Question

how to graph g(x)=3x^4=3x^2

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

why do u have 2 equal signs to one function?

Answer 5

Oops.

Answer 6

g(x)=3x^4-3x^2 My bad.

Answer 7

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

Answer 8

use a graphing calc

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

we get to use graph calcs on my exams

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

Lucky Mario.

Answer 17

yet i still did poorly lol

Answer 18

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

Answer 19

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 ...

Answer 20

\[ g'(x) = 12x^3 - 6x = 0 \]
if and only if
\[ 6x(2x^2 - 1) = 0 \]
i.e.,
\[ x = 0, \pm 1/\sqrt{2} \]

The second derivative is
\[ g''(x) = 36x^2 - 6 = 6(6x^2 - 1) \]

...

Answer 21

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?

Answer 22

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?

Answer 23

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.