Vertex of a Parabola
AlgebraThe vertex of a parabola is its highest or lowest point, located at the coordinates (-b/(2a), f(-b/(2a))) for a quadratic in standard form.
Formula
x = \frac{-b}{2a}
Definition
The vertex of a parabola is the turning point: the very top if the parabola opens downward, or the very bottom if it opens upward. It is the single highest or lowest point on the graph.
Example
For $y = x^2 - 4x + 3$, the vertex is at $x = -(-4)/(2 \cdot 1) = 2$. Then $y = 4 - 8 + 3 = -1$. The vertex is $(2, -1)$, the lowest point on this upward-opening parabola.
Key Insight
The vertex is where the parabola changes direction. It is the minimum of an upward parabola and the maximum of a downward parabola.
Definition
For a quadratic in standard form $y = ax^2 + bx + c$, the x-coordinate of the vertex is $x = -b/(2a)$. Substitute back to find the y-coordinate. In vertex form $y = a(x - h)^2 + k$, the vertex is directly $(h, k)$. The vertex is a minimum if $a > 0$ and a maximum if $a < 0$.
Example
$y = -2x^2 + 8x - 5$: $x_{\text{vertex}} = -8/(2 \cdot (-2)) = 2$. $y_{\text{vertex}} = -2(4) + 8(2) - 5 = -8 + 16 - 5 = 3$. Vertex: $(2, 3)$. Maximum point since $a = -2 < 0$.
Key Insight
The vertex formula $x = -b/(2a)$ comes from completing the square or from setting the derivative to zero in calculus. It identifies the axis of symmetry of the parabola.
Definition
The vertex of the parabola $y = ax^2 + bx + c$ is the unique critical point of the quadratic function $f(x) = ax^2 + bx + c$. Setting $f'(x) = 2ax + b = 0$ gives $x = -b/(2a)$, confirming it as the unique extremum. The vertex is a global minimum ($a > 0$) or global maximum ($a < 0$) since $f$ is strictly convex or concave. In vertex form $f(x) = a(x - h)^2 + k$, the vertex $(h, k)$ is the canonical representation, and $k$ is the optimal value.
Example
Optimization: minimize cost $C(x) = 0.5x^2 - 10x + 60$. $x_{\text{vertex}} = 10/(2 \cdot 0.5) = 10$. $C(10) = 50 - 100 + 60 = 10$. Minimum cost is $10$ at $x = 10$ units.
Key Insight
Finding the vertex is equivalent to solving an optimization problem. The vertex formula is a special case of setting the gradient to zero in multivariable calculus, which is the general technique for unconstrained optimization.