Quadratic Functions: Finding the Vertex of a Parabola

Quadratic Functions

Vertex of a Parabola

A quadratic function is a polynomial function of degree 2, typically written in the form:

• Standard form: $f(x) = ax^2 + bx + c$

The graph of a quadratic function is a parabola. The vertex of the parabola is its highest or lowest point, depending on the direction it opens.

• If $a > 0$, the parabola opens upward (vertex is a minimum).

• If $a < 0$, the parabola opens downward (vertex is a maximum).

Finding the Vertex

The vertex $(h, k)$ of the parabola defined by $f(x) = ax^2 + bx + c$ can be found using the formulas:

• $h = -\frac{b}{2a}$

• $k = f(h)$

Thus, the vertex is at $(-\frac{b}{2a}, f(-\frac{b}{2a}))$.

Example

Given the quadratic function:

• $f(x) = -x^2 - 4x + 3$

Identify $a = -1$, $b = -4$, $c = 3$.

• Calculate $h$:

$h = -\frac{-4}{2 \times -1} = -\frac{-4}{-2} = -2$

• Calculate $k$:

$k = f(-2) = -(-2)^2 - 4(-2) + 3 = -4 + 8 + 3 = 7$

Therefore, the vertex is $(-2, 7)$.

• Vertex: $(-2, 7)$

Additional info: The original question asked for the vertex of a quadratic function. The function was interpreted as $f(x) = -x^2 - 4x + 3$ based on standard notation, as the original text had formatting issues.