PreCalculus Honors: Functions and Their Graphs. Unit Overview. Student Focus. Example. Semester 1, Unit 2: Activity 9. Resources: Online Resources:

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1 Resources: SpringBoard- PreCalculus PreCalculus Honors: Functions and Their Graphs Semester 1, Unit 2: Activity 9 Unit Overview In this unit, students study polynomial and rational functions. They graph these functions and find zeros. They eplore comple factors of polynomial functions. Students also solve polynomial inequalities. Online Resources: PreCalculus SpringBoard Tet Unit 2 Vocabulary: Efficiency Relative maimum Relative minimum Turning points Polynomial function End behavior Increasing Decreasing Multiplicity Multiple root Fundamental Theorem of Algebra Linear Factorization Theorem Rational Root Theorem Factor Theorem Remainder Theorem Descartes Rule of Signs Comple Conjugate Theorem Bounded Horizontal asymptote Vertical asymptote Parameter Hole Oblique asymptote Student Focus Main Ideas for success in lessons 9-1 & 9-2: Model data with polynomial functions. Compare models to best fit a data set. Describe and analyze graphs of polynomial functions. Graph polynomial functions and identify key features and end behavior of the graphs of polynomial functions. Eample Lesson 9-1: Page 1 of 24

2 Page 2 of 24 Lesson 9-2:

3 PreCalculus Honors: Functions and Their Graphs Semester 1, Unit 2: Activity 10 Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Tet Unit 2 Vocabulary: Efficiency Relative maimum Relative minimum Turning points Polynomial function End behavior Increasing Decreasing Multiplicity Multiple root Fundamental Theorem of Algebra Linear Factorization Theorem Rational Root Theorem Factor Theorem Remainder Theorem Descartes Rule of Signs Comple Conjugate Theorem Bounded Horizontal asymptote Vertical asymptote Parameter Hole Oblique asymptote Unit Overview In this unit, students study polynomial and rational functions. They graph these functions and find zeros. They eplore comple factors of polynomial functions. Students also solve polynomial inequalities. Student Focus Main Ideas for success in lessons 10-1, 10-2, & 10-3 Use information about end behavior and min/ma to sketch polynomial functions. Use the Fundamental Theorem of Algebra and the Linear Factorization Theorem. Use the Rational Root Theorem to find the zeros of a polynomial function. Eplore the Factor Theorem and the Remainder Theorem. Eample Lesson 10-1: Difference of Squares Difference of Cubes Sum of Cubes Page 3 of 24

4 Lesson 10-2: 1. Find factors of the leading coefficient, and the constant term, q. 2. Write all combinations of. 3. Write all possible rational zeros. Page 4 of 24

5 Page 5 of 24 Use the Rational Root Theorem to find the possible real zeros and the Factor Theorem to find the zeros.

6 Page 6 of 24 Lesson 10-3:

7 Page 7 of 24

8 Resources: SpringBoard- PreCalculus PreCalculus Honors: Functions and Their Graphs Semester 1, Unit 2: Activity 11 Unit Overview In this unit, students study polynomial and rational functions. They graph these functions and find zeros. They eplore comple factors of polynomial functions. Students also solve polynomial inequalities. Online Resources: PreCalculus Springboard Tet Unit 2 Vocabulary: Efficiency Relative maimum Relative minimum Turning points Polynomial function End behavior Increasing Decreasing Multiplicity Multiple root Fundamental Theorem of Algebra Linear Factorization Theorem Rational Root Theorem Factor Theorem Remainder Theorem Descartes Rule of Signs Comple Conjugate Theorem Bounded Horizontal asymptote Vertical asymptote Parameter Hole Oblique asymptote Student Focus Main Ideas for success in lessons 11-1, 11-2, 11-3 Rewrite polynomials in factored form. Find the zeros of a function, including zeros that are comple factors. Solve polynomial inequalities. Eample Lesson 11-1: The Comple Conjugate Theorem states that for a polynomial function with real coefficients, if is a root, with and real numbers, then its comple conjugate is also a root of the polynomial. Page 8 of 24

9 Lesson 11-2: The Zero Product Property says that for any product (a 1 )(a 2 )(a 3 )... (a n ) = 0, a 1 = 0 or a 2 = 0 or a 3 = 0 or... a n = 0. This is true for polynomial factors as well as real numbers. Page 9 of 24

10 Page 10 of 24

11 Page 11 of 24 Lesson 11-3:

12 Resources: SpringBoard- PreCalculus PreCalculus Honors: Functions and Their Graphs Semester 1, Unit 2: Activity 12 Unit Overview In this unit, students study polynomial and rational functions. They graph these functions and find zeros. They eplore comple factors of polynomial functions. Students also solve polynomial inequalities. Online Resources: PreCalculus Springboard Tet Unit 2 Vocabulary: Efficiency Relative maimum Relative minimum Turning points Polynomial function End behavior Increasing Decreasing Multiplicity Multiple root Fundamental Theorem of Algebra Linear Factorization Theorem Rational Root Theorem Factor Theorem Remainder Theorem Descartes Rule of Signs Comple Conjugate Theorem Bounded Horizontal asymptote Vertical asymptote Parameter Hole Oblique asymptote Student Focus Main Ideas for success in lessons 12-1 and 12-2 Will write rational epressions and rational functions that model real-world situations. Eamine asymptotic behaviors and sketch graphs of rational functions. Emphasize end behaviors. Eample Lesson 12-1: Page 12 of 24

13 Lesson 12-2: The denominator will equal zero if were to equal ; therefore there is a vertical asymptote at. Page 13 of 24

14 Resources: SpringBoard- PreCalculus PreCalculus Honors: Functions and Their Graphs Semester 1, Unit 2: Activity 13 Unit Overview In this unit, students study polynomial and rational functions. They graph these functions and find zeros. They eplore comple factors of polynomial functions. Students also solve polynomial inequalities. Online Resources: PreCalculus Springboard Tet Unit 2 Vocabulary: Efficiency Relative maimum Relative minimum Turning points Polynomial function End behavior Increasing Decreasing Multiplicity Multiple root Fundamental Theorem of Algebra Linear Factorization Theorem Rational Root Theorem Factor Theorem Remainder Theorem Descartes Rule of Signs Comple Conjugate Theorem Bounded Horizontal asymptote Vertical asymptote Parameter Hole Oblique asymptote Student Focus Main Ideas for success in lessons 13-1, 13-2, and Graph rational functions and transformations of rational functions. Eamine asymptotic behavior of rational functions. Write functions involving rational equations and solve ration inequalitites. Eample Lesson 10-1: Page 14 of 24

15 Lesson 13-2: Find the equation of the slant asymptote. Analyze the behavior of each function around its slant asymptote. a.) b.) Lesson 13-3: Write a function whose graph has an asymptote, the line asymptotes at., and a vertical Page 15 of 24

16 Name class date Lesson Eamine the data in the table. What type of function could be used to model the data? Eplain your reasoning. University of XYZ Year Number of Enrolled Students (in thousands) Precalculus Unit 2 Practice Model with mathematics. Use the regression capabilities of your graphing calculator to find a model that best represents the data in Item Data are collected on puppies and their growth (by weight) over a 6-month period. What type of function could be used to model the data for the weight as a function of the number of months since the birth of a puppy? Eplain your reasoning. 3. Attend to precision. Graph the equation you found in Item 2. List the important features of the graph. Approimate any values to three decimal places. y Lesson Which of the following functions is a fifth degree polynomial? A. f() 5 ( 2 3) 2 ( 1 2) B. f() ( 1 5)( 2 7) C. f() 5 ( 2 1)( 1 9) 3 ( 2 2) D. f() 5 5 ( 2 11) 4. Use the regression model to predict about how many students will be attending University XYZ in A. 34,000 B. 36,000 C. 38,000 D. 2,899, College Board. All rights reserved. 1 SpringBoard Precalculus, Unit 2 Pratice Page 16 of 24

17 Name class date 7. Make use of structure. List the important features of the graph below. Approimate any values to three decimal places. 5 y Lesson 10-1 Make use of structure. For Items 11 and 12, determine the y-intercept and the end behavior of each function. 11. f() h() 5 4( 2 1) 2 ( 2 3)( 1 2) What are the zeros of the function f() 5 ( 1 2) 4 ( 2 3) 5? A. 5 2, B. 5 22, 3 23 C. 5 4, 5 24 D. 5 24, Attend to precision. Factor and find the zeros of the function. g(m) 5 m 4 2 2m 3 1 8m Without using a calculator, determine the end behavior and - and y-intercepts of the function f () 5 (2 2 1)( 1 1)( 1 3). 15. Graph f() 5 ( 2 1)( 1 5) 2. f() 9. Without using a calculator, find the end behavior, maimum possible zeros, and maimum possible turning points of the function f () Use appropriate tools strategically. Use a graphing calculator to find the zeros, turning points, y-intercepts, and end behavior. y College Board. All rights reserved. SpringBoard Precalculus, Unit 2 Pratice Page 17 of 24

18 Name class date Lesson Which of the following are possible zeros of f() ? A B C D Use the Rational Root Theorem to find the possible real zeros and the Factor Theorem to find the zeros of the function. u(t) 5 3t 3 2 5t 2 1 6t 1 8 Lesson Which function has the greatest number of sign changes? A. f() B. g() C. h() D. j() Make use of structure. For Items 22 and 23, determine the number of positive and negative real zeros. 22. f() Make use of structure. For Items 18 and 19, use the Rational Root Theorem and synthetic division to find the real zeros. 18. f() f() Attend to precision. For Items 24 and 25, sketch a graph of the polynomial function. 24. f() f() 19. g() Use appropriate tools strategically. Use the Rational Root Theorem and a graphing calculator to find the rational root(s) of the polynomial. f() College Board. All rights reserved. SpringBoard Precalculus, Unit 2 Pratice Page 18 of 24

19 Name class date 25. f() f() b. Sketch and label a graph of the volume function. f() Lesson 11-1 Orange Cell Phone Company is creating a package for their newest phone. The bo package is made from cardboard pieces that are 14 inches long by 5 inches wide. The boes are made by cutting a section of size by out of each corner of the cardboard. 26. Model with mathematics. Write the equation that represents the volume of the bo. 28. Use a graphing calculator to find the maimum volume of a bo that the Orange Cell Phone Company can make. What are the dimensions of the bo with a maimum volume, and what is its volume? 29. Which polynomial has degree 6 and zeros , 0, 1 2, 1, 3 2? A. f () 5 ( 2 1) 2 ( 2 3)( 1 1)( 1 6) B. f () 5 ( 1 1 2) ( 1 1) ( 1 3 ( 2 1)( 2 6) 2) C. f () 5 (2 11)( 1 1)(2 1 3)( 2 1)( 2 6) D. f () 5 (2 21)( 2 1)(2 2 3)( 1 1) ( 1 6) 27. a. Determine the possible domain and range for the construction of these boes. 30. Make use of structure. Find a polynomial with real coefficients of given degree with the given zeros. degree: 4; zeros: 5 21, 0, 3, College Board. All rights reserved. SpringBoard Precalculus, Unit 2 Pratice Page 19 of 24

20 Name class date Lesson Which of the following is a factor of the function f () ? A. 1 8 B. 2 4 C. 2 2 D Make use of structure. For Items 32 and 33, rewrite each polynomial function as a product of comple factors. 32. f () g() Use a graphing calculator to determine the interval over which the volume of the frame boes made from the pieces of metal is larger than 28 cubic feet. 38. Use appropriate tools strategically. Using the piece of metal, what are the dimensions of the air conditioner with the largest possible volume? A. 1.7 feet by 4.6 feet by 5.8 feet B. 2 feet by 8 feet by 15 feet C. 2 feet by 4 feet by 3.5 feet D. 1.7 feet by 7.5 feet by 8 feet Solve each inequality and write the solution interval , 6 Attend to precision. For Items 34 and 35, find the zeros of each function. 34. h () $ r () Lesson 11-3 Model with mathematics. MetalBo Manufacturing also makes industrial air conditioning frames from an 8-foot-by-15-foot piece of metal. Square corners of length are cut from each piece. The volume of the frame bo must be at least 28 cubic feet. 36. Write an inequality for the volume that satisfies the constraint. Lesson 12-1 Model with mathematics. Student A and Student B are studying for the same test at the same rate. Both students studied through the dinner hour. Right now, Student B has been studying twice as many hours after the dinner hour as Student A. When both students study 1 more hour, Student B will have studied one and a half times as many hours from the dinner hour as Student A. Let h represent Student A s number of hours studied since the dinner hour. 41. Which equation can be used to find h, the current number of hours Student A has been studying? A. h 5 h 1 1 B. h (h 1 1) C. 2h 5 2h 1 1 D. 2h (h 1 1) College Board. All rights reserved. SpringBoard Precalculus, Unit 2 Pratice Page 20 of 24

21 Name class date 42. Find the number of hours Student A has studied right now since the dinner hour. 43. Let S() represent the ratio of Student A s number of hours studied since the dinner hour to Student B s number of hours studied since the dinner hour, and let represent the number of hours from now, either past or future. Write S as a function of. 44. What appears to happen to the ratio of Student A s number of hours since the dinner hour to Student B s number of hours since the dinner hour as increases? 45. Reason abstractly. If the two students keep studying at the same rate forever, would Student A ever catch up to Student B? Eplain. Lesson Model with mathematics. O() represents the ratio of Amy s age in years to Michael s age in years, where represents the number of years from now, either past or future. 46. Sketch a graph of the function O() for 215,, 15. y 47. Find the equation of the vertical asymptote of O(). A. 5 1 B C D Find the equation of the horizontal asymptote of O(). A. y 5 1 B. y 5 21 C. y 5 10 D. y Demonstrate why the value of O() will never actually reach the value of the horizontal asymptote. 50. Use appropriate tools strategically. Use a graphing calculator to look at the behavior of the O() to the left of the vertical asymptote. Eplain why you are not looking at that part of the graph for the given scenario. y College Board. All rights reserved. SpringBoard Precalculus, Unit 2 Pratice Page 21 of 24

22 Name class date Lesson Consider the graphs for f() 5 and g() Which of the following best describes the difference between the graphs of the two functions? A. g() is a shift of f() to the left 5 units. B. g() is a shift of f() to the right 5 units. C. g() is a shift of f() up 5 units. D. g() is a shift of f() down 5 units u() y For Items 52 and 53, sketch a graph of each function h() y 54. Attend to precision. Write to eplain to another 1 student how to obtain the graph of r() from the graph of f() Epress regularity in repeated reasoning. Write to eplain to another student the similarities in the 1 graphs of m() b, n() 5 ( 1 a a)2 1 b, and o() 5 1a 1 b when compared to the graphs of their respective parent functions College Board. All rights reserved. SpringBoard Precalculus, Unit 2 Pratice Page 22 of 24

23 Name class date Lesson What is the horizontal asymptote of f() ? A. y 5 0 B. y Attend to precision. Sketch a graph of the function without using a graphing calculator p() y C. y D. y What is the horizontal asymptote of g() ? A. y 5 0 B. y C. y D. y Lesson 13-3 Model with mathematics. For Items 61 63, write a possible function whose graph could have the following asymptotes. 61. y 5 24, Make use of structure. Find the equation of the slant asymptote of the function h() y 5 0, Find the vertical asymptote and the point discontinuity in the graph of the function. f () y , College Board. All rights reserved. SpringBoard Precalculus, Unit 2 Pratice Page 23 of 24

24 Name class date 64. Which of the following functions could have a point of discontinuity at 5 1? A. f() B. f() 5 ( 21)( 11) 11 C. f() 5 ( 11)( 21) 2 D. f() Critique the reasoning of others. Kayla says that a function that is a rational epression has only vertical asymptotes where the factors in the denominator equal zero. Is she correct? If not, eplain and correct her error College Board. All rights reserved. SpringBoard Precalculus, Unit 2 Pratice Page 24 of 24