OK, let's start graphing!

Let's get our intercepts and asymptotes down:


intercepts ( 0 , 0 )  ...  asymptotes y = 0 , x = -4  and  x = 2

 

Since we've got that pesky y = 0 horizontal asymptote, we can't use our "nowhere else" info on the x-intercepts to figure out the "upstairs" or "downstairs" stuff.  Until we have Calculus, we're going to have to humble ourselves and plot some points -- kind of!  We're  going to avoid taking that pathetic trip to Sissyville by cheating a little bit...  I call it "quickie plotting."

Four strategically located points will do it and all we really need to know is if he's above or below the x-axis at each point.

We'll be plugging x's into the factored form:

f( x ) = ( 3x ) / ( ( x - 2 )( x + 4 ) )

* We only care about positives and negatives -- See if you can do it!

 
f( x ) = ( 3x ) / ( ( x - 2 )( x + 4 ) )plugging in -5 for x gives ( - ) / ( ( - )( - ) ) = ( - )  ...  below for y  ,  plugging in -1 for x gives ( - ) / ( ( - )( + ) ) = ( + )  ... above for y  ,  plugging in 1 for x gives ( + ) / ( ( - )( + ) ) = ( - )  ...  below for y  ,  plugging in 3 for x gives ( + ) / ( ( + )( + ) ) = ( + )  ...  above for y

Now we've got it!  Remember that he can only cross the x-axis at
x = 0...  So, once he's below, he'll be stuck there...  and, once he's above, he'll have to stay above:


graph of f( x ) = ( 3x ) / ( ( x - 2 )( x + 4 ) )  ...  below the x-axis at x = -5  ...  above the x-axis at x = -1  ...  below the x-axis at x = 1  ...  above the x-axis at x = 3