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?
Hello lost in functions
Welcome to Math Help Forum!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.
Grandad