#1. (3x^3-4x^2-3x+4)/(x^3-5x)

my answers:
y-int: NONE
x-int: 1, -1, and 4/3
x asymtope: x= 0 and +- square root of 5
y asym: y=3

If it crosses horiz asym: idk i need help on this

#2. (x^4-7x^2+12)/(x^2-5x+4)

x-int: -2, 2, and +- square root of 3
y-int: 3
x-asy: x= 4 and 1
y asym: y=-7/4x^2

I cant find if/where it intersects the horiz asym

#3
(x^2-5x^2)/(x^3-3x)
y-int: (0,5/3)
x-int: 0 and +- square root of 5
xasym: x= 0 and +- square root of 3

I cant find the y asym or where it intersects

#4 (1-x)/(x^3-2x^2+x-2)
x asym: x=2
x-int: 1,0
y int: (0, -1)
yasym: y=0

once again i cant find where it intersects

Thankyou soooo much!! i know its longgg....

No problem! I'll help you understand how to find the answers to your questions step by step.

For #1:
To find the y-intercept, substitute x=0 into the equation:
(3(0)^3 - 4(0)^2 - 3(0) + 4) / ((0)^3 - 5(0)) = 4/0, which is undefined. Therefore, there is no y-intercept in this case.

To find the x-intercepts, set the numerator of the fraction equal to zero and solve for x:
3x^3 - 4x^2 - 3x + 4 = 0.
You can either factor this equation or use a calculator or online tool to solve for x. In this case, the solutions are x=1, x=-1, and x=4/3, which are the x-intercepts.

To find the horizontal asymptotes, find the degrees of both the numerator and denominator polynomials. In this case, the degrees are both 3.
If the degrees are equal, divide the corresponding leading coefficients. So, the horizontal asymptote is y = 3/1 = 3.

To find the vertical asymptotes, set the denominator of the fraction equal to zero and solve for x:
x^3 - 5x = 0.
Solving this equation, we get x=0 and x=±√5. These are the vertical asymptotes.

Regarding the crossed horizontal asymptote, if there is an intersection between the graph of the function and the horizontal asymptote, it means that the function doesn't approach a specific y-value as x approaches infinity or negative infinity. In this case, the function does not intersect the horizontal asymptote y=3.

For #2:
To find the x-intercepts, set the numerator of the fraction equal to zero and solve for x:
x^4 - 7x^2 + 12 = 0.
You can either factor this equation or use a calculator or online tool to solve for x. In this case, the solutions are x=-2, x=2, and x=±√3, which are the x-intercepts.

To find the y-intercept, substitute x=0 into the equation:
(0^4 - 7(0)^2 + 12) / (0^2 - 5(0) + 4) = 12/4 = 3. Therefore, the y-intercept is y=3.

To find the horizontal asymptotes, find the degrees of both the numerator and denominator polynomials. In this case, the degree of the numerator is 4, and the degree of the denominator is 2.
If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. However, as x approaches positive infinity or negative infinity, the function approaches positive or negative infinity.

Regarding the intersection with the horizontal asymptote, again, there is no specific value where the graph intersects the horizontal asymptote y=∞ or y=-∞.

For #3:
To find the y-intercept, substitute x=0 into the equation:
(0^2 - 5(0)^2) / (0^3 - 3(0)) = 0/0, which is undefined. Therefore, there is no y-intercept in this case.

To find the x-intercepts, set the numerator of the fraction equal to zero and solve for x:
x^2 - 5x^2 = 0.
Simplifying the equation gives -4x^2 = 0. The only solution is x=0.

To find the horizontal asymptotes, find the degrees of both the numerator and denominator polynomials. In this case, the degree of the numerator is 2, and the degree of the denominator is 3.
If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0.

To find the vertical asymptotes, set the denominator of the fraction equal to zero and solve for x:
x^3 - 3x = 0.
Solving this equation, we get x=0 and x=±√3. These are the vertical asymptotes.

Regarding the y asymptote and intersection with the horizontal asymptote, in this case, the function doesn't have a clear y asymptote or intersect the horizontal asymptote y=0.

For #4:
To find the x-intercepts, set the numerator of the fraction equal to zero and solve for x:
1 - x = 0. Solving this equation, we get x=1.

To find the y-intercept, substitute x=0 into the equation:
(1 - 0) / (0^3 - 2(0)^2 + 0 - 2) = 1 / (-2) = -1/2. Therefore, the y-intercept is y=-1/2.

To find the horizontal asymptotes, find the degrees of both the numerator and denominator polynomials. In this case, the degree of the numerator is 1, and the degree of the denominator is 3.
If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0.

Regarding the x asymptote, if the degree of the numerator is less than the degree of the denominator by only one, there is an x asymptote at x=2 in this case.

Regarding the intersection with the horizontal asymptote, the graph of the function intersects the horizontal asymptote y=0 at the point (0,0).

I hope this explanation helps you understand how to find the different aspects of these rational functions. Let me know if you have any other questions!