# Thread: Characteristics of the graph

1. ## Characteristics of the graph

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?

2. 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.

• If $(x-a)^4$ 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 $(x-a)^5$ is a factor we have 5-point contact and the graph crosses the axis.

... and so on.