Is This Function a Variation of a Hyperbola?

[dagg3r]

hi i got this equation
y= x^2 - 2x + 1 / X^2 -x - 2

how do i sketch this finding all intercepts, and asymptotes with a gfx calculator? Please check if the steps i did below is right

what i did i factorised the equation so i got

y= (x-1)^2 / (x+1)(x-2)

ASYMPTOTES
the bottom line (x+1)(x-2) = 0 gives me the vertical asymptotes thus
x= -1 and x=2

X-Intercept
to find the x-intercept i let y=0 thus (x-1)^2
X=1

Y-Intercept Let X=0
y=-1/2

I got those values but how do i sketch the graph it looks weird on the gfx calc is this a variation of a hyperbola? i also heard you can do stuff like use polydivion and get another equation or break it up and sketch it or something like that?

 

[SpaceTiger]

I got those values but how do i sketch the graph it looks weird on the gfx calc is this a variation of a hyperbola? i also heard you can do stuff like use polydivion and get another equation or break it up and sketch it or something like that?

How precise does the graph need to be? If I were doing this for a class, I would just calculate all of the interesting points and trends and then connect the dots. For example, what happens as x goes to inf and -inf? Does y go to +inf or -inf at the critical points? I'm not sure why it won't plot on your calculator, though. Do the infinities cause problems? If so, have you tried restricting the range?

 

[codyg1985]

The horizontal asymtote needs to be calculated as well.

To do this, I am going to use limit calculus.

$$y = \lim_{x\rightarrow\infty} \frac{x^2 - 2x + 1}{x^2 - x - 2}$$
$$y = \lim_{x\rightarrow\infty} \frac{\frac{x^2}{x^2} - \frac{2x}{x^2} + \frac{1}{x^2}}{\frac{x^2}{x^2} - \frac{x}{x^2} - \frac{2}{x^2}}$$
$$y = \lim_{x\rightarrow\infty} \frac{1 - \frac{2}{x} + \frac{1}{x^2}}{1 - \frac{1}{x} - \frac{2}{x^2}}$$
$$y = \frac{1 - 0 + 0}{1 - 0 - 0}$$
y = 1

Another way to do this would be to switch x and y, solve for y, and see which values of x are undefined for that function, but the calculus method is quicker and easier IMO.