Minimum value

Definition

The minimum value of a quadratic function is the lowest point on its graph, known as the vertex, when the parabola opens upwards. It represents the smallest y-value that the function attains.

5 Must Know Facts For Your Next Test

1. For a quadratic function in standard form $f(x) = ax^2 + bx + c$ with $a > 0$, the minimum value occurs at $x = -\frac{b}{2a}$.
2. The y-coordinate of the vertex gives the minimum value and can be found by evaluating $f(-\frac{b}{2a})$.
3. A quadratic function has a minimum value if and only if its leading coefficient $a$ is positive.
4. In vertex form $f(x) = a(x-h)^2 + k$, where $a > 0$, the minimum value is $k$ at $x = h$.
5. The axis of symmetry of a quadratic function, given by $x = -\frac{b}{2a}$, passes through the vertex.

Review Questions

• How do you determine whether a quadratic function has a minimum or maximum value?
• What formula is used to find the x-coordinate of the vertex for a quadratic function in standard form?
• If given a quadratic function in vertex form, how can you quickly identify its minimum value?

Related terms

Vertex: The point $(h,k)$ on a parabola that represents either its maximum or minimum value.

Axis of Symmetry: A vertical line that divides the parabola into two mirror images, given by $x = -\frac{b}{2a}$ for a quadratic function.

Standard Form: $f(x) = ax^2 + bx + c$, where \(a\), \(b\), and \(c\) are constants and \(a \neq0\).