A polynomial is a sum of many terms.
It is made up of terms that are added, subtracted, or multiplied. NOT divided.
Example: 5xy^2 + 2x - 3
Polynomials have constants (-3) , variable (x and y), and exponents (^2)
These are polynomials: (positive, integer exponents)
f(x) = x^2+3x-2
y =2
y = 12x^5-11x+3
These are not polynomials: (negative exponents, fractional exponents)
f(x) = 4/x
f(x) = 3^x (this is an exponential function)
EXAMPLE: 4x^7 + 2x^3 - 11x +5
leading term: 4x^7
Leading coefficent: 4
Degree: 7
SKETCHING POLYNOMIALS
here is a video: http://www.youtube.com/watch?v=y8iUngkQFwg (her voice is super obnoxious, sorry)
END BEHAVIOR
- an easy way to simply sketch a polynomial is by using its end behavior
we know that simple even functions tend to look parabola like, and that simple odd functions start high and end low or vice versa.
-look at the coefficient, if it is positive then the graph opens upward. If it is negative, it opens downward. and if the exponent is positive (for simple polynomials) it is a parabola, and for odd exponents it is starting high and ending low.
Finding Zeroes On Your Calculator
On a calculator:
plug in equation, y=
then click 2nd then calc
click 2:zero then enter
then left bound, move the cursor to the left of the zero you want to find, then click enter
then do the same but on the right side for right bound
it will calculate your zeroes for you and be displayed at the bottom of the screen.
HERE IS A VIDEO WHICH WILL EXPLAIN FINDING ZEROES ON YOUR CALCULATOR
http://www.youtube.com/watch?v=KVy3h3VUBGg
HERE IS AVIDEO WHICH EXPLAINS FINDING MINIMUMS AND MAXIMUMS
http://www.youtube.com/watch?v=keYBfngzopo
Dear Gretchen,
ReplyDeleteI enjoyed your post! You began your entry with succinct and helpful definitions, and the pictures you provided were also clear and informative.
However, as I got towards the end of the post, the information was less clear. I don't remember hearing the term "end behavior" in class, although I recognized the concept, and it would have been helpful for you to elaborate on the specific term. Also, you could have explained the purposes of finding zeroes and max. and min. terms on our graph, and introduce the "root" value that goes along with these concepts.
Thanks,
Nyanen
I agree with Nyanen. You gave great examples of classifying polynomials and end behavior, but an extra example or two of a more complicated problem would have been nice.
ReplyDeleteI really liked how you covered calculator use, since its such a key part to our class
I agree with Will about the calculator use segment, it was really helpful. However, I feel like you could have added a bit about how the most amount of zeroes/x-intercepts you can find when plugging in a polynomial is equivalent to its degree, and that you can also have less zeroes/x-intercepts than the amount the degree suggests. Overal, it was a really concise post!
ReplyDeleteI think that you did a great job making the first part of the blog very clear and concise but when I got to the part about the "end behavior" I got a little confused. However, as I kept reading it became more understandable.
ReplyDeleteGretchen, Your post starts off well with some information about what the word polynomial means and what a polynomial is, with examples of functions that are and are not polynomials. I like your picture of some basic polynomial shapes, but it can't quite stand alone. It needed a description of how to categorize the sketches. Your video example is nice, although she does go into factoring by grouping and gets ahead of the idea of just basic sketches. I like that you included end behavior in your post - we focused on the shapes of the graphs and his is nice extension. However, in doing so, it would have been helpful to provide a bit more context since you were introducing something new.
ReplyDeleteYour instructions for finding zeros were accurate and helpful, and the "My Secret Math Tutor" video was a nice addition. (I did drive me crazy that he didn't adjust the y-values in his window, though, but that's really feedback for him!). Since x-intercepts and the sketching the graph of a polynomial go hand in hand, your post would have been enhanced by including information about this connection and highlighting it with some examples.
Finally, your post was late, so unfortunately it is out of sequence. Even so, it is a good recap of the information and a reminder to step back and look at each polynomial's overall shape.