Characteristics of Polynomial Functions / Unit 2 – Polynomial Functions - Lesson 9

(A)Lesson Context

BIG PICTURE of this UNIT: /
  • What are & how & why do we use polynomial functions?
  • proficiency with algebraic manipulations/calculations pertinent to polynomial functions
  • proficiency with graphic representations of polynomial functions

CONTEXT of this LESSON: / Where we’ve been
Grade 9,10 IM math & working with polynomial relations / Where we are
Characteristics of polynomial functions from a graphic perspective / Where we are heading
Familiarity with multiple strategies for working with polynomialfunctions to find characteristic features

(B)Lesson Objectives

  1. Recognize basic features of the graphs of polynomial functions
  2. Investigate the basic shape of parent polynomial functions
  3. Investigate the relationship between a polynomial’s end behaviour & its leading coefficient
  4. Find and work with the zeroes of a polynomial
  5. Sketch polynomial graphs using information about zeroes, leading coefficients & end behaviour

(C)Basic Features of Polynomials  Even & Odd Degrees

  1. Use DESMOS grapher view window set to and
  2. Graph each of the following functions and answer the following analysis questions:
  3. ii. iii. iv.
  1. In the interval of , which graph is on the bottom (or which graph is “lower” or “flatter”)?
  2. Outside of the interval , which graph is lower/on the bottom?
  3. Predict the appearance of the function on the interval . Justify your prediction
  1. Graph each of the following functions and answer the following analysis questions:
  2. ii. iii. iv.
  1. In the interval of and then on the interval , which graph is on the bottom (or which graph is “lower” or “flatter”)?
  2. Outside of these intervals , which graph is lower/on the bottom? When?
  3. Predict the appearance of the function on the interval . Justify your prediction

(D)Basic Features of Polynomials  End Behaviour

  1. Each function has already been programmed in DESMOS for you.
  2. View each graph one at a time & then complete each analysis in the table below
  3. ii.
  1. iv.
  1. vi.
  1. viii.

Function / degree / Number of turning points / Leading Coefficient + or -? / Degree: odd or even? / End behaviour / End behaviour
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
  1. Make a conjecture about the maximum number of turning points in the graph of a polynomial with degree of 8,9, or n.
  1. Make a conjecture about the end behaviour of a function with a degree that is (i) odd, (ii) even
  1. Make a conjecture about the end behaviour of a function with a degree that is:
  2. Even and has a positive leading coefficient
  3. Even and has a negative leading coefficient
  4. Odd and has a positive leading coefficient
  5. Odd and has a negative leading coefficient
  1. Example to demonstrate understanding  given the polynomial , without using a GDC, predict the end behaviour & include a rationale for your prediction.

(E)Zeroes of Polynomials

  1. You will work with the polynomialin this investigation.

BEFORE GRAPHING:

  1. Where are the zeroes/roots of ?
  2. Can you predict the end behaviour of ?
  3. Can you predict the number of extrema of?

NOW GRAPH

  1. Graph the polynomial in an appropriate window (one that allows you to see the extremas as well as the zeroes)  sketch it, labelling key points
  1. You will work with the polynomialin this investigation.

BEFORE GRAPHING:

  1. Where are the zeroes/roots of ?
  2. Can you predict the end behaviour of ?
  3. Can you predict the number of extrema of?

NOW GRAPH

  1. Graph the polynomial in an appropriate window (one that allows you to see the extremas as well as the zeroes)  sketch it, labelling key points

(F)Algebra HELP online

  1. Go to and (i) let’s expand and (ii) let’s factor

(G)Zeroes of Polynomials

(a)If a cubic polynomial has zeroes of 2,1 & ½, determine an equation in (i) factored form and in (ii) standard form
(b)If the cubic goes through the point (-1,-36), what is its equation now? / (c)given the graph of , determine its equation in factored form and in standard form.

(H)Multiplicity of Zeroes

  1. Graph the following functions, carefully sketch the graph in the domain of and in the range of
  1. when factored
  2. when factored
  3. when factored
  4. when factored
  1. Use to factor these quartic polynomials
  1. Explain WHY the graphs appear slightly different around the x-intercept of 2

(I)Consolidation Exercises:

  1. Given what you have learned this lesson, prepare a detailed sketch of .

(J)Homework(emphasize that HW is focused on SKILL DVELOPMENT)

From the textbook PRECALCULUS WITH LIMITS – A Graphing Approach (4th ed) by Larson, Hostetler, Edwards, Sec 2.2, p108-109, KUQ1-8,17,19,21,22,27,29,31,39,41,43,48,50,52,55,65,71