Hi, I'm having trouble finding the asymptotes, turning points, inflection points etc to draw these graphs
Hey Goodwil! I'm happy to help, I'll help you find the asymptotes for both, then I'll go through the full process for the second one, hopefully you can apply it to the first
Okay, so vertical asymptotes exist where the function is undefined. In both of these cases, that means when the denominator is equal to zero. So, we can do it by inspection if we want, or alternatively:
These are the vertical asymptotes! Let's do the turning points and such for the second function. First, we differentiate using the quotient rule:
To find the turning points, we put this equal to zero (note in the second step we discard the denominator because it can't be equal to zero):
Now we'll need our second derivative for the next step, but I'll be honest, that derivative looks sort of gross. It would be another application of chain rule, but I'm going to use magic (Wolfram Alpha) to get the answer:
Okay, so we do a few things with this. First, we substitute in our x values from earlier to find the nature of the turning points. Doing so, we find that:
We also need to put the second derivative equal to zero to find any points of inflexion:
We already know about this point, but at least we know there aren't any more
Putting all of that information together, and perhaps plotting a few points and considering some limiting cases near the asymptotes, leads us to this graph here. Notice the asymptotes in the correct spots, the inflexion in the middle, and the two max/min points
So that's the process, it will be an exact replica for the first question! Asymptotes, first and second derivatives, turning points, inflexions, and then sketch
This sort of question, besides the nasty second derivative, is quite standard in MX1 exams so it is handy to know this process really really well does this help?