Hi some more coordinate geometry help needed

[StudentHelp]

Sorry for Spam if not allowed I will delete, I have an exam coming up but do not understand coordinate geometry at all. Struggling so much

I need to
B) sketch the curves y = (1/X-1) + 1 and y = (1/1-x) + 2 on the same set of exes, showing all essential features. Using this diagram or otherwise, find any points of intersection of the two graphs, clearly justifying your answer.

Ideally I would love the explanation of how to answer the question but if easier I can have the answer and try to learn how to get there.

Thanks for your help

 

[raquelc]

I believe the denominator on both is (x-1). The first thing to look at when we have fractions is that we can not have a 0 on the denominator. In both cases that will happen at x=1. you can draw a vertical line where neither of the functions will reach (vertical asymptote)
The second thing is to see the horizontal asymptote. What happens as the x gets bigger? The fraction will approach 0 (1 divided by+- infinity is 0). That will show us that the horizontal asymptote on the first case is y=1 and on the second case y=2.
Looking at the first case, we have a vertical asymptote at x=1 and a horizontal at y=1. You can draw those, making 4 quadrants. When x is bigger than 1, the function is bigger than 1, so we will have a line that goes from the top right of the line x=0 curving down and going to the right approaching to the y=1. When x is smaller than 1, the function is smaller than 1, so you draw a line that goes from the bottom left of x=1 and curving up and to the left approaching to y=1

You will be doing similar for the other function but instead of y=1, the asymptote is at y=2

 

[Dr.Peterson]

B) sketch the curves y = (1/X-1) + 1 and y = (1/1-x) + 2 on the same set of exes, showing all essential features. Using this diagram or otherwise, find any points of intersection of the two graphs, clearly justifying your answer.

There are a couple different ways to work on this kind of problem, so it would help to know what techniques you have been learning. I'm going to guess that these are meant to be approached as transformations of the graph of y = 1/x.

Assuming the first is meant to be [MATH]y = \frac{1}{x-1}+1[/MATH], which should have been written as y = 1/(x-1) + 2, do you see horizontal and vertical shifts (translations)? Sketch the basic function [MATH]y = \frac{1}{x}[/MATH], with its horizontal and vertical asymptotes, then do the appropriate shifts, which will move them.

Assuming the second is [MATH]y = \frac{1}{1-x} + 2[/MATH], I would start by rewriting it as [MATH]y = -\frac{1}{x-1} + 2[/MATH], and then take the same approach. This time there is also a reflection.

To find intersections, just solve the equation [MATH]\frac{1}{x-1}+1= \frac{1}{1-x} + 2[/MATH].