polynomial functions question (images incl)

**finalfantasy** · September 9th 2007, 08:39 AM

I'm having some trouble with polynomial functions.

http://img410.imageshack.us/img410/2941/m1yq7.jpg
If you look on the sheet above (question 9), what do they mean by intervals.. increasing and decreasing? And how do I find the LOCAL maximum and minimum?

http://img509.imageshack.us/img509/5827/...
And in this one, in question #14, how do I determine the equation of each polynomial function from its graph? I'm having trouble on #17 and 2 as well. Any help would be TRULY appreciated!

**topsquark** · September 9th 2007, 10:34 AM

Originally Posted by finalfantasy

I'm having some trouble with polynomial functions.

http://img410.imageshack.us/img410/2941/m1yq7.jpg
If you look on the sheet above (question 9), what do they mean by intervals.. increasing and decreasing? And how do I find the LOCAL maximum and minimum?

An interval is a section of the real number line. Increasing/decreasing means is the function rising/falling as you go to the right? The local maximum/minimum is the point(s) in the domain of the function which are at their highest points over a small interval.

So for example in a) the function is increasing on $( -\infty, -2) \cup (1, \infty)$ and decreasing on $(-2, 1)$ . There is a local maximum at $(-2, 4)$ and a local minimum at $(1, -8)$ .

-Dan

**topsquark** · September 9th 2007, 10:42 AM

Originally Posted by finalfantasy

http://img509.imageshack.us/img509/5827/...
And in this one, in question #14, how do I determine the equation of each polynomial function from its graph?

For 14 you can write the polynomial as
$f(x) = a(x - r_1)(x - r_2)...(x - r_n)$
where $r_1, r_2, ..., r_n$ are the n x-intercepts of the polynomial. (This is not generally true, but none of the zeros here are complex, so we can get away with it.) The only hitches here are b) and d) where the graph touches the x-axis. In this case that intercept is squared in the polynomial.

As an example, take b)
$f(x) = a(x - (-2))(x - 1)^2 = a(x + 2)(x^2 - 2x + 1) = a(x^3 - 3x + 2)$

Now we need to find a. Any point on the function that isn't an x-intercept will do. So I'll use $(-1, -4)$ :
$-4 = a((-1)^3 - 3(-1) + 2) = a(-1 + 3 + 2) = 4a$

So $a = -1$ .

Thus the polynomial is
$f(x) = -(x^3 - 3x + 2) = -x^3 + 3x - 2$ .

17 is done in a similar fashion.

-Dan

**topsquark** · September 9th 2007, 10:44 AM

Originally Posted by finalfantasy

http://img509.imageshack.us/img509/5827/...
I'm having trouble on #17 and 2 as well.

2. Graph a few odd degree polynomials. What do you notice about their behavior at $\pm \infty$ ? Now graph a few even degree polynomials. What do you notice about their behavior at $\pm \infty$ ?

-Dan

**finalfantasy** · September 9th 2007, 12:32 PM

OMG.

AHH.

THANK YOU SOOOO MUCH!

I actually understood all of that! Today is certainly my lucky day!

But one more question if you don't mind, what of question #17? Thank you so much again!

Also, for your first response, is it actually decreasing on 2,1? Seems more like -2,1 .. or I might just be wrong >.<

**topsquark** · September 9th 2007, 08:10 PM

Originally Posted by finalfantasy

OMG.

AHH.

THANK YOU SOOOO MUCH!

I actually understood all of that! Today is certainly my lucky day!

But one more question if you don't mind, what of question #17? Thank you so much again!

Also, for your first response, is it actually decreasing on 2,1? Seems more like -2,1 .. or I might just be wrong >.<

Yes, you are right. It was a typo. I fixed it in my previous post.

-Dan

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