Plot the following function and find the equation of any asymptotes [10 marks]
could somebody please explain the answer and method using repeated factorisation used here in Q7) (pdf file)
It's a less obvious method of doing long division.
Effectively you work out how many times the highest power of the divisor (the bit on the bottom) goes into the highest power of the dividend (the bit on the top).
In this case it's "x" times.
Then you multiply the bottom by that value, and subtract it from the top.
Thus you have "x + ..." where what you've got left after you've subtracted that "x" is a polynomial of a reduced index (because you've extracted as many "x^2 - x - 2" factors out as you can.
Then you do the same thing again, each time you're reducing the order of the polynomial at the top till it has a smaller order than the one at the bottom.
Notice how "x^3" becomes "x(x^2 - x - 2) + x^2 + 2" in the first line, multiply it out and you see it comes to x^3.
Oh yeah, I notice the "polynomial long division" at the bottom as well, that's doing in symbols what I did with words above.
thanks matt, I eventually ended up doing the polynomial long divison method as it seemed easier. One more question about this, my answer looks identical to the polynomial method answer in the link in the first post but now that I worked out the final answer - how do I use (-3x+11) and (x/4 + 3/4) to draw the graph? and how do I conclude that there are vertical asymtotes at x = -1 and 2 from the answer that I got, I just factorise the divisor right? Thanks a lot for your help btw
We were given the "Sidamo" procedure when plotting graphs:
S - Sign of y for given x. Work out where the zeroes are and work out where it changes sign.
I - Intercepts. Work out where it crosses the axes.
D - Derivatives (although you haven't done calculus yet so you won't have this in your toolbox).
A - Asymptotes - that's the lines the function gets near but doesn't reach when the function gets very big in whatever direction.
M and O I can't remember, maybe someone else can.
To get a rough idea of the asymptotes, see what happens when x tends towards plus and minus infinity - in this example the constants 11 and 3/4 can effectively be ignored (as they'll reduce in importance relative to x as x gets bigger). So see what you have left and that may give you an idea as to where the asymptotes are. This is of course a horrribly inexact way of looking at things but it gives you a rough idea.