drawing graphs (1 Viewer)

[mojako]

Originally posted by CM_Tutor
Mojako, it isn't necessary to transform this equation to see a max tp at the origin - Any even function which is continuous and differentiable at x = 0 must have a turning point at its y-intercept. Similarly, any odd function that is continuous and differentiable at x = 0 must have an inflexion at the origin.

Didn't know that (or at least, didn't realise).

Originally posted by CM_Tutor
Also, I'm not a Capital-T Teacher - I don't work at a school. I am a PhD student at USyd, studying Chemistry Education, and at the same time doing a Masters degree in Education in the field of Teaching and Curriculum Studies.

Ohh sorry. I got the impression from one of the other posts that you were a teacher, or a private tutor at least.
What do you study in the Masters degree in Education in the field of teaching and curriculum studies (a long name...)? What can the graduates (I mean, those who complete the Masters degree) work as?

Originally posted by Grey
btw, how do you know if an equation is even or odd? I mean, I know that if f(x) = f(-x) then its even, but how do you see something like x^2 / (x^2 - 4) and know that its an even function?

Here is one way I'm aware of. CM_Tutor may add.
-> Of course you can find f(x) and then find f(-x) and see if f(-x) equals f(x) or -f(x). (Remember the lesson on that?)
-> By observation, a function will definitely be odd if all the x-terms are raised to an even power (2, 4, 6) or to even roots (or whatever it's called) like square roots, fourth root...
-> A function will definitely be odd.. hmm (not sure about "definitely") ... I think.. if all x-terms are raised to odd powers or odd roots (remember, constants like "4" is "4*x^0", and 0 is an even power).

Well, I don't think 0 is even by definition, but just consider it to be so for this purpose.

 

[Grey Council]

but if you take:
y = (x-2)^2
thats not even.

actually, its x^2 - 4x + 4, heh. all x's HAVEN'T been raised to even powers.

i see. Its kinda obvious, if you think about, hey?

 

[CM_Tutor]

Originally posted by Grey Council
*adds the exercise to the to-do list over the weekend*

and i dunno about CM_Tutor not being qualified to teach, he prolly explains stuff in maths better than any person i've ever met. Its a wonder he can manage to do it over the net as well. heheh, so mojako, just take his word like the bible. if he says something, its right. ^____^

Thanks for the compliment, I've had a lot of practice at explaining. I'm glad it shows amd is helpful. However, please don't accept everything I say as true without question - we all make mistakes, and I don't have a problem with people asking me to explain why I said something, nor with pointing out an error / suspected error.

btw, how do you know if an equation is even or odd? I mean, I know that if f(x) = f(-x) then its even, but how do you see something like x^2 / (x^2 - 4) and know that its an even function?

And what about odd functions?

f(x) = x² / (x² - 4)
f(-x) = (-x)² / [(-x)² - 4] = (-1)²x² / [(-1)²x² - 4] = 1 * x² / (1 * x² - 4) = x² / (x² - 4) = f(x)
Hence, even.

Odd is the same in methodology. For example, you could try showing that f(x) = x³ / (x² - 4) is odd

In general, for straight polynomials:
All degrees zero or even ---> function is even
All degrees odd ---> function is odd
Mixture of odd and even degrees ---> function is neither odd nor even

For products / quotients of odd and even functions (whether polynomials or not):
Even * Even = Even
Even * Odd = Odd
Odd * Odd = Even

Even / Even = Even
Even / Odd = Odd
Odd / Even = Odd
Odd / Odd = Even

For calculus:
If f(x) is even, then f'(x) is odd
If f(x) is odd, then f'(x) is even

If f'(x) is odd, then f(x) is even
If f'(x) is even, then f(x) may or may not be odd

If you aren't sure about these, try proving them using functions f(x) and g(x).

 

[Grey Council]

^__^

Thanks, I see now. I have been enlightened. hehe, hopefully i'll redeem my lazy ways, and be able to conquer graphing once and for all.

 

[CM_Tutor]

Originally posted by mojako
Ohh sorry. I got the impression from one of the other posts that you were a teacher, or a private tutor at least.
What do you study in the Masters degree in Education in the field of teaching and curriculum studies (a long name...)? What can the graduates (I mean, those who complete the Masters degree) work as?

I am a private tutor - I have thousands of hours of experience in tuition, which is why I know what I'm talking about with the HSC.

As for the MEd (shorter name! ), I'm doing it for reasons of credibility. I'm doing a PhD in Science, and have no fromal education qualification, but am researching an education topic. Getting an MEd - which is a post-graduate qualification normally only open to people with an undergraduate education qualification - will give me a qualification that will make me credible to both education and science researchers. Most of the candidates are Teachers (in the capital T sense) looking to upgrade their qualifications, either for promotion, or to move into administration (or in a couple of cases, to do a doctorate and get out of the teaching profession). Some are researchers, looking to do an education PhD or an EdD. Some may want to go on to University positions, and figure it will help them be more employable.

The reason for the long name is that you can get an MEd in a variety of fields - like TESOL, or Higher Education, or Research Methodology, or Educational Psychology, or ..., or even get a general MEd, which means combining subjects from a number of areas, but not enough from any one area to be qualified in that area. My area is Teaching and Curriculum Studies - which covers pretty much what it sounds like - theory of curriculum design and syllabus structure, some educational psychology theory (like psychology of learning, thinking and knowing, and the teaching of thinking skills), and I'm also doing some research methodology stuff, as I haven't done qualitative research before. Hope this answers your question.

 

[mojako]

>> If f'(x) is even, then f(x) may or may not be odd <<
Why? because of the possible constant term of integration.

Also, regarding my previous post, I think the x-term rule is only true for positive integers powers (i.e. polynomials). (ignore the things about roots...)
If it's a general polynomial function (this is a syllabus term, btw, which means polynomial with any rational indices... [normal] polynomial must have indices positive integers), then the rule about indices of x-terms doesn't always hold. eg, y=square-root-of(x), ie y=x^0.5, is of course not odd and not even.
Also, y=square-root-of(x^2) will be even, btw. Just an interesting point as it doesn't equal y=x lol.
This'll be my last post today... got exam 2morow and havent realy studied I'm not very efficient in time management.

 

[mojako]

In reply to CM_Tutor:
Ok I get it. And will try to remember that
need to study for chem exam now
BTW, the reason I know about the maths syllabus is that I'm making a formula summary for maths (and will put in in boredofstudies), and in the Extension2 class we always go over the syllabus at the end of lesson. That's when I first pay attention to "that wordy docuement".

 

[CM_Tutor]

>> If f'(x) is even, then f(x) may or may not be odd <<
Why? because of the possible constant term of integration.

Yes - and I've seen this asked on (I think) a recent HSC, but I don't remember which one...

Also, y=square-root-of(x^2) will be even, btw. Just an interesting point as it doesn't equal y=x lol.

Absolutely correct, as y = sqrt(x²) = |x|, which is even.