How to Find the Vertex of a Parabola (Formula & Examples)

The vertex of a parabola is the point where it turns — its maximum or minimum. For a parabola in standard form y = ax² + bx + c, the fastest method is the vertex formula: the x-coordinate is x = -b/(2a), and you plug that back in to get the y-coordinate. For vertex form y = a(x - h)² + k, the vertex is simply (h, k).

Below we work two full examples, then list the mistakes students make most often.

Worked example: y = 2x² − 12x + 16

This parabola is in standard form with a = 2, b = -12, c = 16.

1. Apply the vertex formula. The x-coordinate is x = -b/(2a) = -(-12)/(2·2) = 12/4 = 3.
2. Plug x = 3 back into the equation. y = 2(3)² - 12(3) + 16 = 18 - 36 + 16 = -2.
3. Write the vertex. The vertex is (3, -2).

You can verify this by completing the square: y = 2(x² - 6x) + 16 = 2(x - 3)² - 18 + 16 = 2(x - 3)² - 2. Vertex form confirms (h, k) = (3, -2).

The vertex formula, step by step

1. Identify a, b, c from y = ax² + bx + c. Example: for y = x² - 4x + 1, a = 1, b = -4, c = 1.
2. Compute x = -b/(2a). For that example, x = 4/2 = 2.
3. Substitute back to get y. y = (2)² - 4(2) + 1 = -3.
4. State the vertex as an ordered pair (x, y). Here, (2, -3).
5. Check the direction. If a > 0, the vertex is a minimum; if a < 0, it's a maximum.

Reading the vertex directly from vertex form

If the equation is already y = a(x - h)² + k, read the vertex off as (h, k). Watch the signs: y = 3(x + 5)² - 7 means h = -5, not +5, so the vertex is (-5, -7).

Another worked example: y = −x² + 6x − 5

Here a = -1, b = 6, c = -5. Because a < 0, this parabola opens downward and the vertex will be a maximum.

1. Vertex x-coordinate: x = -b/(2a) = -6/(2·(-1)) = -6/(-2) = 3.
2. Vertex y-coordinate: y = -(3)² + 6(3) - 5 = -9 + 18 - 5 = 4.
3. Vertex: (3, 4) — the maximum point of the parabola.

Cross-check by completing the square: y = -(x² - 6x) - 5 = -((x - 3)² - 9) - 5 = -(x - 3)² + 4. Vertex is (3, 4). ✓

Common mistakes to avoid

• Forgetting the negative sign in −b/(2a). If b = -12, then -b = +12. Sign errors here send the vertex to the wrong side of the y-axis.
• Only finding x and stopping. The vertex is a point, not a number. You must plug x back in to get y.
• Misreading vertex form signs. In (x - h), the vertex x is +h. So (x + 5)² means h = -5.
• Confusing vertex with center. Parabolas have a vertex; circles and ellipses have a center. Don't mix the terms.
• Dropping the leading coefficient when completing the square. Factor a out of the x-terms first, or your k value will be wrong.

Now that you've seen the method, the fastest way to lock it in is to work a few problems with a tutor that checks your steps — not just your final answer. Sign up for LernOS to try our guided Socratic tutor, which walks you through vertex problems (and the rest of Algebra 2) one prompt at a time.