How do you graph cubics and polynomial functions?

C

Chaim

Junior Member
Joined
Oct 18, 2011
Messages
58
I understand how to graph something like ax2+bx+c
Just by simply using the vertex formula (-b/2a)
Which allows me to find the certain
Therefore I can plug it into a(x-h)2+k and find out how to graph it

But for something like 7x3-21x2-91x+104
How would you graph this?
What are you suppose to look at.

or 7(x-3)2(x+5)4
I understand the zeros are 3, and -5
The problem I have with is that fourth degree.

May anyone help please? :)
Thanks.
 
Not really. Just find the interesting parts and chart points around there. The behavior as x increases, either positive or negative, eventually becomes less than interesting.

y = 7x3-21x2-91x+104

Roughly, 104/7 = 14ish, so it seems unlikely that we will find much of interest for x < -15 (it just goes down without bound) or x > 15 (it just goes up without bound).

A little investigation places zeros around x = 5, 1, and -3. This further narrows the interesting portion of the graph to about [-4,6]

The tiniest bit of calculus would pin mountains and valleys at around x = 3.309 and -1.309, so you'll want to plot some around there. It is a little astounding that we get (3.309,-173.435) and (-1.309,171.435), so the vertical scale might be a bit oppressive.

There you go. Plot things around those five places on the Domain, 5, 1, -3, 3.309, -1.309 and you will find just about everything interesting that there is to find about this particular polynomial.
 
tkhunny said:
Not really. Just find the interesting parts and chart points around there. The behavior as x increases, either positive or negative, eventually becomes less than interesting.

y = 7x3-21x2-91x+104

Roughly, 104/7 = 14ish, so it seems unlikely that we will find much of interest for x < -15 (it just goes down without bound) or x > 15 (it just goes up without bound).

A little investigation places zeros around x = 5, 1, and -3. This further narrows the interesting portion of the graph to about [-4,6]

The tiniest bit of calculus would pin mountains and valleys at around x = 3.309 and -1.309, so you'll want to plot some around there. It is a little astounding that we get (3.309,-173.435) and (-1.309,171.435), so the vertical scale might be a bit oppressive.

There you go. Plot things around those five places on the Domain, 5, 1, -3, 3.309, -1.309 and you will find just about everything interesting that there is to find about this particular polynomial.
Click to expand...
Oh ok thanks :) so for the 2nd one: 7(x-3)2(x+5)4
Would this one be easier to graph?
Since we know the zeroes are 3, and -5
Is it possible to graph this without using a graphing calculator.
 
JeffM said:
Yes, but you need to look at the first (and perhaps second) derivative. Where is the first derivative always positive? Where always negative? What are the zeros of the first derivative? The zeros of the first derivative will give you local minima and maxima (unless the second derivative is zero at the same point.)
Click to expand...
Um... I don't think we studied about deriavatives yet :(
 
Chaim said:
Oh ok thanks :) so for the 2nd one: 7(x-3)2(x+5)4
Would this one be easier to graph?
Since we know the zeroes are 3, and -5
Is it possible to graph this without using a graphing calculator.
Click to expand...

You seem to be suffering from Difference-itis. What disease is this? It is the very strange disease where every new problem poses an entirely different set of challenges. There is a cure for this disease. It is called remembering what you learn. EVERYTHING you learn may be applicable to the next thing you do.

Example: If knowing the zeroes in the last problem was useful, it will be useful in THIS problem. Guess what? It will be useful in the next problem, too.

Case in point: Explore on [-5,3]. Plot just a few points, maybe -4, -3, -2, -1, 0, 1, 2. Anything interesting? Looks to me like a little more exploring on [0,1] might be interesting. Just for fun, check out x = -6 and x = 4 so you can be sure which way things go on the outside.

Play with it!! This will cure Difference-itis. Go boldly. It's an equation or a graph. You won't break it.
 
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