How to Use Calculus to Develop the Formula for a Vertex

If you want to graph a parabola, you need to know the vertex. The vertex is either the lowest point of the curve, if the coefficient of x^2 is positive, or the highest point of the curve, if the coefficient of x^2 is negative. You can create the formula for computing the vertex by using either algebra or calculus. To find the vertex using calculus, set the derivative of the function equal to zero.

Instructions

1. Compute the derivative of the function. For example, consider the parabola:

   f(x) = 2x^2 + 8x + 1

   and calculate the derivative:

   f'(x) = 4x + 8

2. Set the derivative of the function equal to zero. For example:

   f'(x) := 0

   0 = 4x + 8

3. Find the x coordinate of the vertex by subtracting 8 from both sides of the equation and dividing both sides of the equation by 4. For example:

   0 = 4x + 8

   -8 = 4x

   -8/4 = x

   -2 = x

4. Find the y coordinate of the vertex by substituting the value of x into the original equation. For example:

   f(x) = 2(-2)^2 + 8(-2) + 1

   f(x) = 8 - 16 + 1

   f(x) = -7

   vertex = (-2, -7)

Tips & Warnings

• For a parabola f(x) = ax^2 + bx + c, the vertex will always be (-b/2a, f(-b/2a)).

References

• Oaklahoma City Community College: Quadratic Equations