Math Is Fun Forum

Daniel123

Member

Registered: 2007-05-23

Posts: 663

graph sketching

How would you go about sketching:

and

???

Thanks.

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: graph sketching

I can see, for the first, that as x approaches 1, y approaches infinity... and it would cut the x axis at 0 and -1? what would the shape of the curve be?

Last edited by Daniel123 (2007-12-16 10:04:23)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: graph sketching

The main things you need to know to sketch a graph are its roots, poles and how its sign behaves (and possibly also turning points).

You've started well by saying that it reaches the x-axis at 0 and -1, and saying that it approaches infinity as x approaches 1, so the next thing to do is find out how its sign behaves.

To do that, split it up into different cases, where the roots and poles are the boundaries.
Here, we have four cases:

(i)     x<-1
(ii)    -1<x<0
(iii)    0<x<1
(iv)   1<x

By working out whether x-1, x, and x+1 are positive or negative in each of those cases, you can work out whether the function is positive or negative.

Using that, you can find what the function does as x goes to the infinities (+ and -), whether the function cuts through the axis or just touches it at each root, and which infinity it goes to at either side of each asymptote.

If you've learnt differentiation, you can also find the turning points of the function. It's tricky to find them without that though.

---

Why did the vector cross the road?
It wanted to be normal.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: graph sketching

You can check if your sketch is right here.

As for the y² one, that's basically the square root of the function you just drew.
Therefore, the curve will be roughly the same as last time (the roots and asymptotes will be in the same places, and the values mostly don't matter because it's a sketch) but the important part is that wherever the original function was negative, now it won't exist.

---

Why did the vector cross the road?
It wanted to be normal.

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: graph sketching

Aah right thank you, that really helped.

I have learnt differentiation, but only basic differentiation - I wouldn't know how to differentiate that equation.

thanks again.