Copy anything you do not already have memorized into your notes.

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2 Copy anything you do not already have memorized into your notes.

3 A monomial is the product of non-negative integer powers of variables. ONLY ONE TERM. ( mono- means one) No negative exponents which means -> a monomial has NO variable in its denominator No fractional exponents which means -> a monomial has can not be something within a square root, cube root, etc. Examples: 13, 3x, -57, x², 4y², -2xy, or 520x²y²

4 A binomial is the sum of two monomials. It has two unlike terms. TWO TERMS ( bi- means two) Cannot be simplified any further. Examples: 3x + 1, x² - 4x, 2x + y, or y - y²

5 A trinomial is the sum of three monomials. It has three unlike terms. TWO TERMS ( tri- means three) Cannot be simplified any further. Examples: x 2 + 2x + 1, 3x² - 4x + 10, 2x + 3y + 2

6 A polynomial is the sum of 2 or more monomials. ( poly- means many) Cannot be simplified any further. Examples: x 2 + 2x, 3x 3 + x² + 5x + 6, 4x - 6y + 8

7 Classification of a polynomial by number of terms: monomial binomial trinomial polynomial polynomial polynomial

8 The DEGREE of a monomial = the sum of the exponents of it s variables. Examples: 13 degree = 0 3x degree = 1-2xy degree = 2 520x²y² degree = 4 The DEGREE of a polynomial = the degree of the term with the greatest degree. Examples: x 2 + 2x 1 degree = 2 3x 3 + x² + 5x + 6 degree = 3

9 Polynomials are in simplest form when they contain no like terms. x 2 + 2x x² - 4x when simplified becomes 4x 2-2x + 1 Polynomials are generally written in descending order. Descending: 4x 2-2x + 1 exponents of variables decrease from left to right

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13 Exploring Graphs of Polynomial Functions

14 To get an idea of what we will be working with in the next unit Use a graphing calculator to complete the activity, Exploring Graphs of Polynomial Functions shown on the next few slides. Link to free, on-line graphing calculator on class website. Copy the table and complete it on your own piece of paper.

15 Conjecture is math for HYPOTHESIS. Write your conjecture using an equation or one complete sentence. Need help entering it in the calculator? Look at the next slide

16 Can t remember what buttons to push??? Remember! When things start acting weird on the calculator, you can always reset the memory. 2 nd + 7 Enter 2

17 Write your new conjecture using an equation or one complete sentence. Don t forget to EXPLAIN. If you can t explain it, you don t understand it.

18 To help with #17 19 Do this on the same paper as you did Part 1.

19 Local maxima & local minima are also known as turning points. In general, a polynomial of degree n, has n-1 turning points.

20 Local Maximum Local Minimum In English now A local maximum is the top point of a peak. In English now A local minimum is the bottom point of a valley.

21 To help with #1-8 all They are just using the TRACE button (like we did on Graphing Secrets ) to find the max and min (estimating to the nearest tenth as best they can). On #s 5 8, the inequalities that come after each function are just telling you what part of the graph to limit your answers to so -6<x<6 is just saying to only look at the graph between -6 and 6 on the x-axis, and give the answer for what you see there ignore the other areas so the prob. doesn t take all day. Do this on the same paper as you did Parts 1 & 2.

22 Exploring End Behavior of Monomial Functions

23 Make a table on your paper to help you organize your answers. See next slide

24 # Equation Degree Leading Coefficient Left side Right Side 1 y = x 2 Even Positive Rise Rise 2 y = x 4 Even Positive 3 y = 2x 2 4 and on and on for each problem. Remember that this is what determines the patterns you find next Why remember this? Bc it is the leading coefficient that you will need to look at on a polynomial equation to use your patterns (& this is what your last assignment will be on).

25 THE MOST IMPORTANT PART!!! Time to use some brain power what patterns do you see??? These patterns work for any POLYNOMIAL!!! If you don t discover these, the last part of your assignment will take you a lot longer to do! (PS You will be required to memorize the patterns you are supposed to discover here so git er done! )

26 Last definition The leading coefficient of a polynomial is the coefficient of the term with the greatest degree in the polynomial.

27 Don t forget the hint given 2 slides back!!!