In mathematics, a parabola is a curve that is formed by the intersection of a plane and a right circular cone. It is a symmetrical curve, and it is known for its remarkable properties and applications in various fields of study. One crucial aspect of a parabola is its vertex, which plays a fundamental role in determining its shape, position, and behavior.

Identifying the vertex of a parabola is essential since it serves as a reference point for understanding and analyzing the curve further. It is widely used in algebra, geometry, calculus, physics, and engineering, to name a few disciplines. The vertex represents the highest or lowest point on the parabola, depending on its orientation (opens upward or downward).

To identify the vertex of a parabola, we employ a simple and straightforward method known as completing the square. This method involves transforming a given quadratic equation into its standard form, where the vertex and axis of symmetry can be easily determined. Let’s consider the general equation of a parabola in standard form:

y = a(x – h)^2 + k

Here, (h, k) represents the coordinates of the vertex, and a determines the vertical stretch or compression of the parabola. To identify the vertex, we need to determine the values of h and k.

To start, we focus on the quadratic term, (x – h)^2. By expanding and simplifying this term, we obtain x^2 – 2hx + h^2. Comparing this simplified term with the general form, we notice that 2h represents the coefficient of the linear term, and h^2 corresponds to the constant term. Thus, by equating these coefficients, we can derive the values of h and k.

For example, let’s consider the equation y = 2x^2 + 4x + 5. To identify the vertex, we need to rewrite this equation in standard form. We begin by factoring out the common factor of 2 from the quadratic term:

y = 2(x^2 + 2x) + 5

Next, we complete the square by adding and subtracting the square of half the coefficient of the linear term (in this case, 2x):

y = 2(x^2 + 2x + 1 – 1) + 5

Simplifying further, we obtain:

y = 2[(x + 1)^2 – 1] + 5

At this point, we can identify the vertex coordinates: h = -1 and k = -1. Therefore, the vertex is located at (-1, -1).

By analyzing the sign of the coefficient ‘a,’ we can determine whether the parabola opens upward or downward. If a is positive, the parabola opens upward, and if a is negative, the parabola opens downward.

In conclusion, identifying the vertex of a parabola is crucial for understanding its properties and behavior. Through the process of completing the square, we can transform a given quadratic equation into standard form and effortlessly determine the coordinates of the vertex. This process allows mathematicians, scientists, and engineers to analyze and interpret the parabolic curve, enabling them to make accurate predictions and calculations in various fields of study. So the next time you come across a parabola, remember that its vertex holds the key to unlocking its secrets.

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