The horizontal inflection point (orange circle) has a horizontal tangent line (orange dashed line).
A horizontal tangent line is parallel to the x-axis and shows where a function has a slope of zero. You can find these lines either by looking at a graph (which usually gives an approximation) or by setting an equation to zero to find maximums and minimums.
The graph of f(x) = x3 + 2x2 + 3 has two (blue dashed) horizontal tangent lines at y = 4.185 and y = 3 (Graph: Desmos.com).
How to Find Horizontal Tangent Lines
1. From a Graph
Look for places on a graph where the slope (a.k.a. the derivative) is zero. In other words, look for where the slope is horizontal or flat and parallel to the x-axis. If you have a trace function on your calculator, you should be able to pinpoint the exact coordinates. However, if you have a graph on paper or without that “trace” ability, the position of the horizontal tangent line will usually be an estimate. For example, the graph below appears to have a horizontal tangent at y = 3 (at the graph’s low point). However, the tangent line is actually at y = 3.025:
2. With an Equation
A function or graph has a horizontal tangent line when the first derivative is zero. Another way to think about it: if you find all of the critical points of a differentiable function (i.e. one that has a derivative), a horizontal tangent line occurs wherever there is a relative maximum (a peak) or relative minimum (a low point).
Example question: Find the horizontal tangent line(s) for the function f(x) = x3 + 3x2 + 3x – 3.
Step 1: Find the derivative of the function. Using the power rule, the function has a derivative of:
f′(x) = 3x2 + 6x – 3
Step 2: Set the derivative equal to zero.
3x2 + 6x – 3 = 0.
Step 3: Solve for x:
Which simplifies to x = -1 ± √2.
Therefore, the function has two horizontal tangent lines:
- x = -1 + √2.
- x = -1 – √2.
Note that all we’re really doing here is finding critical points, so you may want to check out the article How to Find Critical Points.
References
Desmos graphing calculator.
Morris, J. Section 3.1 – Powers and Polynomials
Stephanie Glen. "Horizontal Tangent Line" From CalculusHowTo.com: Calculus for the rest of us! /horizontal-tangent-line/
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