Graphing functions with x in the denominator (1 Viewer)

[nickyroony]

The thing that I have most trouble with in the extension topic is graphing. It's never been my strong point, but I do have some idea of how to do it. I can graph easy ones no problems, but when it comes to harder ones I struggle a bit. My teacher gave up step by step outline on how to tackle the question (eg: 1. finding domain and range. 2. finding asymptotes etc). Is that the right way to do it? Any tips on how you guys figure those graphs out?

I want to be able to get to the stage where I can just look at the equation and have some idea of what it would look like.

 

[Riviet]

Here are some techniques that you can use (preferably in the order given) when sketching any curve y=f(x):

1. Finding the domain and range is always a good start. For functions with x in the denominator, try factorising it if it is of degree 2 or higher, e.g x/(x²-1) can be factorised to x/[(x+1)(x-1)], and therefore the domain would be x=/=+1. If it can't be factorised let the entire denominator equal zero to find domain in which x is undefined.

Once you have the domain, consider the range, usually harder to spot from first glance, there are tricks to finding the range such as making x the subject.

2. Find any x or y intercepts by letting x=0 to find y intercepts or y=0 to find any x intercepts. If doing this results in a zero in the denominator, then you have an asymptote at x or y=0

3. Find out whether the function is even, odd or neither. For a random function y=f(x), if f(-x)=f(x), then the function is even. If f(-x)=-f(x), then the function is odd. If it's not odd or even, then it has to be neither.

So if the function is even, then the graph is symmetrical about the y-axis and this helps immensely because you only need to sketch half the graph and can then reflect the whole graph about the y-axis.

If the function is odd, then the graph has rotational symmetry. This means if you take one side of the function and rotate it about the origin, it will map onto itself. A good example is y=x³

4. Find the stationary points by f'(x)=0 and determine their nature by testing the points with the first derivative or substituting into the second derivative.

5. If there are parts of the graph you are unsure, substitute points in to find out where the graph goes!

I hope that helps.

 

[bananasmoothy]

To summarise what Riviet said (ok, I admit it, I just skimmed it. Sorry! )

• where function is +ve/-ve/0
• domain/range
• increasing/decreasing
• odd/even
• asymptotes
• any points of discontinuity?

Basically, just look at equation, figure out odd/even, asymptotes, and intercepts. Sketch it. If it seems good, you're pressed for time, and it's a low-mark question, leave it. It's good enough.

If there's a particular question bothering you, post it up.

Oh, just realised: thread topic is about x in the denominator. That will have an asymptote - work out the equation so that there's no x in the numerator, then take denominator to other side. Did you want me to post an example question of this?