Unit I - Chapter 3 Polynomial Functions 3.1 Characteristics of Polynomial Functions

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1 Math 3200 Unit I Ch 3 - Polnomial Functions 1 Unit I - Chapter 3 Polnomial Functions 3.1 Characteristics of Polnomial Functions Goal: To Understand some Basic Features of Polnomial functions: Continuous Ma. turns: n 1 Leading coefficient effect/end behavior Even vs odd degree Comparing f() = 4 flatter than f() = 2 Identifing Polnomial Functions (I) Identifing Polnomial Functions and Graphs of Polnomial Functions Terminolog End Behavior of a polnomial function graphicall refers to what is happening to the function as approaches + or End Behavior Curve etends up into quadrant 1 (as approaches + ) and down into quadrant 3 (as approaches ). Degree of a polnomial function refers to the highest eponent on a variable E. f() = The Degree is

2 Math 3200 Unit I Ch 3 - Polnomial Functions 2 Leading Coefficient is the number in front of the term with the highest eponent E. f() = The Leading Coefficient is Identifing Polnomial Functions

3 Math 3200 Unit I Ch 3 - Polnomial Functions 3 1. Use graphing technolog (using graphing software. to graph each function and complete the table. Function Shape End Behaviour Degree # of Turns Leading Coefficient Number of -Intercepts Ma/Min? -int. = + 2 = 3+1 = 2 4 = = 3 4 = =

4 Math 3200 Unit I Ch 3 - Polnomial Functions 4 Function Shape End Behaviour Degree # of Turns Leading Coefficient Number of -Intercepts Ma/Min? -int. = 3 4 = = = 5 1 = = = ( + 1) 2 ( + 4) 2 2. What does the degree indicate about the behavior of the graph to the left and right?

5 Math 3200 Unit I Ch 3 - Polnomial Functions 5 3. How is the sign of the leading coefficient and the end behavior related? 4. Can ou predict the number of turns from the equation? Wh or wh not? 5. Which feature of the equation relates to the number of -intercepts? 6. Does ever function have either a maimum or minimum? Wh or wh not? 7. How is the -intercept determined? (P #1 - #5, #13)

6 Math 3200 Unit I Ch 3 - Polnomial Functions Continued Analzing Polnomial Functions Algebraicall and Graphicall (I) Rules for graphing odd or even polnomial functions On the same graph sketch each pair of polnomial functions. (A) f() = 2 and g() = 4 (B) h() = 3 and p() = Even Degree Functions f() = a n Odd Degree Functions f() = a n With positive leading coefficients ehibit behavior in the and quadrants With positive leading coefficients ehibit behavior in the and quadrants Graphicall there is a effect when the value of increases over the interval. Graphicall there is a effect when the value of increases over the interval. What end behavior is ehibited when leading coefficients are negative for even and odd functions?

7 Math 3200 Unit I Ch 3 - Polnomial Functions 7 (II) Graph each of the given functions and answer the indicated characteristics. Function Graph End Behavior Degree (Odd/Even) # of -Intercepts Constant Term -int. f() = f() = f() = f() = f() =

8 Math 3200 Unit I Ch 3 - Polnomial Functions 8 Summar of Characteristics of Polnomial Functions (A) Degree of Polnomial Function Odd Degree Functions f() = a n Even Degree Functions f() = a n With negative leading coefficients ehibit behavior in the and quadrants With negative leading coefficients ehibit behavior in the and quadrants With positive leading coefficients ehibit behavior in the and quadrants With positive leading coefficients ehibit behavior in the and quadrants (B) Constant Term of a Polnomial Function For odd/even functions, the corresponds to the constant term. (C) The Number of Real -intercepts Odd Degree Functions f() = a n +. + c Even Degree Functions f() = a n +. + c At least one -intercept to a maimum of intercepts Zero -intercepts to a maimum of intercepts No ma or min points Domain Range Ma or min point depends on direction Domain Range depends on

9 Math 3200 Unit I Ch 3 - Polnomial Functions 9 (III) Review Questions: (a) Identif the features of the graph related to the function f() = Leading Coefficient Degree End Behavior -intercept Number of possible -intercepts Ma or min values? (b) Match the functions with the appropriate graphs.

10 Math 3200 Unit I Ch 3 - Polnomial Functions 10 (c) How man turns can the graph of a polnomial function of degree 5 have? Eplain. (d)

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16 Math 3200 Unit I Ch 3 - Polnomial Functions 16 The Factor Theorem (Part I) Determining the Remainder for a Factor of a Polnomial Eample: Determine the remainder when ( ) is divided b ( + 2). Conclusion: Since P( ) = ( + 2) is a Factors ( - a) of a Polnomial Epression and Zeros of a Polnomial Function (I) Eample: Attaining Linear Factors ( - a) of a Polnomial Epression If ( + 2) is a factor of ( ) then determine all other linear factors.

17 Math 3200 Unit I Ch 3 - Polnomial Functions 17 (II) Attaining Zeros of a Polnomial Function Eample: If a polnomial function P() = epressed in factored form as P() = ( + 2) 2 ( - 1) then determine the zeros. Linear Factors and Zeros of a Function The Factor Theorem ( - a) is a factor of P() if and onl if Eample: Verif if + 3 is a factor of P() =

18 Math 3200 Unit I Ch 3 - Polnomial Functions 18 Integral Zero Theorem How can we determine all of the factors of a polnomial function? Eample: If + 1 is a factor of P() = then determine all other factors. Which term in the polnomial function P() = has factors, and?

19 Math 3200 Unit I Ch 3 - Polnomial Functions 19 Eample: Full factor the given the polnomial function P() = (i) List all possible integral zeros. (Appl Integral Zero Theorem) (ii) Verif one factor. (Appl Factor Theorem) (iii) Reduce polnomial to a lesser degree. (Appl Snthetic Division) (iv) Epress P() in factored form

20 Math 3200 Unit I Ch 3 - Polnomial Functions 20 Factoring Higher Degree Polnomials (I) Factoring Cubic (degree 3 with 4 terms) Polnomials b Grouping Eample: Factor full. (a) (b) (II) Factoring Quartics (degree 4) Polnomials Eample: Factor full Tet Questions: #1b,c #2a,c #3b,d #4a,c #5b,c,d #6a,c,d #7b,c

21 Math 3200 Unit I Ch 3 - Polnomial Functions 21 The Factor Theorem (Part II) Factoring polnomials P(), that contain non - integer zeros b appling IZT, FT and SD Modelling and solving problems involving polnomial functions (I) Factoring a Polnomial P() that also contains non - integer zeros Eample: Given the polnomial function P() = (a) Use the Integral Zero Theorem to list all possible integral factors (b) Verif one of the factors using the factor theorem (c) Appl snthetic division to determine the remaining factors (d) Epress P() = in factored form Factored Form P() =

22 Math 3200 Unit I Ch 3 - Polnomial Functions 22 Note: The linear factors will now produce zeros since the is no longer 1. Summar - To solve polnomial functions: List all possible integral zeros Use the factor theorem to verif a zero Use snthetic division to reduce the polnomial Repeat the above process to determine the remaining factors or the polnomial is reduced to a trinomial that can factor. Eample: Full factor the polnomial function f() =

23 Math 3200 Unit I Ch 3 - Polnomial Functions 23 (II) Modelling and Solving Problems involving Polnomial Functions Eample: P.134 of Tetbook P #9, #11, #13, #16, C1

24 Math 3200 Unit I Ch 3 - Polnomial Functions 24 Section 3.4 (Part I) Equations and Graphs of Polnomial Functions (I) Investigating the relationship between zeros, - intercepts and roots Sketching the graph of polnomial functions Modelling and solving problems involving polnomial functions Investigating the relationship between zeros, - intercepts and roots Eample: Given the polnomial function f() = (a) Use graphing technolog to sketch the graph and determine the - intercepts from the graph. - intercepts of the graph are:

25 Math 3200 Unit I Ch 3 - Polnomial Functions 25 Attaining the Zeros from a Function (b) Factor the polnomial function f() = then use the factors to determine the zeros. (c) Solve the equation = 0 to determine the roots. What do ou notice about the - intercepts of the graph, zeros of the function and roots of the equation?

26 Math 3200 Unit I Ch 3 - Polnomial Functions 26 (II) Investigating the Graphs of Polnomial Functions and the Multiplicit of a Zero For each Polnomial Function: State the intercepts Use technolog to sketch the graph State the multiplicit of each zero Indicate the intervals where the function is positive (above the ais) or negative (below the ais) Multiplicit of a Zero The number of times the zero of a polnomial occurs The shape of a graph at a zero depends on the multiplicit

27 Math 3200 Unit I Ch 3 - Polnomial Functions 27 Function -intercepts Graphs/Multiplicit of zeros Intervals 1. f()=(+1)(-1)(+2) f()=(-1) 2 (+2)

28 Math 3200 Unit I Ch 3 - Polnomial Functions 28 Function -intercepts Graphs/Multiplicit of zeros Intervals 3. f()=(-1) f()=

29 Math 3200 Unit I Ch 3 - Polnomial Functions 29 Function - intercepts Graphs/Multiplicit of zeros Intervals 5. f()= 3 6. f()= 4 7. f()= (+1) 3 ( 2)

30 Math 3200 Unit I Ch 3 - Polnomial Functions 30 NOTE: Multiplicit of a zero and characteristics of the graph (I) Zero of multiplicit One (II) Zero of multiplicit Two (III) Zero of multiplicit Three Odd/Even Multiplicit (i) (ii) The graph of a polnomial function the -ais where the intercepts have odd multiplicit. The graph of a polnomial function the -ais where the intercepts have even multiplicit.

31 Math 3200 Unit I Ch 3 - Polnomial Functions 31 (III) Sketching the Graph of a Polnomial Function Eamples: For each of the given the polnomial functions (i) (ii) determine the degree, leading coefficient, end behaviour, zeros/ - intercepts, - intercept and interval where the function is positive or negative. use the information above to sketch the graph. (a) f() = -( + 2) 2 ( - 1) 2 (b) = ( - 2) 3

32 Math 3200 Unit I Ch 3 - Polnomial Functions 32 (c) = (IV) Determining the Equation of a Polnomial Function from a Graph Eamples: Use the graph to determine the equation of the given polnomial function. (a)

33 Math 3200 Unit I Ch 3 - Polnomial Functions 33 (b) P P.149 #1a, #2c, #3, #4b,c #7a, b, c #8, #9c, d, e #10a, c