# Horizontal Tangent Line

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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).

## 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

CITE THIS AS:
Stephanie Glen. "Horizontal Tangent Line" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/horizontal-tangent-line/
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