the title says it all. I'm pretty sure this is a hyperbola, but is there an alternative way of doing this besides a table of values?

"Graph the equation $y=\frac {x-2}{x+1}$"

I know that $x$ cannot equal $-1$ but I'm not sure how to carry on from there.

Any help would be appreciated. If you could provide a step by step explanation, and a picture of the graph itself, that would be great :D!

  • $\begingroup$ Do you mean to show it? $\endgroup$ – Ofir Attia May 20 '13 at 8:36
  • $\begingroup$ I just need to graph the equation, but an explanation on how to do it would be nice too :D $\endgroup$ – missiledragon May 20 '13 at 8:36
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    $\begingroup$ $y$ can indeed equal 0, i.e, when $x = 2$. ;) $\endgroup$ – user49685 May 20 '13 at 8:38
  • $\begingroup$ OH, I didn't realise that :P. Thanks @user49685 $\endgroup$ – missiledragon May 20 '13 at 8:40

When you want draw the graph of the function, you need to check a couple of things:

  1. Check if there are any points in the domain where the function is undefined, you know there's something happening there. For instance here you saw that when $x=-1$ it is undefined.

  2. Then, $y$ can be equal to $0$, there's no problem with that, and you should find all the $x$ such that $y=0$. In this case, when $x=2, y=0$. Thus you know that the function goes through the point $(2,0)$.

  3. Now you have all the points where there's something special happening, you can check where the function is positive, and where it is negative. You need to look at all the intervals between the special points. In your case : $(-\infty,-1),\,(-1,2),\,(2,\infty)$. Take $(-\infty,-1)$ for example. On this interval, you know that the numerator is always negative and the denominator is always negative as well, so the value for $y\in(-\infty,-1)$ will always be positive. Do that for all the intervals.

  4. Finally, you need to check some limits to see what happens at those special points and also when you tend to infinity (in your case only $-1$ is a special point because you already now that the value of your function when $x=2$ is $0$). So you should check :

$$\lim_{x\rightarrow\infty}f(x),\; \lim_{x\rightarrow-\infty}f(x),\; \lim_{x\rightarrow-1^+}f(x),\; \lim_{x\rightarrow-1^-}f(x),\;$$

That's it, you have everything you need to graph your function! (If you want to go into further details you can also check the derivative for extremas and the second derivative for inflexion points. This means that you have to compute the derivatives and check for which values of $x$ they are equal to $0$).

  • $\begingroup$ Thanks! that helped me a lot :D! $\endgroup$ – missiledragon May 20 '13 at 9:06

Rewrite as $y=1-3/(x+1)$, or $(y-1)= -3/(x+1)$

This is simply the regular recriprocal equation, with the axies rescaled. The standard $Y$ axis, representing $X=0$, is now marked $x=-1$. That is, the x-numbers are moved one to the right. The old X axis of 0,1,2,3 become -1, 0, 1, 2...

The old $X$ axis, or $Y=0$ becomes $y=1$, and we have to rescale by a factor of 3. So the old Y coordinates 0, 1, 2, 3... become 1, -2, -5, -8...

And that's it.


Full Function investigation, check of asymptotic (infinite and non setting points), extreme points and inflection points. nice tool to work with is desmos:
$y=(x−2)/(x+1)$ - Click to view

In addition : Investigating Functions - Click to view

hope its helped you.


Just to add to what Oliver and Wendy have said, To Draw any function:
1.)Check Domain and critical points.
2.)Check intercepts.By putting $x=0 ,y=0$
3.)Check Asymptodes around infinity and critical points by taking $$\lim_{x\rightarrow\infty}f(x),\; \lim_{x\rightarrow-\infty}f(x),\; \lim_{x\rightarrow-1^+}f(x),\; \lim_{x\rightarrow-1^-}f(x),\;$$ 4.)Check Symmetry about axis by putting $x = -x$ and $y=-y$
5.)Intervals of increase/ decrease by equating first derrivative to zero.
If you know all this information you can draw any graph manually.

Hope this helps :)


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