Hello lost in functions

Welcome to Math Help Forum!If a function has a factor then the equation has a root (a solution) at . So in this case, when , the graph is 'at' the -axis, because the -axis is where . OK so far?

Now I say 'at' the -axis because various things might happen:

- The graph may cut the -axis, starting on one side of the axis, crossing it at and emerging on the other side. This will happen if the factor appears just once in .

- The graph may be a tangent to the -axis where . So the graph comes up to the -axis, touches it and then returns on the same side of the axis from whence it came. This corresponds to being a factor of . A tangent has what's called '2-point contact' with the line or curve that it touches.

- The graph may do both of the above! It may be a tangent at (in other words it is horizontal at this point) but it may also cross the axis and emerge on the other side. This is what happens if is a factor of . This is a
point of inflexion, and is sometimes referred to as 3-point contact.

- If is a factor then we have 4-point contact, and the curve touches the x-axis and returns on the side from whence it came.

- If is a factor we have 5-point contact and the graph crosses the axis.

... and so on.

Grandad