Characteristics of the graph

**lost in functions** · September 14th 2009, 10:30 AM

Discuss the characteristics of the graph that arise when a function has a factor the appears twice, three times, four times, etc...

No idea what I'm suppose to do

any help?

**Grandad** · September 15th 2009, 02:16 AM

Hello lost in functions

Welcome to Math Help Forum!

Originally Posted by lost in functions

Discuss the characteristics of the graph that arise when a function has a factor the appears twice, three times, four times, etc...

No idea what I'm suppose to do

any help?

If a function $f(x)$ has a factor $(x-a)$ then the equation $f(x) = 0$ has a root (a solution) at $x = a$ . So in this case, when $x = a$ , the graph $y = f(x)$ is 'at' the $x$ -axis, because the $x$ -axis is where $y = 0$ . OK so far?

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

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

The graph may be a tangent to the $x$ -axis where $x = a$ . So the graph comes up to the $x$ -axis, touches it and then returns on the same side of the axis from whence it came. This corresponds to $(x-a)^2$ being a factor of $f(x)$ . 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 $x = a$ (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 $(x-a)^3$ is a factor of $f(x)$ . This is a point of inflexion, and is sometimes referred to as 3-point contact.