8- Attributes of Polnomial Functions TEKS FCUS Etends TEKS ()(A) Graph the functions f () =, f () =, f () =, f () =, f () = b, f () =, and f () = log b () where b is,, and e, and, when applicable, analze the ke attributes such as domain, range, intercepts, smmetries, asmptotic behavior, and maimum and minimum given an interval. TEKS ()(F) Analze mathematical relationships to connect and communicate mathematical ideas. Additional TEKS ()(D) VCABULARY Degree of a monomial the sum of the eponents of the variables Degree of a polnomial for a polnomial in one variable, the greatest degree of the monomial terms End behavior the direction of the graph to the far left and to the far right Monomial a real number, a variable, or a product of a real number and one or more variables with whole-number eponents Polnomial a monomial or a sum of monomials Polnomial function A polnomial in the variable defines a polnomial function of. Standard form of a polnomial function the polnomial function with the terms arranged b degree in descending numerical order Turning point a point where the graph changes direction from upwards to downwards or from downwards to upwards Analze closel eamine objects, ideas, or relationships to learn more about their nature ESSENTIAL UNDERSTANDING A polnomial function has distinguishing behaviors. You can look at its algebraic form and know something about its graph. You can look at its graph and know something about its algebraic form. Ke Concept Standard Form of a Polnomial Function The standard form of a polnomial function arranges the terms b degree in descending numerical order. A polnomial function P () in standard form is P () = a n n + a n- n- + g + a + a, where n is a nonnegative integer and a n, c, a are real numbers. P() = + + - Cubic term Quadratic term Linear term Constant term Lesson 8- Attributes of Polnomial Functions
Ke Concept Classifing Polnomials You can classif a polnomial b its degree or b its number of terms. Polnomials of degrees zero through five have specific names, as shown in this table. Degree Name Using Degree constant linear quadratic cubic quartic quintic Polnomial Eample Number of Terms Name Using Number of Terms monomial binomial monomial trinomial binomial polnomial of terms Ke Concept Polnomial Functions = + 6 - =- + End Behavior: Up and Up Turning Points: ( -.7, -.), ( -.7,.7), and (., -.) The function is decreasing when 6-.7 and -.7 6 6.. The function increases when -.7 6 6-.7 and 7.. End Behavior: Down and Down Turning Point: (, ) The function is increasing when 6 and is decreasing when 7. = =- + End Behavior: Down and Up Zero turning points. The function is increasing for all. End Behavior: Up and Down Turning Points: ( -.8, -.9) and (.8,.9) The function is decreasing when 6-.8 and when 7.8. The function is increasing when -.8 6 6.8. PearsonTEXAS.com
Ke Concept Determining End Behavior You can determine the end behavior of a polnomial function of degree n from the leading term a n of the standard form. End Behavior of a Polnomial Function With Leading Term a n a Positive a Negative n Even (n ) Up and Up Down and Down n dd Down and Up Up and Down In general, the graph of a polnomial function of degree n (n Ú ) has at most n - turning points. The graph of a polnomial function of odd degree has an even number of turning points. The graph of a polnomial function of even degree has an odd number of turning points. Problem Classifing Polnomials How do ou write a polnomial in standard form? Combine like terms if possible. Then, write the terms with their degrees in descending order. Write each polnomial in standard form. What is the classification of each polnomial b degree? B number of terms? A + 9 + 9 + + The polnomial has degree and terms. It is a quadratic trinomial. B 6 + + + + - The polnomial has degree and terms. It is a quartic polnomial of terms. Problem Describing End Behavior of Polnomial Functions What do a and n represent? a is the coefficient of the leading term. n is the eponent of the leading term. Consider the leading term of each polnomial function. What is the end behavior of the graph? Check our answer with a graphing calculator. A = The leading term is. Since n is odd and a is positive, the end behavior is down and up. B = + 8 8 + The leading term is -. Since n is even and a is negative, the end behavior is down and down. Lesson 8- Attributes of Polnomial Functions
Problem TEKS Process Standard ()(F) How can ou graph a polnomial function? Make a table of values to help ou sketch the middle part of the graph. Use what ou know about end behavior to sketch the ends of the graph. Graphing Cubic Functions What is the graph of each cubic function? Describe the graph, including end behavior, turning points, and increasing/decreasing intervals. A = Step Step.. B = Step Step Step Problem The end behavior is down and up. There are no turning points. The function is increasing for all. Step The end behavior is up and down with turning points at (-, -) and (, ). The function is decreasing when 6- and when 7. It is increasing when - 6 6. TEKS Process Standard ()(D) Using Differences to Determine Degree How do ou find the second differences? Subtract the consecutive first differences. What is the degree of the polnomial function for the data? With consecutive input values that differ b a constant, ou can analze the output differences to find the least-degree polnomial for the data. If the first differences are constant, the function is linear. If the second differences (but not the first) are constant, the function is quadratic. If the third differences (but not the second) are constant, the function is cubic, and so on. -value st difference nd difference rd difference 6 7 8 6 6 8 6 6 6 6 9 7 The third differences are constant. 7 9 7 The degree of the polnomial function is. PearsonTEXAS.com
NLINE H M E W R K PRACTICE and APPLICATIN EXERCISES Scan page for a Virtual Nerd tutorial video. For additional support when completing our homework, go to PearsonTEXAS.com.. Appl Mathematics ()(A) The data show the power generated b a wind turbine. The column gives the wind speed in meters per second. The column gives the power generated in kilowatts. What is the degree of the polnomial function that models the data? Classif each polnomial b degree and b number of terms. Simplif first if necessar.. a + a - a. 7. (). (a - )a - 6. -8d - 7 + -d - 6 7. b(b - ) 6 7 8 9 7.8 7..96 8. 8. Cop and complete the table, which shows the first and second differences in -values for consecutive -values for a polnomial function of degree. 9. Use Multiple Representations to Communicate Mathematical Ideas ()(D) The outputs for a certain function are,,, 8, 6,, and so on. a. Find the first differences of this function. b. Find the second differences of this function. c. Find the tenth difference of this function. d. Can ou find a polnomial function that matches the original outputs? Eplain our reasoning.. Connect Mathematical Ideas ()(F) A cubic polnomial function f has leading coefficient and constant term 7. If f () = 7 and f () = 9, what is f (-)? Eplain how ou found our answer. Describe the shape of the graph of each cubic function including end behavior, turning points, and increasing/decreasing intervals.. = - -. =-9 - + +. = + 9. =. =- - 6. = 8 Determine the sign of the leading coefficient and the least possible degree of the polnomial function for each graph. 7. 8. 9. st diff. nd diff. 6 6 8 Lesson 8- Attributes of Polnomial Functions
Determine the end behavior of the graph of each polnomial function.. =-7 + 8 +. =- + 6 -. = - - 6-6. = 8-9 + 6 +. =- - 7-9. =--6-9 8 6. Connect Mathematical Ideas ()(F) Write an equation for a polnomial function that has three turning points and end behavior up and up. 7. Show that the third differences of a polnomial function of degree are nonzero and constant. First, use f () = - - - 6. Then show third differences are nonzero and constant for f () = a + b + c + d, a. 8. Suppose that a function pairs elements from set A with elements from set B. A function is called onto if it pairs ever element in B with at least one element in A. For each tpe of polnomial function, and for each set B, determine whether the function is alwas, sometimes, or never onto. a. linear; B = all real numbers b. quadratic; B = all real numbers c. quadratic; B = all real numbers greater than or equal to d. cubic; B = all real numbers 9. Make a table of second differences for each polnomial function. Using our tables, make a conjecture about the second differences of quadratic functions. a. = b. = 7 c. = 7 + d. = 7 + + Determine the degree of the polnomial function with the given data... 6 7 9 9 9 TEXAS Test Practice. Which epression is a cubic polnomial? A. B. + C. + - D.. Which equation has - { i as its solutions? F. + 6 =- G. + 6 =- H. + = J. + =. What is the discriminant of q + r + s =? A. qrs B. q - rs C. r - qs D. s - qr. What is a simpler form of ( - ) -? Classif the polnomial b degree and b number of terms. PearsonTEXAS.com