Vertex of a Parabola — Definition, Formula & Examples

Q: How do you find the vertex of a parabola from standard form?

Given y = ax² + bx + c, compute the x-coordinate with h = −b/(2a). Then plug h back into the equation to get the y-coordinate k = a·h² + b·h + c. The vertex is the point (h, k).

Q: Is the vertex of a parabola always a minimum?

No. When a > 0 the parabola opens upward and the vertex is a minimum. When a < 0 the parabola opens downward and the vertex is a maximum. The sign of the leading coefficient a determines which case applies.

Q: What is the difference between vertex form and standard form of a parabola?

Standard form is y = ax² + bx + c, where the vertex requires calculation. Vertex form is y = a(x − h)² + k, where the vertex (h, k) can be read directly. You can convert between the two by completing the square.

Vertex of a Parabola

The point at which a parabola makes its sharpest turn. The vertex is halfway between the directrix and the focus.

Upward-opening parabola curve with the lowest point labeled "vertex" at its bottom center.

Example:

Graph of upward-opening parabola with vertex at (3,-2), focus above vertex, vertical axis of symmetry at x=3, and horizontal...

This is a graph of the parabola with all its major features labeled: axis of symmetry, focus, vertex, and directrix.

Key Formula

h = -\frac{b}{2a}, \quad k = f(h) = a h^2 + b h + c

Where:

$h$ = The x-coordinate of the vertex
$k$ = The y-coordinate of the vertex, found by substituting h back into the equation
$a$ = The coefficient of x² in the standard form y = ax² + bx + c
$b$ = The coefficient of x in the standard form
$c$ = The constant term in the standard form

Worked Example

Problem: Find the vertex of the parabola y = 2x² − 8x + 3.

Step 1: Identify the coefficients a, b, and c from the equation y = ax² + bx + c.

a = 2, \quad b = -8, \quad c = 3

Step 2: Calculate the x-coordinate of the vertex using the formula h = −b/(2a).

h = -\frac{-8}{2(2)} = \frac{8}{4} = 2

Step 3: Substitute h = 2 back into the original equation to find the y-coordinate k.

k = 2(2)^2 - 8(2) + 3 = 8 - 16 + 3 = -5

Step 4: Write the vertex as an ordered pair.

(h, k) = (2, -5)

Answer: The vertex of the parabola is (2, −5). Since a = 2 > 0, the parabola opens upward and this vertex is a minimum point.

Another Example

This example uses vertex form y = a(x − h)² + k, where the vertex can be read directly without calculation. It also illustrates a downward-opening parabola with a maximum vertex, contrasting the first example's minimum vertex.

Problem: Find the vertex of the parabola y = −(x + 3)² + 7, which is already in vertex form.

Step 1: Recognize that the equation is already in vertex form y = a(x − h)² + k.

y = -1(x - (-3))^2 + 7

Step 2: Read the vertex directly from the equation. Here h = −3 and k = 7.

(h, k) = (-3, 7)

Step 3: Since a = −1 < 0, the parabola opens downward, so the vertex is a maximum point.

a = -1 < 0 \implies \text{maximum}

Answer: The vertex is (−3, 7), and it is the highest point on the parabola.

Frequently Asked Questions

How do you find the vertex of a parabola from standard form?

Given y = ax² + bx + c, compute the x-coordinate with h = −b/(2a). Then plug h back into the equation to get the y-coordinate k = a·h² + b·h + c. The vertex is the point (h, k).

Is the vertex of a parabola always a minimum?

No. When a > 0 the parabola opens upward and the vertex is a minimum. When a < 0 the parabola opens downward and the vertex is a maximum. The sign of the leading coefficient a determines which case applies.

What is the difference between vertex form and standard form of a parabola?

Standard form is y = ax² + bx + c, where the vertex requires calculation. Vertex form is y = a(x − h)² + k, where the vertex (h, k) can be read directly. You can convert between the two by completing the square.

Vertex Form vs. Standard Form

	Vertex Form	Standard Form
Equation	y = a(x − h)² + k	y = ax² + bx + c
Finding the vertex	Read directly: vertex is (h, k)	Calculate: h = −b/(2a), then k = f(h)
Finding the y-intercept	Expand and set x = 0, or substitute directly	Read directly: y-intercept is c
When to use	When you need the vertex or axis of symmetry quickly	When you need the y-intercept or are factoring

Why It Matters

The vertex appears throughout algebra and precalculus whenever you graph quadratic functions or solve optimization problems. For instance, finding the maximum height of a projectile or the minimum cost in a business model both reduce to finding a vertex. Standardized tests like the SAT and ACT frequently ask you to identify the vertex from an equation or graph.

Common Mistakes

Mistake: Forgetting the negative sign in h = −b/(2a) and computing b/(2a) instead.

Correction: The formula has a negative sign in front: h = −b/(2a). For example, if b = −8 and a = 2, then h = −(−8)/(2·2) = 2, not −2.

Mistake: Misreading the sign of h from vertex form y = a(x − h)² + k.

Correction: The form uses (x − h), so y = (x + 3)² means h = −3, not +3. Whatever value makes the expression inside the parentheses equal zero is h.

Related Terms

Parabola — The curve whose turning point is the vertex
Axis of Symmetry of a Parabola — Vertical line passing through the vertex
Focus of a Parabola — Point above or below the vertex defining the curve
Directrix of a Parabola — Line on the opposite side of the vertex from the focus
Vertex — General term for a corner or turning point
Vertices of an Ellipse — Analogous special points on an ellipse
Vertices of a Hyperbola — Analogous special points on a hyperbola
Graph of an Equation or Inequality — Visual representation where the vertex appears