back into the original equation:

y = 1/2(-2)² + 2(-2) + 3 = 1/2(4) - 4 + 3 = 2 - 4 + 3 = 1

Therefore, the vertex of the parabola is (-2, 1).

2. Finding the Roots (x-intercepts)

The roots of a quadratic equation are the x-values where the parabola intersects the x-axis (where y = 0). We can find these by solving the quadratic equation 1/2x² + 2x + 3 - y = 0 for x when y = 0:

1/2x² + 2x + 3 = 0

This equation can be solved using several methods:

• Quadratic Formula: The quadratic formula is a general solution for finding the roots of any quadratic equation in the form ax² + bx + c = 0:

  x = (-b ± √(b² - 4ac)) / 2a

  In our case, a = 1/2, b = 2, and c = 3. Substituting these values into the quadratic formula, we find:

  x = (-2 ± √(2² - 4*(1/2)3)) / (2(1/2)) = (-2 ± √(4 - 6)) / 1 = (-2 ± √(-2)) / 1

  Since we have a negative value under the square root, there are no real roots. This means the parabola does not intersect the x-axis. The roots are complex numbers.

• Completing the Square: This method involves manipulating the equation to create a perfect square trinomial. While more algebraically intensive, it can provide a deeper understanding of the equation's structure.

3. Graphing the Parabola

Knowing the vertex and whether or not the parabola intersects the x-axis allows us to accurately graph the parabola. Since the coefficient of x² (a = 1/2) is positive, the parabola opens upwards. The vertex (-2, 1) is the lowest point on the graph. The y-intercept is (0,3). Because there are no real roots, the parabola remains entirely above the x-axis.

Further Exploration and Advanced Concepts

This detailed analysis covers the fundamental aspects of converting the given equation to standard form and extracting crucial information. However, several advanced concepts can be explored further:

• Discriminant: The expression b² - 4ac (the term under the square root in the quadratic formula) is called the discriminant. Its value determines the nature of the roots:

  • If the discriminant is positive, there are two distinct real roots.
  • If the discriminant is zero, there is one real root (a repeated root).
  • If the discriminant is negative, there are no real roots (as in our example), indicating complex roots.

• Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is simply x = -b / 2a (the x-coordinate of the vertex).

• Concavity: The concavity of a parabola describes whether it opens upwards (positive 'a' value) or downwards (negative 'a' value).

Conclusion

Converting the equation y = 1/2x² + 2x + 3 into a form similar to the standard quadratic equation, while not exactly the same due to the presence of 'y', allows us to gain significant insights into the parabola's behavior. By calculating the vertex, identifying the roots (or lack thereof), and understanding the concept of the discriminant, we can confidently graph and analyze the quadratic function. This comprehensive guide provides a solid foundation for tackling more complex quadratic equations and related problems. Remember, understanding the underlying mathematical concepts is key to successfully navigating these types of problems.