Minimum value

Definition

The minimum value of a quadratic function is the lowest point on its graph, which occurs at the vertex if the parabola opens upwards. It is found using the vertex form or by completing the square.

5 Must Know Facts For Your Next Test

1. The minimum value occurs at $x = -\frac{b}{2a}$ for a quadratic function in standard form $ax^2 + bx + c$ where $a > 0$.
2. The y-coordinate of the vertex gives the minimum value of the quadratic function.
3. If a quadratic function opens upwards ($a > 0$), it has a minimum value; if it opens downwards ($a < 0$), it has a maximum value instead.
4. To find the minimum value, you can rewrite the quadratic equation in vertex form $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex.
5. The axis of symmetry for a parabola given by $ax^2 + bx + c$ is $x = -\frac{b}{2a}$. This line passes through the vertex.

Review Questions

• What is the x-coordinate of the vertex for the quadratic function $3x^2 - 6x + 1$?
• How do you determine if a quadratic function has a minimum or maximum value based on its leading coefficient?
• Write the standard form equation and identify its minimum value: $f(x) = 2(x-3)^2 + 4$.

Related terms

Vertex:

The highest or lowest point on a parabola. For a quadratic function, it represents either the maximum or minimum value.

Axis of Symmetry:

A vertical line that divides a parabola into two mirror images. For a quadratic equation $ax^2 + bx + c$, it is given by $x = -\frac{b}{2a}$.

Parabola:

The graph of a quadratic function. It is U-shaped and can open either upwards or downwards depending on the sign of its leading coefficient.