help creating rational function impossible

[jbirdwell]

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these are the characteristics it has to follow:
lim f(x) = 2 x -> infinty

lim f(x) = -2 x-> - infinty

lim f(x) = - infinty x-> -4

lim f(x) = - infinty x-> 2-

lim f(x) = infinty x-> 2+


relative min of 0 at x=2
relative max of -0.900466 at x=0.442818
concave down (-infinty, -4) (-4,-2) (6.835351, infinty)
concave up (2,6.835351)
x-inter (4,0)
y-inter (0,-1)
vertical asymptotes at x=2, x=4

[Martingale]

[img]H:\Photo0217.jpg[/img]

these are the characteristics it has to follow:

lim f(x) = 2 lim f(x) = -2 lim f(x) = - infinty lim f(x) = - infinty lim f(x) = infinty
x -> infinty x-> - infinty x-> -4 x-> 2- x-> 2+

relative min of 0 at x=2
relative max of -0.900466 at x=0.442818
concave down (-infinty, -4) (-4,-2) (6.835351, infinty)
concave up (2,6.835351)
x-inter (4,0)
y-inter (0,-1)
vertical asymptotes at x=2, x=4

Hi jbirdwell,

I Cant see your picture/image

[stapel_eliz]

Are you supposed to come up with "the" function, or just "a" function, that fits the listed criteria?

[jbirdwell]

come up with a function that fits all of the characterics

[stapel_eliz]

these are the characteristics it has to follow:
lim f(x) = 2 x -> infinty
lim f(x) = -2 x-> - infinty

This suggests that the degrees of the numerator and denominator need to be the same.

lim f(x) = - infinty x-> -4

This says that there is a vertical asymptote at x = -4, so there is a factor of x + 4 in the denominator. Also, the sign on f(x) will not change as x passes -4. (Hint: What would happen to the sign of, say, (x + 4)²?)

lim f(x) = - infinty x-> 2-
lim f(x) = infinty x-> 2+

This says that there is a vertical asymptote at x = 2, so there is a factor of x - 2 in the denominator. Also, the sign on f(x) will change as x passes 2.

relative min of 0 at x=2

How can you have a max/min point where the function is not defined?

relative max of -0.900466 at x=0.442818
concave down (-infinty, -4) (-4,-2) (6.835351, infinty)
concave up (2,6.835351)

Do you really have to get these exact values???

x-inter (4,0)

So the function is equal to zero at x = 4, so the numerator is zero at x = 4, so there is a factor of x - 4 in the numerator.

y-inter (0,-1)

So the function is equal to -1 when x = 0.

You should be able to at least get started with the above. Please reply with your thoughts and work so far.