Parabola Maximum and Minimum: Example 8

Topic

Quadratics

Description

The maximum or minimum value of a parabola occurs at its vertex. For a parabola opening upward (a > 0), this point is a minimum. For a parabola opening downward (a < 0), it's a maximum. The vertex lies on the axis of symmetry.

For a quadratic function in standard form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by x = -b/(2a), and the y-coordinate (maximum or minimum value) can be calculated by substituting this x-value into the original function.

For f(x) = x² + 4x + 5, a = 1 is positive, so the parabola opens upward with a minimum value. The x-coordinate of the vertex is x = -b/(2a) = -2. The minimum y-value is f(-2) = 1. Thus, the minimum value is 1 at x = -2. The vertex (-2, 1) lies on the axis of symmetry x = -2.

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