Write down the function that defines the graph where you must find the tangent line. The equation is usually expressed as f(x) = x^z at x= a number. For example, f(x) = x^2 - 1 at x=2.
Solve for f(x) to identify the y coordinate of point where the tangent line makes contact. Insert 2 for x in the equation and solve (x^2 - 1 = 2^2 - 1= 4 -1= 3). In this example f(x), or the y coordinate is equal to 3.
Graph the original function on a sheet of paper. In this case, the graph is a parabola with the origin at -1 and the curve passing through the x axis at -2 and 2.
Mark the point of contact for the tangent line as calculated earlier. In this case at (2,3) or 2 on the x axis and 3 on the y axis.
Draw the tangent line on the graph. Line up the ruler so that it is parallel to the curve at the point of contact for the tangent line. Draw a straight line to mark the tangent line.
Calculate a second set of coordinates present on the tangent line. Examine the graph and choose any x value that corresponds to a point on the tangent line. The closer to the point of contact, the more accurate your final estimate will be. For example, use 2.1 for x. If x is equal to 2.1, then y is equal to 3.41 (f(x) = x^2 - 1 = 2.1^1 - 1=4.41 - 1 =3.41). The second set of coordinates is therefore (2.1, 3.41).
Estimate the slope of the tangent line with the quotient of the change in y over the change in x from one set of coordinates to the other. Slope is equal to the second y coordinate minus the first y coordinate divided by the second x coordinate minus the first x coordinate (slope= y2 - y1/x2 - x1). For the example, y2 = 3.41, y1 = 3, x2 = 2.1 and x1 = 2. Thus, y2 - y1/x2 - x2 = 3.41 - 3/2.1 - 2= 4.1. The estimated slope of the tangent line in this example is approximately 4.