Equation of a Tangent Line

Tangent line equation: method

A tangent line touches a curve at one point and has the same slope as the curve there. To write its equation, you usually need two things:

1. The point of tangency Find the point on the curve where the line touches. If the x-value is given, substitute it into the function to get the y-value.

2. The slope at that point Compute the derivative of the function and evaluate it at the x-value of tangency. That derivative value is the slope of the tangent line.

3. Use point-slope form Once you know a point ((x_0, y_0)) and slope (m), write [ y - y_0 = m(x - x_0). ] Then simplify if needed.

Check your answer

• Make sure the line passes through the tangency point.
• Verify that its slope matches the derivative at that point.
• If you simplify to slope-intercept form, keep the same line, just rewritten.

Common mistake to avoid

Do not use the original function’s value as the slope. The slope comes from the derivative, not from the function itself.