IM 3 Lesson 4.2 Day 3: Graphing Reflections and Connections Unit 4 Polynomial Functions

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1 (A) Lesson Context BIG PICTURE of this UNIT: What is a Polynomial and how do they look? What are the attributes of a Polynomial? How do I work with Polynomials? CONTEXT of this LESSON: Where we ve been We have discussed the basics: degree, type, and operations (+,, x) Where we are What are the key attributes of a polynomial and how do these affect the shape? Where we are heading What are the key attributes of a polynomial and how do these affect the shape? (B) Lesson Objectives: a. Begin to analyze the the attributes of a polynomial function and it s effect on the graph. b. Observations and patterns in the graphs of polynomials. c. Solidify our perdictions of how polynomials behave.

2 (C) Drawing Challenge #1: The following challenge is based on the END BEHAVIOURS and MULTIPLICITIES of a polynomial function. From the factored form of each polynomial function, sketch the graph of each function showing clearly the end behaviour and what the function looks like at each root. f (x) = (x 8)(x 2)(x + 4)(x + 9) f (x) = 2(x + 2) 2 (x 2) 2 f (x) = (x 3) 2 (x 1)(x + 5) 2 f (x) = (x + 3)(x 6)(x + 7)(x 1)(x + 4) f (x) = (x 5) 3 (x +1) f (x) = (x + 4) 3 (x 2) 2 (x +1) f (x) = 1 2 (x 4)4 (x + 3) 2 (x 8) f (x) = (x + 6)(x 2) 3 (x +1) 5 *CHALLENGE* f (x) = (x + 5) 2 (2 x)(x + 7) *CHALLENGE* f (x) = (3 x) 2 (x + 9) 2 (6 x) 3

3 (D) Drawing Challenge #2: The following challenge is based on the EXTREMA and ZEROS of a polynomial function. From the description of each polynomial function, sketch a possible graph of each function clearly showing the correct number of extrema or zeros. DO NOT repeat any graphs! IF it is not possible to sketch a function fitting the criteria, EXPLAIN why. (HINT: there are only 2 below that are not possible). 5 th degree polynomial with 4 x-intercepts 5 th degree polynomial with 2 extrema 4 th degree polynomial with 2 x-intercepts 6 th degree polynomial with 3 extrema 4 th degree polynomial with 2 extrema 6 th degree polynomial with 2 x-intercepts 6 th degree polynomial with 1 extrema 5 th degree polynomial with 2 extrema 6 th degree polynomial with 4 x-intercepts 5 th degree polynomial with 1 extrema

4 (E) Matching Challenge #3 Now that we have a grasp on what a polynomail is and we have begun to develop the key vocabulary we can now apply our knowledge WITHOUT the use of technology to see if we can make connections between the two forms of a polynomial and their graphs. You are given a set of 6 factored form equations, 6 standard form equations, and 6 graphs. Your goal is to MATCH a factored form equation to a standard form equation to a graph. Record the combinations below: Factored Form Standard Form Graph

5 Connections and Reflections: 1. Please reflect upon and write about any connections between the factored form equation and the graph. Picture or sketch to support your thinking. Writen Response: 2. Please reflect upon and write about any connections between the standard form equation and the graph. Picture or sketch to support your thinking. Writen Response: 3. Please reflect upon and write about any connections between the x-intercpts equation and the equation. Picture or sketch to support your thinking. Writen Response:

6 4. Please reflect upon and write about any connections between the y-intercpts equation and the equation. Picture or sketch to support your thinking. Writen Response: 5. Please reflect upon and write about any connections between the degree of the polynomial and the shape of the graph. Picture or sketch to support your thinking. Writen Response: 6. Please reflect upon and write about any connections between the sign of the leading coefficient and the end behaviour of the graph. Picture or sketch to support your thinking. Writen Response:

7 Final Consolidaiton Degree and family name 1 st Degree Name: 2 nd Degree Name: 3 rd Degree Name: 4 th Degree Name: 5 th Degree Name: Graph Shape Graph Shape Graph Shape Graph Shape Graph Shape + Leading Coefficient - Leading Coefficient EXT 1. Given the following factored form equation without technology put into standard form and draw a sketch of the graph. f (x) = (x 1)(x 7)(x + 2) EXT 2. Given the following Graph write a possible factored form equaiton or standard form equation.

8 Matching Graphs M

9 Matching Graphs N

10 Matching Graphs O

11 Matching Graphs P

12 Matching Graphs Q

13 Matching Graphs R