Characteristics of the graph

• Sep 14th 2009, 11:30 AM
lost in functions
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?
• Sep 15th 2009, 03:16 AM
Hello lost in functions

Welcome to Math Help Forum!
Quote:

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.