Vertex of a Parabola - Find Max & Min Points Easily

The vertex is the most important point on a parabola. It's the peak of a hill or the bottom of a valley — and in real-world problems, it's usually the answer you're looking for.

Here's how to find it from any form of the quadratic equation.

Finding the Vertex from Standard Form

Given f(x) = ax² + bx + c:

Vertex x-coordinate: x = −b / (2a)

Then plug back in for y: y = f(−b/2a)

Example: f(x) = −3x² + 12x − 5

1. x = −12 / (2·(−3)) = −12/(−6) = 2
2. f(2) = −3(4) + 12(2) − 5 = −12 + 24 − 5 = 7
3. Vertex: (2, 7) — a maximum since a = −3 < 0

Finding the Vertex from Vertex Form

Given f(x) = a(x − h)² + k, the vertex is simply (h, k).

No calculation needed — just read it off.

| Equation | Vertex | |----------|--------| | f(x) = 2(x − 3)² + 1 | (3, 1) | | f(x) = −(x + 4)² − 7 | (−4, −7) | | f(x) = 0.5(x − 0.5)² + 2 | (0.5, 2) |

Watch the signs: (x − 3)² means h = +3, not −3.

Converting Standard Form to Vertex Form

Complete the square to convert ax² + bx + c → a(x − h)² + k.

Example: f(x) = 2x² − 8x + 5

1. Factor out a from the first two terms: 2(x² − 4x) + 5
2. Complete the square: (−4/2)² = 4 → add and subtract 4 inside
3. 2(x² − 4x + 4 − 4) + 5 = 2(x − 2)² − 8 + 5
4. f(x) = 2(x − 2)² − 3
5. Vertex: (2, −3)

Maximum vs. Minimum — Which Is It?

| Condition | Vertex Type | Real-World Meaning | |-----------|------------|-------------------| | a > 0 | Minimum (opens up) | Minimum cost, minimum time | | a < 0 | Maximum (opens down) | Maximum profit, maximum height |

Optimization Applications

Physics: Maximum Height of a Projectile

A ball is launched: h(t) = −16t² + 64t + 8

• Vertex at t = −64/(2·(−16)) = 2 seconds
• h(2) = −16(4) + 64(2) + 8 = 72 feet
• Maximum height: 72 feet at t = 2 seconds

Economics: Maximum Revenue

Revenue: R(p) = −5p² + 200p (price p in ₹)

• Vertex at p = −200/(2·(−5)) = ₹20
• R(20) = −5(400) + 200(20) = −2000 + 4000 = ₹2,000
• Maximum revenue: ₹2,000 at price ₹20

Geometry: Maximum Area

A farmer has 100m of fencing for a rectangular pen against a wall:

• A = x(100 − 2x)/2 = 50x − x²
• Vertex at x = −50/(−2) = 25m
• A(25) = 50(25) − 625 = 625 m²
• Maximum area: 625 m² when width is 25m

The Trench Truth: Optimization problems on exams always ask for the vertex — they just disguise it as "maximum profit" or "minimum cost." Find the quadratic, identify a and b, compute −b/2a. That's the answer 90% of the time. The remaining 10% requires checking endpoints.

Use our quadratic formula calculator to find the vertex of any quadratic instantly.

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