1 Math 301 UNIT 5: Polnomial Functions NOTES Section 5.1 and 5.: Characteristics of Graphs and Equations of Polnomials Functions What is a polnomial function? Polnomial Function: - A function that contains onl the operations of multiplication and addition with real-number coefficients, whole-number eponents, and two variables. Polnomial functions are named according to their degree. Degree of a Function: - Is equal to the highest eponent in the polnomial. Eamples of Polnomials: Tpe of Polnomial Standard Form Specific Eample Degree Graph Constant f() = a f() = 5 0 0 Horizontal line Linear f() = a + b f() = 1 + 1 1 Line with slope a Quadratic f() = a + b + c f() = + 1 Parabola Cubic f() = a 3 + b + c + d f() = 3 + 3 +1 3?? Will eplore! NOTES: 1. The terms in a polnomial function are normall written so that the powers are in descending order.. Vertical lines are not considered here because the are not functions since the do not satisf the vertical line test. Vertical lines are not part of the polnomial famil!
In previous grades, we have alread eplored Constant (Gr.9), Linear (Gr. 9 and L1), and Quadratic (L) functions. We will review these functions and then eplore cubic functions. First however, we will need a few new terms: Constant Term: - The term in the polnomial that does not have a variable. Leading Coefficient: - The coefficient (the number in front) of the term with the greatest degree in a polnomial function in standard form. This is denoted as a. - For eample, the leading coefficient of f() = 3 + 3 +1 is. Question 1: For each polnomial, determine the degree, leading coefficient and constant term. a) f() = 3 + 5 Degree = Leading coefficient = Constant term = b) f() = + 7 Degree = Leading coefficient = Constant term = c) f() = 3 + 6 9 Degree = Leading coefficient = Constant term =
3 End Behaviour: - The description of the shape of the graph, from left to right, on the coordinate plane. - The behaviour of the -values as becomes large in the positive or negative direction. - Remember the coordinate plane is divided into four quadrants. Q Q1 Q3 Q This cubic function etends from Q3 to Q1 This quadratic function etends from Q to Q1 Turning Point: - An point where the graph of a function changes from increasing to decreasing or from decreasing to increasing. - For eample, This curve has turning points This curve does not have an turning points
So let s review! 1. Constant Function, f() = a (Degree = 0) Possible Sketches If a is positive (a > 0) If a is negative (a < 0) Horizontal Lines Q Q1 Q3 Q End Behaviour Line etends from Q to Q1 Line etends from Q3 to Q Number of -intercepts 0 (eception: = 0 in which case ever point is on the -ais) Number of -intercepts 1 ( = a) Number of Turning Pts 0 Domain Range a,
5. Linear Functions, f() = a + b (Degree = 1) Possible Sketches Oblique Lines If leading coefficient, a, is positive (a > 0) If leading coefficient, a, is negative (a < 0) Q1 Q Q3 Q End Behaviour Line falls to the left and rises to the right Line etends from Q3 to Q1 Line falls to the right and rises to the left Line etends from Q to Q Number of -intercepts 1 Number of -intercepts 1 ( = b) Number of Turning Pts 0 Domain Range
6 3. Quadratic Functions, f() = a + b + c (Degree = ) Possible Sketches Parabolas If leading coefficient, a, is positive (a > 0) Parabola opens up and has a minimum Q Q1 If leading coefficient, a, is negative (a < 0) Parabola opens down and has a maimum End Behaviour Parabola rises to the left and rises to the right Etends from Q to Q1 Q3 Parabola falls to the left and falls to the right Etends from Q3 to Q Q NOTE: The graphs have the SAME behavior to the left and right. Range (depends on verte and opening) min, ma, Number of -intercepts, 1 or 0 Number of -ints 1 ( = c) Number of Turning Points Domain 1
7 Let s now eplore Cubic Functions of the form 3 a b c d! 1. 3 - - Leading Coefficient: Constant Term: End Behaviour: # of Turning Points: # of -intercepts: -intercept:. 3 6 Leading Coefficient: - - - 6-8 Constant Term: End Behaviour: # of Turning Points: # of -intercepts: -intercept: 3. 3 Leading Coefficient: - - Constant Term: End Behaviour: # of Turning Points: # of -intercepts: -intercept:
8. 3 3 3 - - Leading Coefficient: Constant Term: End Behaviour: # of Turning Points: # of -intercepts: -intercept: 5. 3 1 - - Leading Coefficient: Constant Term: End Behaviour: # of Turning Points: # of -intercepts: -intercept: 6. 3 3 Leading Coefficient: - - Constant Term: End Behaviour: # of Turning Points: # of -intercepts: -intercept:
9 Function Sketch Leading Coeff.(a) Cons. Term (d) End Behav. Number Turn. Pts # of -ints. -int. a) 3 - - b) 3 6 - - - 6-8 c) 3 - - d) 3 3 3 - - e) 3 1 - - f) 3 3 - -
10 Investigation Questions: 1. How are the sign of the leading coefficient and the end behavior related?. How are the degree of the function and the number of -intercepts related? 3. How is the -intercept related to the equation?. How man turning points can a cubic function have? 5. In general, how are the number of turning points and the degree related? 6. What is the domain and range for cubic functions? 7. Eplain wh quadratic functions have maimum or minimum values, but cubic polnomial functions have onl turning points?
11. Cubic Functions, f() = a 3 + b + c + d (Degree = 3) Possible Sketches Sidewas S If leading coefficient, a, is positive (a > 0) Q1 If leading coefficient, a, is negative (a < 0) Q End Behaviour Q3 Curve falls to the left and rises to the right Etends from Q3 to Q1 Q Curve rises to the left and falls to the right Etends from Q to Q NOTE:The graphs have OPPOSITE behaviors to the left and right. The have similar behavior to the behavior of linear functions. Number of - intercepts Number of - intercepts Number of Turning Points 1 ( = d) Domain or 0 Range
1 *** Refer to page 76 and page 86 for a summar.*** Summar Points for Polnomials of Degree 3 or less: - The graph of a polnomial function is continuous - Degree determines the shape of graph - Degree = ma # of -intercepts - There is onl one -intercept for ever polnomial and it is equal to the constant term - The maimum number of turning points is one less than the degree. That is, a polnomial of degree n, will have a maimum of n 1 turning points. - The end behavior of a line or curve is the behavior of the -values as becomes large in the positive or negative direction. For linear and cubic functions the end behavior is opposite to the left and right, while it is the same for quadratic functions. ASSIGN: p. 77, #1 (You can refer to the graphs at the end of the tet to answer question 3 if ou do not have graphing technolog)
13 EXAMPLES: 1. Determine the following characteristics of each function: - number of possible -intercepts - -intercept - domain and range - number of possible turning point - end behavior a. f ( ) 3 b. f ( ) 6 3 c. 3 f ( ) 3 1 d. 3 f ( )
1. Match each graph with the correct polnomial function. a 3 ( ) c 3 ( ) b ( ) d 1 3 ( ) 3 e( ) 1 f ( ) 3 3. Determine the degree, the sign of the leading coefficient, and the constant term for the polnomial function represented b each graph. a. b. Degree: Sign: Constant Term: Degree: Sign: Constant Term:
15. Sketch a possible graph of polnomial functions that satisf each set of characteristics. a. Degree, one turning point which is a minimum, constant term of b. Two turning points (one in Q3 and one in Q1), negative leading coefficient, constant term of 3 c. Degree 1, positive leading coefficient, constant term of d. Cubic, three -intercepts, positive leading coefficient
16 5. Write a polnomial function that satisfies each set of characteristics. a. Etending from QIII to QIV, one turning point, -intercept of 5 b. Etending from QIII to QI, -intercept of - c. Degree 1, increasing, -intercept of -3 d. Two turning points, -intercept of 7 e. Range of and -intercept of ASSIGN: p. 87, #1, 6 13