# Characteristics of the graph

• Sep 14th 2009, 10:30 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, 02:16 AM
Hello lost in functions

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

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

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

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

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

... and so on.