To find the equation of a line that is tangent to a curve, you must find a line with the same slope as that point of the curve. This can be done with a graphing calculator that can give the slope at a particular point or by hand using the derivative of a curve function.
- Take the first derivative of the function
The first derivative of a curve (f(x)) gives a function for the rate of change of that curve (f'(x)). For example, if the function of the original curve is a parabola defined by the equation f(x) = 4x^2 - 6x + 4, the rate of change is defined by f'(x) = 8x - 6.
- Find the slope at your given point
If you are trying to find the line tangent to f(x) when x=1, input 1 into f'(x) to calculate the slope at that point. In our example rate-of-change function f'(1) = 8(1) - 6 = 2, which means that the slope of the original function is 2 when x=1.
- Write the tangent in point-slope form
Point-slope form of a line is written y – y1 = m(x – x1), where you know the slope (m) and a coordinate (x1,y1). In the example, the slope at x=1 is 2. To find the y-value, take f(1). For example, f(1) = 4(1)^2 - 6(1) + 4 = 2. At the point (1,2) the slope is 2. So the line tangent to f(x) at x=1 is (y - 2) = 2(x - 1) or y = 2x.