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AQA FP1 - Graphs of rational functions

I don't have a graphing calculator and GraphSketcher online doesn't always give the full graph.

How do you know what shape the graph is between the asymptotes or if, indeed, there is one?

I haven't looked at any past papers yet. Perhaps the bit in the middle isn't important as long as all the axes/asymptotes intersection points are labelled and the shapes OK.

I look forward to advice and info. I am self studying so can only be guided by TSR.

Thanks
Reply 1
You need to learn the shapes of the graphs of different types of function: reciprocal, quadratic, cubic etc. This will allow you to identify the shape of the graph by identifying the type of equation that you have.

As for intersections with the axes, they are found by setting x = 0 or y = 0 and solving for the equation for the other variable.

Asymptotes are usually present in a 'division by 0' situation, where the function has no defined value. By setting a value of the variable in a reciprocal function such that you have a 'division by 0', the asymptotes can be found.

In some functions such as cubic and quartic functions, it can be difficult to know whether the graph is 'rising' or 'falling', and simply finding the intersections will not help you with this. Consider the value of y when x is very large and negative and very large and positive to figure this out .

Learning the general shapes of different types of rational function and finding the asymptotes, intersections and general shape in the way described is usually all you need to know sketch a graph of a rational function.
(edited 10 years ago)
Original post by VannR
You need to learn the shapes of the graphs of different types of function: reciprocal, quadratic, cubic etc. This will allow you to identify the shape of the graph by identifying the type of equation that you have.

As for intersections with the axes, they are found by setting x = 0 or y = 0 and solving for the equation for the other variable.

Asymptotes are usually present in a 'division by 0' situation, where the function has no defined value. By setting a value of the variable in a reciprocal function such that you have a 'division by 0', the asymptotes can be found.

In some functions such as cubic and quartic functions, it can be difficult to know whether the graph is 'rising' or 'falling', and simply finding the intersections will not help you with this. Consider the value of y when x is very large and negative and very large and positive to figure this out .

Learning the general shapes of different types of rational function and finding the asymptotes, intersections and general shape in the way described is usually all you need to know sketch a graph of a rational function.


I'm looking at rational functions with 2 distinct linear factors in the denominator. I've got how to find the asymptotes and then I'm able to use use the x = 0 and y = 0 to find other points and then I can also find the crossing point(s) of the horizontal asymptote and, generally, I get the shape OK. However, in the book there is sometimes a graph between the vertical asymptotes and this is the one that I need help with.

So far, in the book, I've see one that looks like the graph of a tan and another that has an upside down quadratic. Would you think that I just need to work out the y values of x values the lie between the asymptotes and sketch from that?

Thanks
Reply 3
Original post by maggiehodgson
I'm looking at rational functions with 2 distinct linear factors in the denominator. I've got how to find the asymptotes and then I'm able to use use the x = 0 and y = 0 to find other points and then I can also find the crossing point(s) of the horizontal asymptote and, generally, I get the shape OK. However, in the book there is sometimes a graph between the vertical asymptotes and this is the one that I need help with.

So far, in the book, I've see one that looks like the graph of a tan and another that has an upside down quadratic. Would you think that I just need to work out the y values of x values the lie between the asymptotes and sketch from that?

Thanks


Post a few questions with your attempts and someone will give some pointers.
Here are two questions that I've had a go at.

maths.png




In these question should there be any graph between the vertical asymptotes?

I'm also attaching the example from the book which mentions all the points that I have tried to find but it also has a curve between the asymptotes which it does not account for in the workings.

maths.png

Thanks
Original post by maggiehodgson
I don't have a graphing calculator and GraphSketcher online doesn't always give the full graph.

How do you know what shape the graph is between the asymptotes or if, indeed, there is one?

I haven't looked at any past papers yet. Perhaps the bit in the middle isn't important as long as all the axes/asymptotes intersection points are labelled and the shapes OK.

I look forward to advice and info. I am self studying so can only be guided by TSR.

Thanks


Look to see what happens on either side of the asymptote.
i.e. for a vertical asymptote, does the function tend to positive or negative infinity on each side.
See pages 13-15 in Notes for C1-4 and pages 12-13 in notes for FP1-3 below.
See also worked example
Reply 6
You don't need a graphics calculator to use a calculator to find the shape of a function. Use the input your function into the table mode of your calculator and examine the values.
Original post by brianeverit
Look to see what happens on either side of the asymptote.
i.e. for a vertical asymptote, does the function tend to positive or negative infinity on each side.
See pages 13-15 in Notes for C1-4 and pages 12-13 in notes for FP1-3 below.
See also worked example



Thank you so much for those documents. It'll take a while to digest them but from a quick glance they seem to be addressing my problem.
Original post by VannR
You don't need a graphics calculator to use a calculator to find the shape of a function. Use the input your function into the table mode of your calculator and examine the values.



Thanks - another good idea that I'm going to try out.

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