## IsTim 4 years ago how to graph g(x)=3x^4=3x^2

1. IsTim

I'm looking thru my old notes now, but I can't find anything that could help.

2. IsTim

I simplified the equation so: g(x)=3x^2(x^2+1)

3. IsTim

I was thinking apq, but I don't know if that applies to this.

4. sasogeek

why do u have 2 equal signs to one function?

5. IsTim

Oops.

6. IsTim

7. IsTim

$g(x)=3x ^{4}-3x ^{2}$ If you want a cleaner version.

8. anonymous

use a graphing calc

9. anonymous

3x^2(x^2-1) you plug in points frankly :-/ it looks like a x^2 graph.

10. IsTim

So I just plug in values? That's feels "brute". There's no other way?

11. IsTim

@ Mario; I'm studying for an exam. I don't get those.

12. anonymous

we get to use graph calcs on my exams

13. anonymous

plug in pounts. their are zeroes at 0, 1,-1.

14. anonymous

the rest is just brute force plugging. yes. that's how it works.

15. IsTim

Oh well. I was looking for some equation rearranging. Would that work?

16. IsTim

Lucky Mario.

17. anonymous

yet i still did poorly lol

18. JamesJ

19. IsTim

IF possible, please give an explaination of how to dervie the graph from teh equatioon.

20. JamesJ

Ok. If g(x)=3x^4-3x^2 first we'd like to know the zero; i.e., the intercepts on the x-axis. Setting g(x) = 0, we have $3x^2(x^2 - 1) = 0$ Hence $x = 0, \pm 1$ Next, what's the y-intercept: y = g(0) = 0. Next, critical values ...

21. JamesJ

$g'(x) = 12x^3 - 6x = 0$ if and only if $6x(2x^2 - 1) = 0$ i.e., $x = 0, \pm 1/\sqrt{2}$ The second derivative is $g''(x) = 36x^2 - 6 = 6(6x^2 - 1)$ ...

22. JamesJ

It's not hard now to show that x = 0 must be a local max and $x = \pm 1/\sqrt{2}$ are local mins. So now we have the behavior of the function in the interval [-1,1] What happens outside that?

23. JamesJ

The next thing we observe is that g(x) is an even function, g(-x) = g(x) meaning the function is symmetric about the y-axis. As $g(x) = 3x^2(x^2 - 1)$ it is clear that for $$x > 1$$, $$g(x) > 0$$ and as $$x \rightarrow \infty$$, $$g(x) \rightarrow \infty$$. We now have all the information we need to draw the graph and it's consistent with the picture I posted above. Make sense?

24. IsTim

It's higher level information that I don't understand, but I think if I read thru it a bit more, I'll understand. Thank you very much.