How Do You Calculate a Horizontal Tangent Line?

A tangent is a line that intersects a curve at only one point and does not pass through it, such that its slope is equal to the curve’s slope at that point. To find a horizontal tangent, you must find a point at which the slope of a curve is zero, which takes about 10 minutes when using a calculator.

  1. Determine the nature of the function

    Analyze your function. Determine whether or not it has any maximums or minimums, also known as extrema. For example, a parabola (or x^2 function) has one maximum or minimum, and at that point has a horizontal tangent. Trigonometric functions, such as sin(x), have an infinite number of maximums and minimums.

  2. Find the extrema

    Choose a point of extrema that seems easiest to calculate or find on a graph. Extrema can be found in many different ways. One of the easiest methods is using a graphing calculator’s TRACE tool and looking for the maximums and minimums. Taking the derivative of a function and solving for dy/dx=0 is a precise method for finding extrema. For example, this function can be used: f(x) = 3x^2 – 12x + 4, f'(x) = 6x – 12. f'(x) = 0 when x = 2 f(2) = 3(2)^2 – 12(2) + 4 = 3*4 – 24 + 4 = 12 – 20 = -8

  3. Write the formula for the tangent

    The equation for a horizontal tangent line is given as a function that relates y to a constant value. In the example f(x) function, the line y = -8 forms a horizontal tangent.