Minimum (Quadratic Functions)
See It in Action
Two examples of the idea done right—and one common look-alike that is not right.
EXAMPLE
EXAMPLE
NON-EXAMPLE
Definition
The minimum of a quadratic function refers to the lowest point on the graph of the parabola. This occurs when the parabola opens upward, which happens when the coefficient a in the quadratic equation y = ax^2 + bx + c is positive.
The minimum value corresponds to the y-coordinate of the vertex, which is the point where the parabola reaches its lowest value. The x-coordinate of the vertex gives the value of x at which the minimum occurs.
Key points about the minimum:
A quadratic has a minimum if it opens upward (i.e., a > 0).
The minimum value is the y-coordinate of the vertex.
The x-coordinate of the vertex gives the value of x at which the minimum occurs.
Example:
For the quadratic equation y = 2(x - 1)^2 + 3, the vertex form shows that:
The parabola opens upward because a = 2 is positive.
The vertex is at (1, 3), so the minimum value is 3, and it occurs at x = 1.