Graphs: help! (1 Viewer)

[big crocodile]

Hi there, I'm a newbie here so if i say anything wrong pls don't kill me. I'm doing ext 2 this year. Well, I'm not so good at maths but I really like it. Now in ext 2 we are doing graphs and I am just plain lost.

I'm confused how to start when doing the graphs questions. Like do you need to find 1st and second differentiation for the function, or find its symmetry or what else should you do. I don't know how the other can figured out the shapes of the curve so quickly, especially the trig and exponential ones.
Also, when we do the addition/subtraction functions, some people just draw two initial graphs then figure out the graphs right away!

So, I just wanna ask if anybody has any tips on this topic. Thanks!

 

[Shadowdude]

Hi there, I'm a newbie here so if i say anything wrong pls don't kill me. I'm doing ext 2 this year. Well, I'm not so good at maths but I really like it. Now in ext 2 we are doing graphs and I am just plain lost.

I'm confused how to start when doing the graphs questions. Like do you need to find 1st and second differentiation for the function, or find its symmetry or what else should you do. I don't know how the other can figured out the shapes of the curve so quickly, especially the trig and exponential ones.
Also, when we do the addition/subtraction functions, some people just draw two initial graphs then figure out the graphs right away!

So, I just wanna ask if anybody has any tips on this topic. Thanks!

Uhh, it's really just common sense. What you do is, you know your basic graphs, and then you fudge and tinker with them to get the graph you want.

Ideally, for most graphing questions, you use no calculus at all. You just 'know' things about basic graphs that you can apply.

Also trig and exponential curves? They don't "know them" quickly, they learn them off by heart. You should too.


Post an example question.

 

[jnney]

I had the same problem.

You need to stop freaking out and start thinking logically. Take it slow. You use whatever method that you think you need to sketch that graph. Sometimes calculus is necessary to locate the exact point/s of max/min or POI. Sometimes if a graph is an even function, it would be easier to figure out what the left/right hand side looks like, and then reflect it instead of doing extra calculations.

People can figure out the shapes of curves easily because they've done a lot of questions. With practice, you can too.

 

[Shadowdude]

If you have to identify min-max points and the like, then yes - you use calculus. But 95% of the questions I got were just "sketch the graph".

You figure out the shapes of curves easily by knowing lots of 'basic' curves. Like if you have 1/x^6 , that's going to be like 1/x, for example.

 

[TheCardician]

I agree with Shadow - you need to know your basic graphs i.e. trig functions, quadratic, exponential, cubic, logarithmic, hyperbolic etc. If the graphing happens to be quite obscure, you can just plug in x values and form a pattern. Also take into account as the graph goes to infinity (x -> infinity); this will tell you where your asymptotes are. If you want to draw a 'rough' diagram of the function, what you can do is plug in values close to an asymptote or axis, and values further away from them; eg. if the asymptote is at x = 2, try sub in x = 2.001 and x = 1.999 to help see the pattern. Also finding x and y intercepts are of help.

For graphing, it's all about practice and using your understanding of manipulating algebra to solve equations in a particular fashion and graph them.

 

[Shadowdude]

I agree with Shadow - you need to know your basic graphs i.e. trig functions, quadratic, exponential, cubic, logarithmic, hyperbolic etc. If the graphing happens to be quite obscure, you can just plug in x values and form a pattern. Also take into account as the graph goes to infinity (x -> infinity); this will tell you where your asymptotes are. If you want to draw a 'rough' diagram of the function, what you can do is plug in values close to an asymptote or axis, and values further away from them; eg. is the asymptote is at x = 2, try sub in x = 2.001 and x = 1.999 to help see the pattern. Also finding x and y intercepts are of help.

For graphing, it's all about practice and using your understanding of manipulating algebra to solve equations in a particular fashion and graph them.

^ That, pretty much.

 

[tohriffic]

Since no one has mentioned this yet, when you add/subtract graphs you can draw the two graphs you need to add/subtract and from there you manipulate the y values. In addition, you add the y values together at that x point, though I recommend only adding the "nice" y values. In subtraction, you minus the y values according to what is subtracting what.

Remember, it's all about practice so don't fret if you don't get it straight away, we all started somewhere.

Good luck!

 

[Kimyia]

1. x-intercepts, y-intercepts
2. odd/even function
3. vertical/horizontal asymptotes
4. domain and range
5. first and second derivatives
6. if all else fails, table of values.
...mightn't be so good for ext2 but hey, its a start
Note: not in any particular order..'cept no.6

 

[big crocodile]

Whoa.. Thanks you guys so much. Its been a great help and Ive read through all the replies.
And yeah, what I do is making a huge table of values (which takes a lot of time) then therefore figuring out the trend of the graphs. I've learned the shapes of basic functions. Yet I still dont know how to combine and subtract them without making the table of values, like how to do know it in the middle of the 2 initial graphs of on the left/right hand side of them. Also, there are some asymptotes which are not straight lines - how can you knows the asymptotes if they are not straight lines? And in order to find the asymptotes, you have to do a bit of calculus, dont you?.. well Im still practicing now and assessment is next week! so nervous ..
and again, thank you guys all. im really appreciate

 

[nightweaver066]

Whoa.. Thanks you guys so much. Its been a great help and Ive read through all the replies.
And yeah, what I do is making a huge table of values (which takes a lot of time) then therefore figuring out the trend of the graphs. I've learned the shapes of basic functions. Yet I still dont know how to combine and subtract them without making the table of values, like how to do know it in the middle of the 2 initial graphs of on the left/right hand side of them. Also, there are some asymptotes which are not straight lines - how can you knows the asymptotes if they are not straight lines? And in order to find the asymptotes, you have to do a bit of calculus, dont you?.. well Im still practicing now and assessment is next week! so nervous ..
and again, thank you guys all. im really appreciate

Asymptotes are not found by calculus.

To find vertical asymptotes, write down the natural domain noting down any restrictions (0 in denominator, negative in square roots and anything else).

To find horizontal asymptotes (only applicable if it is in the form f(x)/g(x) where degree of f(x)<= g(x) ), take the limit to positive and negative infinity.

To find oblique or any other kind of asymptotes (only applicable if it is in the form f(x)/g(x) where degree of f(x)>g(x) ), perform long division.

I probably missed a few things so wait for someone else to clear up what i've said or to add more..

 

[D94]

You are rarely asked to find the derivatives, because the whole point of 4U curve sketching is to avoid differential and integral calculus.

Just consider:
1. Vertical Asymptote
2. Horizontal Asymptote
3. Oblique Asymptote
4. Intercepts
5. Nature of curve as it approaches the Vertical Asymptotes on both positive and negative sides
6. Check if the curve cuts the Horizontal Asymptote

 

[big crocodile]

Thanks guys! I think I need more practice BTW, do you have to show all the working out when finding asymptotes and intercept?

 

[D94]

Thanks guys! I think I need more practice BTW, do you have to show all the working out when finding asymptotes and intercept?

When you practice these questions, it becomes second nature to write out the working out methodically, so yeah, show all working (even though it's like 2 lines for each criteria).

 

[big crocodile]

When you practice these questions, it becomes second nature to write out the working out methodically, so yeah, show all working (even though it's like 2 lines for each criteria).

but for the vertical and horizontal asymptotes you can figure out right away but to show the working you have to go through the limits - which is time consuming , rite?

 

[D94]

but for the vertical and horizontal asymptotes you can figure out right away but to show the working you have to go through the limits - which is time consuming , rite?

Hardly time consuming. Sure, you can just write them out, eg. VA: x = +/-5; HA: y = 2, just don't write them out incorrectly. Depends on the question; if you're just starting out, you'd be drawing some pretty basic graphs, but they can get pretty large.

What if the denominator of the expression was some quadratic which required the quadratic formula? I reckon it would be faster doing the working out than doing it in your head.

 

[big crocodile]

Here, I would do an extremely simple question to depict my problematic problem




<a href="http://s172.photobucket.com/albums/w31/nguyenthienkhue/?action=view&amp;current=1-2.jpg" target="_blank"><img src="http://i172.photobucket.com/albums/w31/nguyenthienkhue/1-2.jpg" border="0" alt="Photobucket"></a>

okay, this is EXACTLY the 3 units method which is really time consuming. Also when I find the horizontal asymptotes, I cant figure it out right away. How would you do that?
And whenever I try to use to 4 units method, all my attempts are fail. For example in this question, after finding the asymptotes and 2 function and I have no idea how to make division between them:

<a href="http://s172.photobucket.com/albums/w31/nguyenthienkhue/Maths%20ect/?action=view&amp;current=2.jpg" target="_blank"><img src="http://i172.photobucket.com/albums/w31/nguyenthienkhue/Maths%20ect/2.jpg" border="0" alt="Photobucket"></a>

yo guys have any idea to help me out? thanks

 

[nightweaver066]

Shortcut for horizontal asymptotes.

If the highest degree in the numerator is the same in the denominator, take the coefficients of them and there's your horizontal asymptote.

If the highest degree of the numerator is lower than the denominator, the horizontal asymptote will be y = 0.

Also, you have your vertical and horizontal asymptotes said the wrong way around lol.

I don't think you need to go in to that much detail unless it asks for all important features.. Not too sure though.

 

[cutemouse]

okay, this is EXACTLY the 3 units method which is really time consuming. Also when I find the horizontal asymptotes, I cant figure it out right away. How would you do that?

If deg of numerator = deg of denominator then a horizontal asymptote exists. It is then given by y=(coeff of highest term in numerator)/(coeff of highest term in the denominator). So in your example it would be y=(-1)/1 ie. y=-1.

And you don't do division of graphs usually. In Graphs you mainly look at the transformations y=(f(x))^2, y^2=f(x), y=g(f(x)) (where g(x) is usually sinx, cosx or ln x) and y=1/f(x).

You however, also could be asked to do addition or multiplication of ordinates.

 

[big crocodile]

Also, you have your vertical and horizontal asymptotes said the wrong way around lol.

yeah I just fixed it.

 

[SpiralFlex]

Another method to further lessen the time, when differentiating do it quickly in your head like this.





As tends to positive and negative infinite, you will see your horizontal asymptote to be




Also, another tip throughout my time of graphs. When you have or an "always positive" value as your denominator. Your graph will usually generally look like a "mountain".


For your division problem,

It is not a good idea to do so, but if you really want...and have lots of spare time.