How Tangent Lines Look
The tangent line to f at a is the line approached by the secant lines between a and a + h as h approaches 0. This applet lets us watch the secant line approach the tangent line as we drag the point at a + h closer to a:
Tangent lines and secant lines are (usually) different things.
A secant line must hit two points on a graph. A secant line between the points on the graph where x = a and x = b will hit the graph at x = a and x = b.
A tangent line occurs at one single point on a graph and has the same slope as the graph at that point.
Visually, the tangent line to f at a bounces off the graph at x = a:
A tangent line may also cross through the graph somewhere else. The important thing that makes a tangent line a tangent line is that it grazes the graph at that one special point. This line is tangent to f at x = a because it bounces off there:
However, this line is also a secant line between x = a and x = b:
A tangent line usually doesn't "cross over" the graph from one side to the other. However, it may cross over the graph at x = a in cases like this:
Here, the tangent line isn't so much "bouncing off" as it is "laying along" the graph of f.
This phenomenon has to do with something called concavity.
When looking to see if a line is tangent to f at a, we're looking to see if the line "bounces off" or "lays along" the graph of f at a.
This function has a tangent line with infinite slope at x = a:
If we have a function f that's already a line, the tangent line to f at any point a will be f again: