How to Find the Vertex of a Quadratic Function: A Step‑by‑Step Guide

Understanding the Vertex

The vertex is the highest or lowest point on the graph of a quadratic equation, depending on whether the parabola opens upward or downward. It is a crucial feature for solving optimization problems, sketching graphs, and analyzing real‑world data.

Method 1: Using the Formula ‑b⁄(2a)

For a quadratic expressed in standard form y = ax² + bx + c, the x‑coordinate of the vertex can be found directly with the formula:

xv = ‑b ⁄ (2a)

Once xv is known, substitute it back into the original equation to obtain the y‑coordinate:

yv = a(xv)² + b(xv) + c

Example: For y = 2x² – 8x + 3, a = 2 and b = –8. Then xv = –(‑8) ⁄ (2·2) = 2. Plugging in, yv = 2·(2)² – 8·2 + 3 = –5. The vertex is (2, –5).

Method 2: Completing the Square

Another reliable technique is to rewrite the quadratic in vertex form y = a(x – h)² + k, where (h, k) is the vertex.

1. Factor out the leading coefficient a from the x‑terms.
2. Inside the parentheses, add and subtract ((b⁄2a)²) to create a perfect square.
3. Simplify to reveal the vertex coordinates h = –b⁄(2a) and k (the constant term after simplification).

Example: Starting with y = 3x² + 12x + 5:

Factor 3: y = 3(x² + 4x) + 5

Add & subtract ((4⁄2)² = 4) inside: y = 3[(x² + 4x + 4) – 4] + 5

Rewrite: y = 3(x + 2)² – 12 + 5 = 3(x + 2)² – 7

Thus the vertex is (‑2, –7).

Quick Tips for Accuracy

• Check the sign of a: If a > 0, the vertex is a minimum; if a < 0, it is a maximum.
• Always verify by plugging the vertex back into the original equation.
• Use a graphing calculator or software to confirm your result visually.

Why Knowing the Vertex Matters

Finding the vertex quickly enables you to determine the range of a quadratic function, solve real‑world optimization tasks (like maximizing profit or minimizing distance), and graph parabolas with confidence.

By mastering both the formula method and completing the square, you’ll be equipped to handle any quadratic problem that comes your way.

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