How to Use Calculus to Prove the Vertex Formula

The vertex is the highest or lowest point of a parabola. The formula used to calculate the vertex of a parabola is "x = -b/2a". This formula is usually learned by rote memorization in pre-calculus. Pre-calculus teachers use a set of algebraic steps to show the derivation of the vertex formula. You can derive the vertex formula in only five steps.

Instructions

    • 1

      Write down the equation of a parabola, which is "f(x) = ax^2 + bx + c".

    • 2

      Take the derivative of the parabolic equation with respect to x. Using the normal derivation rules, you will then have "f '(x) = 2ax + b".

    • 3

      Set the derivative equal to zero. In other words, assume f'(x) = 0.

    • 4

      Substitute the right hand side of your derivative into the left hand side of the equation from step 2. Then, you will have: "2ax + b = 0".

    • 5

      Solve for x. You need basic algebra to solve this equation. The steps are to first subtract "b" from both sides, and then divide both sides by "2a". In the end, you will have "x = -b/(2a)". This is the vertex formula.

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