Tangent Lines:

If Java worked here, you'd see something like this. At each point on a graph there are many straight lines passing through that point. Most of them cut right through the graph, but one special line just brushes the graph. This one line is said to be tangent to the graph at that point.

Click twice on the graph to draw a line that intersects the graph at the chosen points. (This is called a secant line for the graph.) As you bring the two points together, the secant line gets closer and closer to a tangent line, which cuts the graph only at one point.

Slope:

One way to measure the direction of a line is to measure the angle it makes with the horizontal. It's often more useful to use the tangent of this angle, which is defined using a little right triangle with its hypotenuse lying on the line, as in the diagram. The tangent is the ratio of the vertical to the horizontal side, and this is called the slope of the line.

The diagram shows the slope m of a tangent line to the graph of an exponential function N=b^t.