The vertex of a parabola is the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. In the context of a quadratic function, the vertex provides crucial information about the function’s maximum or minimum value. There are a couple of methods to find the vertex of a parabola represented by the quadratic function y = ax² + bx + c.
Methods to Find the Vertex of a Parabola
The vertex can be found using either the vertex formula or by completing the square.
Vertex Formula
The vertex (h, k) of a parabola y = ax² + bx + c can be found using the formula: h = -b / (2a) and k = c – b² / (4a).
Completing the Square
This method involves rewriting the quadratic equation in the form y = a(x – h)² + k, where (h, k) is the vertex.
Examples of Finding the Vertex
Example 1 (Vertex Formula): Find the vertex of y = 2x² + 8x + 3. Here, a = 2, b = 8. So, h = -8 / (2 × 2) = -2, k = 3 – 8² / (4 × 2) = -5. Therefore, the vertex is (-2, -5).
Example 2 (Completing the Square): For y = x² – 6x + 5, rewrite it as y = (x – 3)² – 4. So, the vertex is (3, -4).
Significance of the Vertex
Finding the vertex is important in various applications, such as optimizing areas and volumes, solving maximum and minimum problems, and analyzing the motion of objects under gravity in physics.
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