In mathematics, the vertex of a parabola is the point where the curve reaches its highest or lowest point and can be used as a key point in graphing and analyzing quadratic functions. The vertex can be determined through various methods including completing the square or using a formula. In this article, we will discuss both approaches for finding the vertex of a parabolic function.

1. Understanding Quadratic Functions:

A quadratic function is represented by the equation:

f(x) = ax² + bx + c

where a, b, and c are constants.

The graph of a quadratic function is a parabola. If “a” is positive, the parabola opens upwards, making it a minimum point (lowest point). If “a” is negative, the parabola opens downwards, making it a maximum point (highest point).

2. Finding the Vertex Using Completing the Square Method:

Completing the square involves rewriting the quadratic function in vertex form:

f(x) = a(x – h)² + k

where (h, k) represents the coordinates of the vertex. Follow these steps to find the vertex using completing the square method:

– Make sure your quadratic equation is in standard form: ax² + bx + c.

– Factor “a” from x² and x terms.

– Add and subtract a term within parentheses to complete the square.

– Rewrite in vertex form.

– Identify h and k as coordinates of the vertex.

3. Finding the Vertex Using Vertex Formula:

A quicker way to find the vertex without completing the square involves using this formula:

(h, k) = (-b/2a, f(-b/2a))

The coordinates (h,k) represent the vertex.

– Calculate h by plugging in values for b and a: -b/2a.

– Substitute h back into f(x) to get value of k: f(-b/2a).

– The vertex coordinates are (h, k).

Example:

Consider the quadratic function: f(x) = 3x² – 6x + 4.

Using the vertex formula:

h = –(-6) / (2 * 3) = 1

k = f(1) = 3(1)² – 6(1) + 4 = 1

The vertex of the parabola is (1, 1).

In summary, calculating the vertex of a parabolic function is an essential skill for graphing and analyzing quadratic functions. Completing the square or using the vertex formula are two methods to find the coordinates (h, k), which indicate the highest or lowest point on a parabolic curve.