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.

This on the SAT

The Digital SAT tests vertex questions in its Advanced Math domain. Typical wording: What is the y-coordinate of the vertex of the graph of y = 2x² − 12x + 16 in the xy-plane? Or the multiple-choice version: Which of the following equivalent equations displays the coordinates of the vertex as constants or coefficients? The second phrasing is really asking you to convert standard form into vertex form — the same completing-the-square move shown above.

Bluebook (the digital testing app) ships with Desmos built in, so on test day you can graph the parabola, click the turning point, and read the vertex coordinates directly. The trap the SAT sets is the sign inside vertex form: y = 3(x + 5)² − 7 puts the vertex at (−5, −7), not (+5, −7). That sign flip is the single most-missed step on this question type.

For an SAT-scoped walkthrough of this exact question type, see How to Find the Vertex of a Parabola on the SAT. Working through quadratics more broadly? Our page on how to find the discriminant of a quadratic equation covers what b² − 4ac tells you about the number of solutions — another Advanced Math fact the SAT tests directly.

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.