This is a long one ok  

f(x)=x^2-6x+9 over x^2

1.  Is there a horizontal asymptote? If yes, what is it?

2. Vertical asymptote? If yes, what is it?

3. Is there a slant/oblique asymptote for this function?  If yes what is it?

4. What is the x intercepts if any?  Y intercepts if any?

5. On what intervals is this function positive (so that the graph is
above the x axis)?

Bart22 --

What we need here is a good graphing function.  Try this one, clicking
on the "Plot" link on the right-hand side to bring up a Java applet:
http://www.math.com/students/solvers/online_solvers.htm

Your function simplifies to:
Y = (x-3)^2/x^2

Clearly there are some key points in this graph:
X = 0
X = 3

So, we want to know what this graph looks for a range from x = -5 to
about x = 7; we'd like to see if y is ever negative (duh, it's not)
and look at a range of -1 to about 10.  So set up the plot that way.

What do you see?

2.  Vertical asymptote: x = 0

4.  X intercept at x = 3; no Y intercept (because both L and R sides
of this discontinuous function approach infinity as x approaches 0)

5.  This function is positive for all x except x = 3

---

1.  Now the horizontal asymptotes: for x > 0, what limit is there for
y in this function?  Trying some samples will give you a pretty good
idea:

x = 10; y = 0.49
x = 100; y = 0.9409
x = 200; y = 0.9702

So, for x > 0, the asymptote is y = 1 -- which is pretty intuitive
when you see that (any positive number minus 3) squared will always be
less than (any positive number) squared

     For x < 0, you'll see that as x increases, y approaches 0 -- so y
= 0 is the asymptote on the L side

3.  A little harder is the slant/oblique asymptote.  A note from this
definition of an oblique asymptote on the Mathwords page:
"Oblique asymptotes always occur for rational functions which have a
numerator polynomial that is one degree higher than the denominator
polynomial."

Mathwords
"Oblique Asymptote"
http://www.mathwords.com/o/oblique_asymptote.htm

Your numerator and denominator are both second-order functions -- so
you won't have an oblique asymptote.

Google search strategy:
"graphing online"
"oblique asymptote"



Best regards,

Omnivorous-GA