2.1 Evaluate and Graph Polynomial

Transcription

1 2. Evaluate and Graph Polnomial Functions Georgia Performance Standard(s) MM3Ab, MM3Ac, MM3Ad Your Notes Goal p Evaluate and graph polnomial functions. VOCABULARY Polnomial Polnomial function Degree of a polnomial function Leading coefficient Standard form of a polnomial function Snthetic substitution End behavior Even function Odd function Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 75

2 Your Notes Eample Identif polnomial functions Decide whether the function is a polnomial function. If so, write it in standard form and state its degree and leading coefficient. a. f() b. f() Solution a. The function a polnomial function because the term has an eponent that is. b. The function a polnomial function written as in its standard form. It has degree and a leading coefficient of. Checkpoint Complete the following eercise.. State the degree and leading coefficient of f () Eample 2 Evaluate b snthetic substitution Use snthetic substitution to evaluate f() when 5 2. Write the coefficients of f() in order of eponents. Write the value of to the left. Bring down the leading coefficient. Multipl the leading coefficient b and write the product under the second coefficient.. Multipl the previous sum b and write the product under the third coefficient. Add. Repeat for all of the remaining coefficients coefficients f (2) 5 76 Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

3 Your Notes END BEHAVIOR OF POLYNOMIAL FUNCTIONS For the graph of f() 5 a n n a n 2 n 2... a a 0 : Ifn is odd and a n > 0, then f() as ` and f() as 2`. Ifn is odd and a n < 0, then f() as ` and f() as 2`. Ifn is even and a n > 0, then f() as ` and f() as 2`. Ifn is even and a n < 0, then f() as ` and f() as 2`. Eample 3 Graph and analze a polnomial function (a) Graph the function f() , (b) find the domain and the range of the function, (c) describe the degree and leading coefficient of the function, and (d) decide whether the function is even, odd, or neither and describe an smmetries in the graph. Solution a. Make a table of values and plot the corresponding points. Connect the points with a smooth curve f() b. The domain is and the range is. c. The degree is and the leading coefficient is. d. The function is because f(2) 52(2) 3 2(2) 2 2(2) 2 5 which is not equal or. The graph has. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 77

4 Your Notes Checkpoint Complete the following eercises using the function f() Evaluate f() for 5 22 using snthetic substitution. 3. Graph f(). Homework 78 Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

5 Name Date LESSON 2. Practice Decide whether the function is a polnomial function. If it is, write the function in standard form.. f() f() g() h() 5 } State the degree and leading coefficient of the polnomial. 5. g() h() f() 5 2 } g() Ï } 2 Use direct substitution to evaluate the polnomial function for the given value of. 9. f() ; 5 0. g() ; 5 2. h() ; f() ; 5 22 Use snthetic substitution to evaluate the polnomial function for the given value of. 3. g() ; 5 4. h() ; 5 23 Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 79

6 Name Date LESSON 2. Practice continued Use what ou know about end behavior to match the polnomial function with its graph. 5. f() f() f() A. B. C. Decide whether the function is even, odd, or neither. Describe an smmetries in the graph. 8. f() f() g() Business The cost of manufacturing a product can be modeled b the function C(n) 5 0.2n 3 2 7n 2 08n 00 where C is the cost in dollars and n is the number of units of the product in thousands. a. State the degree of the function. b. Complete the table of values for the function. n Cost (dollars) C n Units (thousands) C c. Use our table to graph the function. 80 Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

7 2.2 Translate Graphs of Polnomial Functions Georgia Performance Standard(s) MM3Aa, MM3Ac, MM3Ad Your Notes Goal Eample p Graph translations of polnomial functions. Translate a polnomial function verticall Graph g() 5 4. Compare the graph with the graph of f() Make a table of values and plot the corresponding points f 2. Connect the points with a smooth curve and check the end behavior. The degree is and the leading coefficient is. So, g() as and g() as. 3. Compare with f() 5 4. The graph of g() 5 4 is the graph of f() 5 4 translated up unit. The domains of f and g are. The range of f is 0 and the range of g is. The function f has - and -intercepts of 0 and g has a -intercept of. Notice that both f and g are smmetric with respect to the and are functions because f(2) 5 (2) 4 5 and g(2) 5 (2) 4 5. Checkpoint Complete the following eercise.. Graph g() Compare the graph with the graph of f() 5 4. f Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 8

8 Your Notes Eample 2 Translate a polnomial function horizontall Graph g() 5 } 2 ( 2) 3. Compare the graph with the graph of f() 5 } Make a table of values and plot the corresponding points. 2 f Connect the points with a smooth curve and check the end behavior. The degree is and the leading coefficient is. So, g() as and g() as. 3. Compare with f() 5 } 2 3. The graph of g() 5 } 2 ( 2) 3 is the graph of f() 5 } 2 3 translated to the left units. The domains and ranges of f and g are. The function f has - and -intercepts of 0 and g has an -intercept of and a -intercept of. Notice that f is smmetric with respect to the and is an function because f(2) 5 } 2 (2) 3 5. Notice that g is because g(2) 5 } 2 (2 2) 3, which is not equal to or. Checkpoint Complete the following eercise. 2. Graph g() 5 2( 2 3) 3. Compare the graph with the graph of f() f Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

9 Your Notes Eample 3 Translate a polnomial function Graph g() 52( 2 2) 4 3. Compare the graph with the graph of f() Make a table of values and plot the corresponding points f 2 2. Connect the points with a smooth curve and check the end behavior. The degree is and the leading coefficient is. So, g() as and g() as. 3. Compare with f() The graph of g() 52 ( 2 2) 4 3 is the graph of f() 52 4 translated units and. The domains of f and g are. The range of f is and the range of g is. The function f has - and -intercepts of 0 and g has -intercepts of about and and a -intercept of. Notice that f is smmetric with respect to the and is an function because f(2) 52(2) 4 5. Notice that g is because g(2) 52(2 2 2) 4 3, which is not equal to or. Checkpoint Complete the following eercise. 3. Graph g() 52( 4) 3 2. Compare the graph with the graph of f() f Homework 2 Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 83

10 Name Date LESSON 2.2 Practice Match the function with its graph ( ) 3 A. B. C. 2 Eplain how the graphs of f and g are related. 4. f() 5 3, g() f() 5 4, g() 5 ( 2 2) 4 6. f() 5 2 3, g() f() 5 2 4, g() 5 2 ( 3) 4 Graph the function. Compare the graph with the graph of f() g() 5 ( 2 ) 3 9. g() f f 84 Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

11 Name Date LESSON 2.2 Practice continued Graph the function. Compare the graph with the graph of f() g() g() 5 ( 2) 4 f f 2. Geometr The volume of a cube with side length inches is given b V 5 3. The volume of a cube with side length ( ) inches is given b V 2 5 ( ) 3. a. Cop and complete the table V V 2 b. Use the table from part (a) to graph V and V 2. Eplain how the graphs are related. Volume (cubic inches) V Side length (inches) c. Find the volume of each cube when 5 8. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 85

12 2.3 Factor and Solve Polnomial Equations Georgia Performance Standard(s) MM3A3b, MM3A3d Your Notes Goal p Factor and solve polnomial equations. VOCABULARY Factored completel Factor b grouping Quadratic form SPECIAL FACTORING PATTERNS Sum of Two Cubes a 3 b 3 5 (a b)(a 2 2 ab b 2 ) Eample ( 2)( ) Difference of Two Cubes a 3 2 b 3 5 (a 2 b)(a 2 ab b 2 ) Eample (2 2 )( ) 86 Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

13 Your Notes Eample Factor the sum or difference of two cubes Factor the polnomial completel. a. z z 3 2 Difference of 5 (z 2 )( ) two cubes b ( ) Factor common monomial. 5 3[ ] Sum of two cubes 5 3( )( ) Checkpoint Complete the following eercise.. Factor the polnomial completel. Eample 2 Factor b grouping Factor the polnomial completel ( ) 2 9( ) Factor b grouping. 5 Distributive propert 5 Difference of two squares Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 87

14 Your Notes Eample 3 Factor polnomials in quadratic form Factor completel: (a) and (b) a ( ) b ( ) 5 Checkpoint Factor the polnomial completel Eample 4 Find real-number solutions Find the real-number solutions of the equation Write original equation. 5 0 Write in standard form. 5 0 Factor trinomial. 5 0 Difference of two squares 5, 5, 5, 5 Zero product propert The solutions are. Homework Checkpoint Find the real-number solutions Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

15 Name Date LESSON 2.3 Practice Find the greatest common factor of the terms in the polnomial n n 3 6n p 6 2 5p 4 2 0p c 9 3 Match the polnomial with its factorization A. 2 3 ( 2)( 2 2)( 2 3) B. 2( 4)( 2 4) C. (3 2)( 3) D. ( 2 4)( 2 4) E. 2 2 ( 2 2 2)( 2) F. (5 2 6)( ) Factor the sum or difference of cubes. 3. s q a h Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 89

16 Name Date LESSON 2.3 Practice continued Factor the polnomial b grouping z 3 2 z 2 5z f 3 4f 2 f m 3 2 2m 2 4m t 3 2 2t 2 2 9t 8 Find the real-number solutions of the equation. 25. w 2 2 3w v 3 5v d s s Match the equation for volume with the appropriate solid. 3. V V V A. 2 2 B. C Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

17 2.4 Solve Polnomial Inequalities Georgia Performance Standard(s) MM3A3c Your Notes Goal p Solve polnomial inequalities. VOCABULARY Polnomial inequalit Intervals INEQUALITY AND INTERVAL NOTATION In the intervals below, the real numbers a and b are the endpoints of each interval. Bounded intervals: Inequalit a b a < < b a < b a < b Notation Unbounded intervals: Inequalit a > a b < b Notation Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 9

18 Your Notes Eample Solve a polnomial inequalit algebraicall Solve < 2 algebraicall. First, write and solve the equation obtained b replacing with Write equation that corresponds to original inequalit Write in standard form. 5 0 Factor. 5, 5, or 5 Zero product propert The numbers 0, 4, and 23 are the of the inequalit < 2. Plot 0, 4, and 23 on a number line, using because the values do not satisf the inequalit. The critical -values partition the number line into four intervals. Test an -value in each interval to see if it satisfies the inequalit. Test 5 2: (2) 3 2 (2) 2 5, Test 5 5: , Test 5 24: (24) 3 2 (24) 2 5, Test 5 : , The solution set consists of all real numbers in the intervals and. Checkpoint Complete the following eercise.. Solve > 0 algebraicall. 92 Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

19 Your Notes Eample 2 Solve a polnomial inequalit b graphing Solve b graphing. The solution consists of the -values for which the graph of lies or the -ais. Find the graph's -intercepts b letting 5 0 and solve for Set equal to Factor. 5, 5, or 5 Zero product propert Graph the polnomial and plot the the -intercepts,, and. The graph lies on or above the -ais between (and including) 5 and 5 and to the right of 0 (and including) 5. The solution set consists of all real numbers in the intervals and. Checkpoint Complete the following eercise. 2. Solve b graphing. Homework Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 93

20 Name Date LESSON 2.4 Practice Represent the inequalit using interval notation.. > < 6 Represent the inequalit using inequalit notation. 4. [22, 3] 5. (2, ) 6. [5, `) Tell whether the given -value is a solution of the inequalit < 0; ; 5 } > 0; ; ; < 22; 5 0 Solve the inequalit algebraicall < > < Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

21 Name Date LESSON 2.4 Practice continued Use the graph of the corresponding equation to determine the solution set of the inequalit > < Solve the inequalit using a graph > Geometr The volume V of a cube with a side length of inches is greater than 27 cubic inches. a. Cop and complete the table to determine the possible side lengths of the cube V 5 3 b. Check the solution ou found in part (a) b solving the inequalit 3 > 27 algebraicall. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 95

22 2.5 Appl the Remainder and Factor Theorems Georgia Performance Standard(s) MM3A3a Your Notes Goal p Use theorems to factor polnomials. VOCABULARY Polnomial long division Snthetic division Eample Use polnomial long division Divide f() b Write polnomial division in the same format ou use when dividing numbers. Include a 0 as the coefficient of 3. At each stage, divide the term with the highest power in what is left of the dividend b the first term of the divisor. This gives the net term of the quotient q www You can check the result of a division problem b multipling the quotient b the divisor and adding the remainder. The result should be the dividend. Write the result: }} Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

23 Your Notes REMAINDER THEOREM If a polnomial f() is divided b 2 k, then the remainder is r 5. Eample 2 Use snthetic division Divide f() b 2. Solution }} 2 5 FACTOR THEOREM A polnomial f() has a factor 2 k, if and onl if f(k) 5. Eample 3 Factor a polnomial Factor f() completel given that 2 3 is a factor. Solution Because 2 3 is a factor of f(), ou know that f(3) 5. Use snthetic division to find the other factors Use the result to write f() as a product of two factors and then factor completel. f() ( )( ) 5 ( )( )( ) Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 97

24 Your Notes Eample 4 Finding zeros of functions Find the other zeros of f() given that f(2) 5 0. Solution Because f(2) 5 0, is a factor of f. Use snthetic division to find the other factors Use the result to write f() as a product of two factors and then factor completel. f() ( )( ) 5 ( )( )( ) The zeros are. Checkpoint Complete the following eercises.. Use long division to divide b Use snthetic division to divide b 3. Homework 3. Factor f() given that 2 is a factor. 4. Find the other zeros of f() given that f(2) Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

25 Name Date LESSON 2.5 Practice Write the divisor, dividend, quotient, and remainder represented b the snthetic division Divide using polnomial long division. 3. ( ) 4 ( 2 ) 4. ( ) 4 ( 2) 5. ( ) 4 ( 6) 6. ( ) 4 ( 5) 7. ( ) 4 ( 2 4) 8. ( ) 4 ( 3) 9. ( 2 4) 4 ( 2 2) 0. ( ) 4 ( 7) Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 99

26 Name Date LESSON 2.5 Practice continued Divide using snthetic division.. ( ) 4 ( 9) 2. ( ) 4 ( ) 3. ( ) 4 ( 2 2) 4. ( ) 4 ( 3) 5. ( ) 4 ( 4) 6. ( ) 4 ( 2 5) 7. ( 3 2) 4 ( 2 ) 8. ( 2 2 7) 4 ( 2) You are given an epression for the area of the rectangle. Find an epression for the missing dimension. 9. A A A ??? Publishing The profit P (in thousands of dollars) for an educational publisher can be modeled b P 5 2b 3 5b 2 b where b is the number of workbooks printed (in thousands). Currentl, the publisher prints 5000 workbooks and makes a profit of $5000. What lesser number of workbooks could the publisher print and still ield the same profit? 00 Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

27 2.6 Find Rational Zeros Georgia Performance Standard(s) MM3A3a, MM3A3d Your Notes Goal p Find all real zeros of a polnomial function. THE RATIONAL ROOT THEOREM If f() 5 a n n... a a 0 has coefficients, then ever rational zero of f has the following form: p } q 5 factor of constant term }}} factor of leading coefficient Eample Find zeros when the leading coefficient is Find all real zeros of f() List the possible rational zeros. The leading coefficient is and the constant term is. So, the possible rational zeros are: 5,,, 2. Test these zeros using snthetic division. Test 5 : is a zero. 3. Factor the trinomial and use the factor theorem. f() 5 ( )( ) 5 The zeros of f are. Checkpoint Find all real zeros of the function.. f() Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 0

28 Your Notes Eample 2 Find zeros when the leading coefficient is not Find all real zeros of f() List the possible rational zeros of f: 2. Choose a reasonable value to check using the graph of the function. 3. Check 5 : is a zero. 4. Factor out a binomial. f() 5 5 Write as a product of factors. Factor out. 5 Multipl b. 5. Repeat the steps above for g() 5. An zero of g will also be a zero of f. Snthetic division shows that is a zero and ields the quotient. Factoring a 4 out of the quotient ields f() Find the remaining zeros b solving Use quadratic formula. Simplif. The real zeros of f are. 02 Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

29 Your Notes Eample 3 Solve a multi-step problem Sandbo You are building a wooden square sandbo for a local plaground. You want the volume of the bo to be 6 cubic feet. You want the height of the bo to be feet and the length of each side of the square base to be 3 feet. What are the dimensions?. Write an equation for the volume of the sandbo. The volume is V 5 Bh where B 5 base area and h 5 height. Volume (cubic feet) 5 Area of base (square feet) p Height (feet) 6 5 ( 3) 2 p 6 5 Write the equation. 6 5 Multipl List the possible rational solutions: Subtract from each side. 3. Test the possible rational solutions. Onl positive -values make sense. 4. Check for other solutions. The other possible rational solutions solutions, so 5 is the solution. The height of the sandbo should be foot and each side of the base should be 5 feet. Homework Checkpoint Find all real zeros of the function. 2. f() Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 03

30 Name Date LESSON 2.6 Practice. Can ou use the rational root theorem to find the zeros of the polnomial function f() ? Eplain wh or wh not. List the possible rational zeros of the function using the rational root theorem. 2. f() g() h() f() h() g() Use snthetic division to decide which of the following are zeros of the function: 23, 2,, f() g() g() h() Find all rational zeros of the function. 2. f() f() Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

31 Name Date LESSON 2.6 Practice continued Find all real zeros of the function. 4. f() g() h() h() g() f() f() g() Crafts You have 8 cubic inches of wa, and ou want to make a candle in the shape of a pramid with a square base as shown. a. Write an equation that shows that the volume of the candle is 8 cubic inches. 3 b. Use the rational root theorem to list all possible rational solutions of the equation in part (a). c. Find all real values of that are valid as a dimension of the candle. d. Find the dimensions of the candle. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 05

32 2.7 Appl the Fundamental Theorem of Algebra Georgia Performance Standard(s) MM3A3a Your Notes Goal p Classif the zeros of polnomial functions. VOCABULARY Repeated solution THE FUNDAMENTAL THEOREM OF ALGEBRA Theorem: If f() is a polnomial of degree n where n 0, then the equation f() 5 0 has at least solution in the set of comple numbers. Corollar: If f() is a polnomial of degree n where n 0, then the equation f() 5 0 has eactl solutions provided each solution repeated twice is counted as solutions, each solution repeated three times is counted as solutions, and so on. Eample Find the number of solutions or zeros Find the number of solutions or zeros for each equation or function. a. Because is a degree polnomial equation, it has solutions. b. Because f() is a degree polnomial function, it has zeros. Checkpoint Complete the following eercise.. State the number of zeros of f() Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

33 Your Notes Eample 2 Find the zeros of a polnomial function Find all zeros of f() Solution. Find the rational zeros of f. Because f is a fifth-degree function, it has zeros. The possible rational zeros are. Using snthetic division, ou can determine that is a zero repeated twice and is also a zero. 2. Write f () in factored form. Dividing f b its known factors gives a quotient of. So, f() Find the comple zeros of f. Use the quadratic formula to factor the trinomial into linear factors. f() 5 The zeros of f are. Checkpoint Find all zeros of the polnomial function. 2. f() COMPLEX CONJUGATES THEOREM If f is a polnomial function with coefficients, and is an imaginar zero of f, then is also a zero of f. IRRATIONAL CONJUGATES THEOREM Suppose f is a polnomial function with coefficients, and a and b are rational numbers such that Ï } b is irrational. If is a zero of f, then is also a zero of f. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 07

34 Your Notes You can check this result b evaluating f() at each of its three zeros. Eample 3 Use zeros to write a polnomial function Write a polnomial function f of least degree that has rational coefficients, a leading coefficient of, and 22 and 3 i as zeros. Because the coefficients are real and 3 i is a zero, must also be a zero. Use the three zeros and the factor theorem to write f() as a product of three factors. f () 5 ( )[ 2 ( )][ 2 ( )] Factored form 5 ( )[ ][ ] Regroup terms. 5 Multipl. 5 Epand, use i Simplif. 5 Multipl. 5 Combine like terms. Checkpoint Complete the following eercise. 3. Write a polnomial function f of least degree that has rational coefficients, a leading coefficient of, and 4 and Ï } 6 as zeros. DESCARTES RULE OF SIGNS Let f() 5 a n n a n 2 n 2... a 2 2 a a 0 be a polnomial function with real coefficients. The number of real zeros of f is equal to the number of changes in sign of the coefficients of or is less than this b an number. The number of real zeros of f is equal to the number of changes in sign of the coefficients of or is less than this b an number. 08 Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

35 Your Notes Eample 4 Use Descartes rule of signs Determine the possible numbers of positive real zeros, negative real zeros, and imaginar zeros for f() Solution f() The coefficients of f() have sign changes, so f has positive real zero(s). f(2) 5 2(2) 5 2 7(2) 4 2(2) 3 2(2) 2 4(2) 6 5 The coefficients of f(2) have sign changes, so f has negative real zero(s). Positive real zeros Negative real zeros Imaginar zeros Total zeros Checkpoint Complete the following eercise. 4. Determine the possible numbers of positive real zeros, negative real zeros, and imaginar zeros for f() Homework Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 09

36 Name Date LESSON 2.7 Practice Identif the number of solutions or zeros f(t) 5 t 3 4t g(z) 5 2z 3 z 2 6z h() f() r 3 2 r 2 5r Given that f() has real coefficients and 5 k is a zero, what other number must be a zero? 9. k Ï } 2 0. k 5 i. k i 2. k 5 Ï } 3 2 i 3. k 5 Ï } 5 i 4. k 5 Ï } 2 Ï } 7 i Find all the zeros of the polnomial function. 5. g() f() h() g() f() h() Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

37 Name Date LESSON 2.7 Practice continued Write a polnomial function f of least degree that has rational coefficients, a leading coefficient of, and the given zeros , , , 0, 25. 2, 2, , 2, The graph f() is shown at the right. How man real zeros does the function have? How man imaginar zeros does the function have? 28. Geometr A square piece of sheet metal is 0 inches b 0 inches. Squares of side length are cut from the corners and the remaining piece is folded to make an open bo. The volume of the bo is modeled b V() What size square(s) can be cut from the corners to give a bo with a volume of 25 cubic inches? 0 in. in. in. 0 in. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3

38 2.8 Analze Graphs of Polnomial Functions Georgia Performance Standard(s) MM3Ab, MM3Ad Your Notes Goal p Use intercepts to graph polnomial functions. VOCABULARY Local maimum Local minimum Multiplicit of a root MULTIPLICITY OF A ROOT For the polnomial equation f() 5 0, k is a repeated solution, or a root with a, if and onl if the factor 2 k has an eponent greater than when f() is factored completel. If the eponent is, the graph of f the -ais at the zero. If the eponent is, the graph of f the -ais at the zero. 2 Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

39 Your Notes Eample Graph the function f() 5 } ( ) 2 ( 2 4). 4 Use -intercepts to graph a polnomial function. Use the intercepts. Because and are zeros of f, plot (, ) and (, ). 2. Plot points between and beond the -intercepts Determine the end behavior and multiplicit. Because f has factors of the form 2 k, and a constant factor of, it is a function with a leading coefficient. So, f() as 2` and f() as `. The zero 2 is repeated. So, the graph of f the -ais at (2, 0). 4. Draw the graph so that it passes through the plotted points and has the appropriate end behavior. Checkpoint Complete the following eercise.. Graph the function f() 5 2( 2 2)( )( 2 ). TURNING POINTS OF POLYNOMIAL FUNCTIONS The graph of ever polnomial function of degree n has at most turning points. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 3

40 Your Notes Eample 2 Find turning points Graph the function. Identif the -intercepts and the points where the local maimums and local minimums occur. a. f() b. f() a. Use a graphing calculator to graph the function. Notice that the graph of f has -intercepts and turning points. Use the graphing calculator s zero, maimum, and minimum features to approimate the coordinates of the points. The -intercepts of the graph are. The function has a local maimum at (, ) and a local minimum at (, ). b. Use a graphing calculator to graph the function. Notice that the graph of f has -intercepts and turning points. Use the graphing calculator s zero, maimum, and minimum features to approimate the coordinates of the points. The -intercepts of the graph are. The function has local maimums at (, ) and (, ). The function has a local minimum at (, ). Checkpoint Complete the following eercise. Homework 2. Use a graphing calculator to identif the -intercepts, local maimums, and local minimums of the graph of f() Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

41 Name Date LESSON 2.8 Practice. True or False If k is a zero of the polnomial function f, then k is an -intercept of the graph of f(). Eplain our answer. Determine the lowest-degree polnomial that has the given graph Estimate the coordinates of each turning point and state whether each corresponds to a local maimum or a local minimum Match the graph with its function. 8. f() f() f() A. B. C. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 5

42 Name Date LESSON 2.8 Practice continued Determine the -intercepts of the function.. g() 5 ( 4)( 2 ) 2. h() 5 ( 2 2)( 2 3) 3. f() 5 ( 4)( 2 5) 4. f() 5 ( 3)( )( 2 8) 5. g() 5 ( 6) 2 6. h() 5 ( 2 )( 2 7) 2 Graph the function. 7. f() 5 ( )( 2 2) 8. g() 5 ( 2 3)( 2 ) 9. h() 5 ( 6)( 7) h() 5 0.9( 5)( 2 2) 2. g() 5 ( 2 3) f() 5 ( )( 2 )( 2 3) Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

43 Name Date LESSON 2.8 Practice continued 23. Let f be a fourth-degree polnomial function with the zeros 22, 6, 2i, and 22i. a. How man distinct linear factors does f() have? b. How man distinct solutions does f() 5 0 have? c. What are the -intercepts of the graph of f? 24. Manufacturing You are designing an open bo from a piece of cardboard that is 8 inches b 8 inches. Squares of side length are cut from the corners and the remaining piece is folded to make an open bo. The volume of the bo is given b the function V Using a graphing calculator, ou would obtain the graph shown below. a. What is the domain of the volume function? Eplain. b. Use the graph to estimate the length of the cut that will maimize the volume of the bo. c. Estimate the maimum volume the bo can have. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 7

44 Words to Review Give an eample of the vocabular word. Polnomial Polnomial function Degree of a polnomial function Leading coefficient Standard form of a polnomial function Snthetic substitution End behavior Even function Odd function Factored completel 8 Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.

45 Factor b grouping Quadratic form Polnomial inequalit Intervals Polnomial long division Snthetic division Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3 9

46 Repeated solution Local maimum Local minimum Multiplicit of a root 20 Georgia Notetaking Guide, Mathematics 3 Copright McDougal Littell/Houghton Mifflin Compan.