Parabola Vertex
x-coordinate of the vertex of a quadratic.
Formula
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Example
For x²-4x: vertex at x=2
How it works
For any quadratic y = ax^2 + bx + c, the graph is a parabola and its turning point is called the vertex. The x-value of that point is x = -b/(2a) because the parabola is symmetric, and this formula comes from completing the square or from finding where the slope changes from decreasing to increasing. Once you know this x-coordinate, you can substitute it back into the quadratic to find the y-coordinate of the vertex.
When to use it
Use this when you need the highest or lowest point of a quadratic graph, such as in optimisation problems, projectile motion, or sketching parabolas. It is also useful when you want to find the axis of symmetry of a quadratic quickly.
Worked example, step by step
Given y = 2x^2 + 8x - 3 Step 1: Identify a and b. a = 2, b = 8 Step 2: Substitute into x = -b/(2a). x = -8 / (2 × 2) x = -8/4 x = -2 Step 3: The vertex has x-coordinate -2. Step 4 (optional): Find the y-coordinate by substituting x = -2 into the equation. y = 2(-2)^2 + 8(-2) - 3 y = 2(4) - 16 - 3 y = 8 - 16 - 3 y = -11 Final answer: the vertex is (-2, -11).
Common mistakes to avoid
- Using the formula on a non-quadratic equation
- Forgetting the negative sign in x = -b/(2a)
- Using c instead of b when identifying the coefficient values
Did You Know?
A parabola's vertex marks its highest or lowest point — the reason satellite dishes and headlights use parabolic shapes.
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