I am working on rational funtions and having problems finding the ordered pairs.

My question has three parts

Step 1 of 3

Given the following rational function:  x^2-x+8/x^2-4x+3

Find the vertical asymptotes, if any, for the rational function.  Separate multiple equations with a comma.

To get step 1 I factored the denominator and got (x-3)(x-1) giving me x=1 and x=3

Step 2 of 3

Find equations for the horizontal or oblique asymptotes, if any for the rational function.  Since the degrees are the same for both denominator and numerator the horizontal or oblique asymptotes is y=1

Step 3 of 3

This is where I get confused

Enter four ordered pairs for the graph of the rational function.

How do I know what x values to plug into the function to get my ordered pairs.  I tried plugging in any number besides the ones above and I got the problem wrong.

I plugged 2 as the x value and got the ordered pair (2,-10)

I plugged 4 as the x value and got the ordered pair (4,5)

I plugged -1 as the x value and got the ordered pair (-1, 10/13)

I plugged -2 as the x value and got the ordered pair (-2, 7/8)

What am I doing wront for part 3 of this question?

Thanks and I appreciate your help

Expert Answers

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Given the function `f(x)=(x^2-x+8)/(x^2-4x+3)` :

(1) Find the vertical asymptotes:

A rational function has a vertical asymptote at any value of x that causes the denominator to be equal to zero while the numerator is nonzero. To find the zeros of the numerator and denominator we can factor:


There are vertical asymptotes at x=1 and x=3.

(2) Since the degree of the numerator is the same as the degree of the denominator, there is a horizontal asymptote at `y=a_n/b_m` where `a_n,b_m` are the leading coefficients of the numerator and denominator respectively when written in standard form.

The horizontal asymptote is y=1.

(3) We are asked to find four ordered pairs that are on the graph of the function.





`f(0)=8/3`  (This is the y-intercept.)



So some points on the graph include:



The graph:

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