Do Now12/04/2015 Solve: (Hint set each factor equal to zero and solve for x). CC4-LT4A: Graphing Polynomials A few things to know before you start today Vocab Degree - the term with the greatest exponent. A polynomial with one variable is in standard form when its terms are written in descending order by degree. Vocab Leading coefficient the coefficient of the first term (IN STANDARD FORM). Identify the leading coefficient and degree. EX) A. 3 5x 2 + 4xB. 8x(2x 4) 2 5x 2 + 4x + 3 Write terms in STANDARD FORMS with descending order by degree. Leading coefficient: 5 Degree: 2 Leading coefficient: 32 Degree: 3 Polynomial graphs look like this: What do you notice? (30 sec) NON-Polynomial graphs look like this: What do you notice? (30 sec) Staple into your notes Standard v. Factored Form Standard Form: L.C.= Degree= Factored Form: L.C.= Degree= What are the differences? (2 mins) Multiply all coefficients Add all exponents 7(1)(1)= =8 DO NOW 12/07/ Describe the end behavior for a) b) 2. Identify whether the function has an odd or even degree and a positive or negative leading coefficient. Identify the leading coefficient, degree, and end behavior. A. Q(x) = x 4 + 6x 3 x + 9 The leading coefficient is 1, which is negative. The degree is 4, which is even. As x +, P(x) , and as x , P(x) . B. P(x) = 6x 4 x x 5 The leading coefficient is 2, which is positive. The degree is 5, which is odd. As x +, P(x) +, and as x -, P(x) -. Ex 2A: Graphs & Key Features Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. As x +, P(x) +, and as x -, P(x) +. P(x) is of positive leading coefficient with an even degree. Right arm is up, LC was positive, Arms together, Degree was even Ex 2B: Graphs & Key Features Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. As x +, P(x) -, and as x -, P(x) +. P(x) is of negative leading coefficient with an odd degree. WordsGraph Roots = zeros = solutions = x intercept Where the y = 0 Whats a root? Where the function crosses the x - axis Pre-Ex) (x+4)(x+2)(x-3)=y Find the roots Ex 1 Solve the polynomial equation by factoring. Set the equation equal to 0. Factor the trinomial in quadratic form. (x 2 25)(x 2 1) = 0 (x 5)(x + 5)(x 1)(x + 1) Solve for x. x 5 = 0, x + 5 = 0, x 1 = 0, or x + 1 =0 The roots are 5, 5, 1, and 1. x = 5, x = 5, x = 1 or x = 1 Ex 2 Solve the polynomial equation by factoring. 2x 6 10x 5 12x 4 = 0 Factor out the GCF, 2x 4. 2x 4 (x 2 5x 6) = 0 Factor the quadratic. 2x 4 (x 6)(x + 1) = 0 Set each factor equal to 0. 2x 4 = 0 or (x 6) = 0 or (x + 1) = 0 Solve for x. x = 0, x = 6, x = 1 The roots are 0, 6, and 1. Vocab Multiplicty - the number of times that x r is a factor of P(x). When a real root has even multiplicity, the graph of y = P(x) BOUNCES off the x-axis (does not cross it). When a real root has odd multiplicity, the graph CROSSES the x-axis. When a real root has odd multiplicity greater than 1, the graph TWITCHES as it crosses the x- axis. root Even Multiplicity: Odd Multiplicity: Odd Multiplicity > 1 BOUNCECROSSTWITCH What will each one look like on a graph? Ex1) Try on your own first Find the multiplicity of each root and describe the behavior of each root on the graph. You can at least find the roots! Goal Problem DOK 1: Find the multiplicity of each root. DOK2: Describe the end behavior of and sketch a prediction of what the graph may look like. What other info would you need to sketch a more accurate graph? 0 (w/ m. of 6), -1/2 (w/ m. of 1), -3 (w/ m. of 1) and 4 (w/ m. of 4) Positive L.C. & Even Degree so End behavior: X 0 (m. of 3), -5 (m. of 1) and 2 (m. of 2) Update your notes! Something you learned? Something old you forgot? Highlight/underline something you didnt notice before Add a side note to a definition or example What do you need to practice more of? Day 5 Try it out Find the zeros. 5 min. DOK 1: DOK 1+: Do Now Find the zeros and multiplicity of each zero: y = 2x 6 22x x x 3 = 2x 3 (x + 1)(x 6) 2 3 Things you need to sketch a polynomial: 1.End behavior (lead coefficient & degree) 2.Zeros (multiplicity: bounce, cross & wiggle) 3.Y int (x = 0) Do notes on graph paper EX) Graph: 1.End behavior 2.Zeros 3.Y int The maximum # of turning points is equal to the polynomials degree 1. Practice: 1.Graph each polynomial from your assignment this week (dont do the table) 2.Start your active practice Update your notes! Something you learned? Something old you forgot? Highlight/underline something you didnt notice before Add a side note to a definition or example What do you need to practice more of? Day 6 Do Now 1. Find the end behavior. 2. Find the zeros and multiplicity 3. Find the y intercept 4. Sketch the polynomial. Active Practice You Choose 10 Pts DOK2s are 1 point each DOK3s are 2 points each Do any problem you want, but enough to earn 10 points. Points only count if the solution AND SOLUTION PATHWAY is correct. Typo: Instead of finding Leading Term, find Lead Coefficient. Where do I start? I still cant do #1 3: Start with DOK2 I can do #1 3 but have a hard time sketching: Do two problems from DOK2 and move on to DOK 3 I can do it all! Start with DOK 3 Goal Problems DOK 2: Sketch DOK3: a)Sketch a graph that has a root at 4 and -2. With a negative coefficient and an odd degree. The y intercept is (0, 7). b) What does the curve look like at the roots of 4 and -2 (bounce, cross or wiggle)? c) What would be a possible equation for the graph you just sketched? Update your notes! Something you learned? Something old you forgot? Highlight/underline something you didnt notice before Add a side note to a definition or example What do you need to practice more of? What would be a possible polynomial for the given graph? Ex) Graph Equation (finding a) Find roots and find the factored form of the polynomial. What do you think the equation (factored form) looks like? Do they even need to know this to graph a polynomial? Save for another lesson?