A quadratic function, represented by the equation f(x) = ax² + bx + c, where a, b, and c are real numbers and a ≠ 0, forms a graphical representation in the shape of a parabola. The central element of this parabola is its vertex, which represents the highest or lowest point of the parabola.
DETERMINING THE VERTEX
The vertex of a quadratic function is the point defined by the coordinates (p, q) and is crucial for understanding the shape and orientation of the parabola. The coordinates of the vertex are calculated using the following formulas:
- p-coordinate of the vertex: p = -b / (2a)
- q-coordinate of the vertex: q = f(p), meaning q = a(p)² + b(p) + c, after calculating p.
INFLUENCE OF COEFFICIENTS ON THE VERTEX
- COEFFICIENT a: Determines the direction the parabola opens (upwards for a > 0 and downwards for a < 0) and affects the width of the parabola.
- COEFFICIENT b: Influences the horizontal position of the vertex.
- COEFFICIENT c: Influences the vertical position of the parabola, specifically it is the y-intercept, which affects where the vertex is relative to the y-axis if 'b' also changes.
EXPLORING THE VERTEX WITH AN EXAMPLE
Given the function f(x) = 2x² – 4x + 1:
- CALCULATE THE p-COORDINATE OF THE VERTEX:
p = -(-4) / (2 * 2) = 4 / 4 = 1 - THEN CALCULATE THE q-COORDINATE OF THE VERTEX:
q = f(1) = 2*(1)² – 4*(1) + 1 = 2 – 4 + 1 = -1
Thus, the vertex of the parabola is the point (1, -1).
SIGNIFICANCE OF THE VERTEX IN PARABOLA ANALYSIS
The vertex of the parabola provides important information about the behavior of the quadratic function:
- If a > 0, the vertex represents the lowest point of the parabola (minimum value of the function).
- If a < 0, the vertex represents the highest point of the parabola (maximum value of the function).
- The horizontal position of the vertex (the p-coordinate) also defines the axis of symmetry of the parabola (x = p).
