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Cache County School District 2013-2014 Secondary II Math Utah Integrated Mathematics Core Teacher Edition Unit 6: Expanding Quadratic Ideas Secondary II Unit 6 – Expanding Quadratic Ideas: Table of Contents Homework Help (QR Codes and links to videos, tutorials, examples)…………………. Section 6.1 – Vertex and Vocabulary, Teacher Notes .............................................................. Notes, Assignment............................................................................................................ Section 6.2 –Graphing in Standard Form, Teacher Notes ....................................................... Notes, Assignment............................................................................................................ Section 6.3 – Identifying x and y Intercepts Task/Notes, Teacher Notes, Assignment……….. Section 6.4 - Parabolas and Vertex Form, Teacher Notes ....................................................... Notes, Assignment ...................................................................................................................... Graphing In Vertex Form (Translations) Review Assignment………………………. Section 6.5 – Three Ways to Represent Quadratic Functions Task, ………………………… Teacher Notes, Assignment……………………………................................................. Section 6.6 – Average Rate of Change Task, Teacher Notes…...…………………………... Assignment……………………………………………………………………….. Section 6.7 –Quadratic Models Task, Teacher Notes ............................................................... Notes, Assignment............................................................................................................ Section 6.8 – Quadratic Practice Your Understanding Task, Teacher Notes………….. Notes, Assignment ………………………………………………………………… Section 6.9 - Comparing Linear, Quadratic, and Exponential Functions Task, ………. Teacher Notes, Notes, Assignment………………………………………………… Section 6.10 – Fundamental Theorem of Algebra Task/Notes, Teacher Notes, Assignment… Section 6.11 –Exponentials Take All Task, Teacher Notes, Task…………………………….. Matching Parabolas to Quadratic Equations Sorting Activity………………………. Additional Quadratic Story Problems WKS/Task……………………………………. Unit 6 Lesson 1 – Vertex and Vocabulary Notes 6.1 Vocabulary Section: For questions 1-8: Fill in each blank using the word bank. See how many you can fill in without help. vertex minimum axis of symmetry x-intercepts parabola maximum zeros/roots ax2 + bx + c 1. Standard form of a quadratic function is y = 2. The shape of a quadratic equation is called a 3. 4. 5. When the vertex is the highest point on the graph, we call that a . 6. When the vertex is the lowest point on the graph, we call that a . 7. Our solutions are the . 8. Solutions to quadratic equations are called . When solving quadratic equations, you found there could different types of solutions. Two real solutions (rational or irrational) – a positive number inside the square root. One (repeated) real solution – zero inside the square root. Two complex solutions - a negative number inside the square root. Two real solutions One (repeated) real solution Two complex solutions two distinct xintercepts one (repeated) x-intercept no x-intercepts Determine whether the quadratic functions have two real roots, one real root, or no real roots. You will learn more about roots in the next section. 9. Number of roots: 10. Number of roots: 11. Number of roots: To find the vertex you can use the formula: You can tell a lot about the graph of a parabola by looking at and using its equation. Three things we will focus on today are…. Axis of Symmetry Vertex Direction of Opening Find the AOS, Vertex, and Direction of Opening for each of the following quadratic equations. Example 1: y 2 x 2 4 x 9 a ______ b ______ c ______ Vertex: ____, ____ Axis of Symmetry:______________ Direction of Opening: ___________ Example 2: y x 2 10 a ______ b ______ c ______ Vertex: ____, ____ Axis of Symmetry:______________ Direction of Opening: __________ Example 3: Vertex: ____, ____ Axis of Symmetry:______________ Direction of Opening: ___________ Example 4: y 2 x2 8x 8 a ______ b ______ c ______ Vertex: ____, ____ Axis of Symmetry:______________ Direction of Opening: ___________ Unit 6 Lesson 1 – Vertex and Vocabulary Ready, Set, Go! - Assignment 6.1 Name_________________________________ Date_________ Hour_______ http://goo.gl/T6h2g Ready 1. Vertex ______________ 2. Zeros_________________________ 3. Axis of symmetry _______________ 4. Does it have a maximum or a minimum?__________ 5. Opening up or down?_______________ 6. Vertex ______________ 7. Zeros_________________________ 8. Axis of symmetry _______________ 9. Does it have a maximum or a minimum? __________ 10. Opening up or down?_______________ 11. Vertex ______________ 12. Zeros_________________________ 13. Axis of symmetry _______________ 14. Does it have a maximum or a minimum? __________ 15. Explain in your own words how to find the vertex of a parabola, when given a quadratic equation in standard form. Set Write the equations in standard form (if needed), identify the coefficients, vertex, axis of symmetry (AOS), direction of opening, and sketch the graph 𝒚 = 𝒙𝟐 + 𝟖𝒙 + 𝟏𝟓 16. Standard form of the equation _________________________________________ 17. a = _________, b = _________, c = __________ 18. Vertex ______________________ 19. AOS _________________________ 20. Direction of opening _________________ 𝟔𝒙 − 𝟓 = 𝒙𝟐 − 𝒚 21. Standard form of the equation _________________________________________ 22. a = _________, b = _________, c = __________ 23. Vertex ______________________ 24. AOS _________________________ 25. Direction of opening _________________ 𝒇(𝒙) = −𝟑(𝒙 + 𝟑)𝟐 + 𝟒 26. Standard form of the equation _________________________________________ 27. a = _________, b = _________, c = __________ 28. Vertex ______________________ 29. AOS _________________________ 30. Direction of opening _________________ Go! Connect the quadratic equation with its parabola by tracing a line. Do this by finding the vertex and observing the sing of the coefficient a. 31. 33. 𝒚 = −𝟐𝒙𝟐 − 𝟖𝒙 − 𝟓 𝒚 = −𝟑𝒙𝟐 + 𝟏𝟐𝒙 − 𝟗 32. 34. 𝒚 = 𝟐x 2 − 3 𝒚 = 𝟑(𝒙 − 𝟑)𝟐 Unit 6 Lesson 2 – Graphing in Standard Form Notes 6.2 We have been spending a lot of time solving quadratic type equations. We have learned a few methods in the last unit: factoring, completing the square, and the quadratic formula. These skills can be used to find SOLUTIONS to quadratic equations. These solutions are closely related to: Today we will be learning how to graph quadratic functions in standard form. Steps for Graphing a Quadratic Function when in Standard Form 1. Find the vertex and axis of symmetry. 2. Plot the vertex on a coordinate plane. 3. Make a table of values, using x-values to the left and right of the vertex. 4. Find the solutions and plot them. 5. Connect the dots with a smooth curve. Example 1 Graphing a Quadratic Function with a Positive a-value: a) Sketch the graph of y x 2 1 a ______ b ______ c ______ M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com 7 y 6 5 4 Vertex: ____, ____ 3 2 x 1 Opens:______________ -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 Min or Max ________ -4 -5 -6 x y # of Zeros: ___________________ Zeros: _______________________ -7 1 2 3 4 5 6 7 a ______ b ______ c ______ b) Sketch the graph of y x 2 4 x 5 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com y 11 10 9 8 Vertex: 7 ____, ____ 6 5 Opens:______________ 4 3 Min or Max ________ 2 x 1 x y -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 -2 # of Zeros: ___________________ -3 Zeros: _______________________ a ______ b ______ c ______ c) Sketch the graph of 𝑓(𝑥) = 5𝑥 2 − 6 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com 5 4 y 3 2 x 1 Vertex: ____, ____ Opens:______________ -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 Min or Max ________ x y # of Zeros: ___________________ Zeros: _______________________ -5 -6 -7 1 2 3 4 5 6 7 8 Unit 6 Lesson 2 – Graphing in Standard Form Ready, Set, Go! - Assignment 6.2 Name_________________________________ Date_________ Hour_______ http://goo.gl/tMMid Ready Answer the following questions. Write your answer in the spaces provided. You may need to do your work on a separate piece of paper. 1. y x2 2x 2 Vertex: ____, ____ 2. y x 2 5 Vertex: ____, ____ # of Zeros: ___________________ # of Zeros: ___________________ Zeros: _______________________ Zeros: _______________________ 3. y 4 x2 8x 1 Vertex: ____, ____ 4. y x 2 2 x 2 Vertex: ____, ____ # of Zeros: ___________________ # of Zeros: ___________________ Zeros: _______________________ Zeros: _______________________ 5. y x 2 6. y x 2 4 x Vertex: ____, ____ Vertex: ____, ____ # of Zeros: ___________________ # of Zeros: ___________________ Zeros: _______________________ Zeros: _______________________ Set Answer the questions and sketch a graph for each of the following. Sketch your own graph in the space provided and be sure to label your axis for full credit. 7. Sketch the graph of y 3x 2 6 x Vertex: ____, ____ Opens:______________ Min or Max” ___________ x y Zeros: _______________________ 8. Sketch the graph of y x 2 3x Vertex: ____, ____ Opens:______________ Min or Max: ___________ x y Zeros: _______________________ 9. Sketch the graph of y x 2 4 Vertex: ____, ____ Opens:______________ Min or Max x y Zeros: _______________________ Go! Sketch the graph of each function. Show your work and label key information about each graph that helped you sketch an accurate curve. Be sure to label your axis and use an appropriate scale. 10. 𝑦 = 𝑥 2 + 7𝑥 + 9 11. 𝑦 = 𝑥 2 − 5𝑥 + 5 12. 𝑦 = 2𝑥 2 − 10𝑥 + 16 13. 𝑦 = −𝑥 2 − 5𝑥 − 7 14. 𝑦 = 𝑥2 15. 𝑦 = 𝑥 2 − 6𝑥 + 7 Unit 6 Lesson 3 – Identifying X and Y intercepts Task/Notes 6.3 Name__________________________________ Date_______ Hour_______ The x-intercept is the point where a graph crosses or touches the x-axis. It is the ordered pair (x, 0), where x is a real number. The x-intercept is where y = 0. The x-intercept of this graph is the point (-1, 0). The y-intercept is the point where a graph crosses or touches they y-axis. It is the ordered pair (0, y), where y is a real number. The y-intercept is where x = 0. The y-intercept of this graph is the point (0, 1). Example 1: Find the x- and y-intercept(s) of each graph. x-int: x-int: x-int: y-int: y-int: y-int: Example 2: Given a table find the x and y intercept(s) x y x y x y -2 1 9 -2 0 -2 -3 -1 1 3 -1 1 -1 0 0 1 0 2 0 -1 1 3 1 3 1 0 2 9 2 4 2 3 Example 3: Find the x and y itnercep(t) of the following equations. a. 𝑦 = 𝑥 2 − 4 b. 𝑓(𝑥) = 3𝑥 2 − 5𝑥 − 2 Example 4: Find the x and y intercepts of the problems in Example 3 using your graphing calculator. a. 𝑦 = 𝑥 2 − 4 b. 𝑓(𝑥) = 3𝑥 2 − 5𝑥 − 2 Unit 6 Lesson 3 – Identifying X and Y intercepts Ready, Set, Go! – Assignment 6.3 Name__________________________________ Date_______ Hour_______ Ready 1. Find the x and y intercepts of each graph below. Set 2. Find the x and y intecepts of each table below. x 0 1 2 3 4 5 6 y 5 0 -3 -4 -3 0 5 x -3 -2 -1 0 1 2 3 y 9 4 1 0 1 4 9 x -1 0 1 2 3 y 5 2 1 2 5 Go! Find the x and y intercept(s) of each equation. 3. 𝑦 = 𝑥 2 − 16 4. 𝑦 = (𝑥 − 1)(𝑥 + 2) 5. 𝑔(𝑥) = (𝑥 − 1)2 + 2 6. 𝑓(𝑥) = 𝑥 2 + 4 7. 𝑓(𝑥) = 𝑥 2 + 3𝑥 − 2 8. 𝑓(𝑥) = −2𝑥 2 − 3𝑥 + 4 9. 𝑦 = −𝑥 2 + 1 10. 𝑓(𝑥) = 2𝑥 2 + 5𝑥 − 3 Unit 6 Lesson 4 – Parabolas and Vertex Form Notes 6.4 Quick Review: Use the equation provided to graph and answer the following questions. 𝟏. 𝒚 = 𝒙𝟐 − 𝟒𝒙 − 𝟓 Vertex: AOS: Zeros: Max or Min? Direction of Opening: Notes about the equivalent form “vertex form”: Complete the square to write in an equivalent form and identify the vertex, if the function has a min or max, and direction of opening for each: 2. 𝑦 = −5𝑥 2 + 9 3. 𝑦 = 𝑥 2 − 2𝑥 − 5 Write each function in its equivalent form of 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. Identify the vertex, if the function has a min or max, and the direction of opening for each: 4. 𝑦 = (𝑥 − 3)2 − 4 5. 𝑦 = −2(𝑥 + 8)2 − 2 6. Describe a few pros and cons to the form 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 𝑎𝑛𝑑 𝑦 = (𝑥 − ℎ)2 + 𝑘 and why you may want to change the equation to the equivalent form. For each of the following parabolas, complete the following: A) Identify the vertex. B) Identify the x-intercept(s), if any. C) Identify the y-intercepts D) Graph. Don’t forget to label your axis and scale. E) Intervals where parabola is increasing and decreasing. 7. 𝑦 = (𝑥 + 2)2 + 3 8. 𝑓(𝑥) = 𝑥 2 − 9 9. 𝑦 = 𝑥 2 − 10𝑥 + 21 A) ____________________ A) ____________________ A) _____________________ B) ____________________ B) B) ____________________ ____________________ C) ____________________ C) ____________________ C) ____________________ D) D) D) ____________________ E) Additional Notes/Examples: E) ____________________ E) ____________________ Unit 6 Lesson 4 – Parabolas and Vertex Form Ready, Set, Go! - Assignment 6.4 Name____________________________________ Date__________ Hour________ http://goo.gl/8njDI Expand each quadratic and write in Standard Form. Identify the Vertex and direction of opening for each: Vertex Form Standard Form Vertex Opening 2 1. y ( x 3) - 10 2 2. y ( x 5) 4 2 3. y 2(x 1) 7 4. y 3( x 4)2 1 5. y 2( x 3)2 6. y ( x 3)2 10 Now, take each of these equations and rewrite in Vertex Form. Then identify the vertex. Standard Form 2 7. y x 8 x 1 2 8. y x 6 x 17 2 9. y x 5 x 11 2 10. y x 10x 11. y 4 x 2 8 x 1 12. y 4 x 2 13. Create your own equation here. Vertex Form Vertex For each of the following parabolas, complete the following: A) Write in an equivalent form. B) Identify the vertex. C) Identify the x-intercept(s), if any. D) Identify the y-intercepts E) Graph F) Intervals where the parabola is increasing and decreasing. 14. 15. 𝑓(𝑥) = 𝑥 2 + 4𝑥 + 4 𝑦 = −2(𝑥 − 3)2 + 1 A) _____________________ A) _____________________ B) B) ____________________ ____________________ C) ____________________ C) ____________________ D) D) ____________________ E) E) M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com 7 y 7 6 6 5 5 4 4 3 3 2 2 1 -7 F) ____________________ -6 -5 -4 -3 -2 -1 1 x 1 2 3 4 5 6 7 y -7 -6 -5 -4 -3 -2 -1 1 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 -7 -7 F) x 2 3 4 5 6 7 16. 𝑓(𝑥) = 𝑥 2 − 𝑥 − 6 17. 𝑦 = −𝑥 2 − 6𝑥 A) _____________________ A) _____________________ B) B) ____________________ ____________________ C) ____________________ C) ____________________ D) D) ____________________ E) E) M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com 7 y 7 6 6 5 5 4 4 3 3 2 2 1 -7 F) ____________________ -6 -5 -4 -3 -2 -1 1 x 1 2 3 4 5 6 7 y -7 -6 -5 -4 -3 -2 -1 1 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 -7 -7 F) x 2 3 4 5 6 7 Graphing Parabolas Using Transformations Review Name____________________________________ Date__________ Hour________ Sketch the following graph without creating a table of values. 1. 𝑦 = (𝑥 − 2)2 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com y 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 1 2 3 4 5 6 8 7 6 5 4 3 2 1 -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com 1 2 3 4 5 6 5. 𝑦 = (𝑥 − 6)2 + 2 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com y x 1 2 3 4 5 6 7 8 8. x 1 2 3 4 5 6 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com 1 2 3 4 5 6 7 8 𝑦 = (−𝑥 + 5)2 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com y 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 1 2 3 4 5 6 6. 𝑦 = (𝑥 − 5)2 x -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8 1 2 3 4 5 6 7 8 9. 𝑦 = −(𝑥 − 3)2 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com y x 1 2 3 4 5 6 y 8 7 6 5 4 3 2 1 x -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8 x -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 y 8 7 6 5 4 3 2 1 y 6 5 4 3 2 1 x -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 7. 𝑦 = −(𝑥 − 2)2 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com 3. 𝑦 = (𝑥 − 3)2 y 6 5 4 3 2 1 x 4. 𝑦 = (𝑥 − 8)2 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com 𝑦 = (𝑥 + 5)2 1. 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 y x 1 2 3 4 5 6 10. 𝑦 = (−𝑥 − 8)2 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com 8 7 6 5 4 3 2 1 -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8 11. 𝑦 = −(𝑥 − 6)2 + 2 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com y x 1 2 3 4 5 6 7 8 -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8 13. 𝑦 = −(𝑥 − 1)2 − 1 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 8 7 6 5 4 3 2 1 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com y M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com 6 5 4 3 2 1 x -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 8 7 6 5 4 3 2 1 x -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8 1 2 3 4 5 6 7 8 14. 𝑦 = (−𝑥 − 2)2 + 1 y 1 2 3 4 5 6 12. 𝑦 = −(𝑥 − 5)2 + 1 x 1 2 3 4 5 6 7 8 15. 𝑦 = −(𝑥 − 3)2 − 2 M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com y x 1 2 3 4 5 6 y 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 y x 1 2 3 4 5 6 Write the equation for the described parabola: 16. Shifted 2 units to the right and 4 units down _____________________________________________ 17. Shifted 3 units up and three units to the left, ______________________________________________ 18. Shifted one unit to the left, three units up, and reflected on the x axis __________________________ Unit 6 Lesson 5 – Three Ways to Represent Quadratic Functions Task 6.5 Name_____________________________________ Date_________ Hour_________ 1. Enter the following functions into your graphing calculator: 𝑝(𝑥) = 2𝑥 2 − 12𝑥 + 16 𝑓(𝑥) = 2(𝑥 − 2)(𝑥 − 4) 𝑣(𝑥) = 2(𝑥 − 3)2 − 2 a. What type of graphs do you think these functions will produce? b. Change your calculator’s viewing window to fit the graph. Sketch the graph on the axis provided. c. What is this type of graph called? d. Prove algebraically that 𝑝(𝑥) and 𝑓(𝑥) are equivalent. e. Prove algebraically that 𝑝(𝑥) and 𝑣(𝑥) are equivalent. f. What are some critical attributes we can identify on a parabola? List them below and label them on your graph on the previous page. 2. Enter the following functions into your graphing calculator: 𝑣(𝑥) = −1(𝑥 − 2)2 + 1 𝑓(𝑥) = −1(𝑥 − 1)(𝑥 − 3) 𝑝(𝑥) = −1𝑥 2 + 4𝑥 − 3 a. How are these functions similar to the ones from question 1 above? b. Make a sketch of the graph of these functions on the axes provided. c. Prove algebraically that 𝑝(𝑥) and 𝑓(𝑥) are equivalent. d. Prove algebraically that 𝑝(𝑥) and 𝑣(𝑥) are equivalent. e. Use the trace/calc function of the graphing calculator to find the values for each of the following: Vertex: ________________________ Roots: _________________________ Y-intercept: _____________________ f. How did your predictions match up with your findings above? 3. Enter the following functions in your graphing calculator. 1 𝑓(𝑥) = (𝑥 + 1)(𝑥 + 3) 2 1 3 2 2 𝑝(𝑥) = 𝑥 2 + 2𝑥 + 1 1 2 2 𝑣(𝑥) = (𝑥 + 2)2 − a. Make a sketch of the graph of these functions on the axes provided. b. Prove algebraically that 𝑝(x) and 𝑓(𝑥) are equivalent. c. Prove algebraically that 𝑝(x) and 𝑣(𝑥) are equivalent. d. Use the trace function of the graphing calculator to find the values for each of the following: Vertex: ________________________ Roots: _________________________ Y-intercept: _____________________ 4. Consider the following functions: 𝑓(𝑥) = 𝑎(𝑥 − 𝑟1 )(𝑋 − 𝑟2 ) 𝑝(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 𝑣(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘 a. Based on the above examples, list any conclusions concerning the attributes of each of the above functions. Unit 6 Lesson 5 – Three Ways to Represent Quadratic Functions Ready, Set, Go! – Assignment 6.5 Name_____________________________________ Date_________ Hour_________ Ready Identify the domain and range of each function. Write answers using interval notation. 1. 2. 3. Domain: Domain: Domain: Range: Range: 4. 5. 6. Domain: Domain: Domain: Range: Range: Range: Range: Set 7. The graph h(t) represents the height of a tennis ball thrown upward. h(t) a) What is the domain of the function? b) What is the range of the function? c) What does the y-intercept represent? d) Evaluate by reading the graph. h(0) = h(0.2) » h(1) = e) What does the expression h(0.2) represent? f) What is the maximum height the tennis ball reaches? g) When does the tennis ball reach its maximum height? h) What does the x-intercept represent? 8. The graph h(t) represents the height of a rocket shot up into the sky. a) What is the domain of the function? Write a sentence describing the meaning of the domain for h(t). b) What is the range of the function? Write a sentence describing the meaning of the range for h(t). c) What does the y-intercept represent? d) Estimate the height of the rocket at 2 seconds. e) Estimate the height of the rocket at 8 seconds. Given the equation of the graph is h(t) 16t2 200t f) Evaluate function for the following values. Show your work! h(0) = h(8) = i) Estimate the times (in seconds) that the rocket is at a height of 450 feet. (Hint: there are two answers) j) What does the x-intercept represent? Go ! 16. The graph represents the height of an air-filled ball thrown in a swimming pool. a) What is the domain of the function? b) What is the range of the function? c) What does the y-intercept represent? d) Determine the minimum height of the ball. g) Determine how long it takes the ball to reach its minimum height. h) Estimate the times (in seconds) that the ball is at a height of -2 feet. i) What does the x-intercept represent? Unit 6 Lesson 6 – Average Rate of Change Task 6.6 Name____________________________________ Date__________ Hour_________ 1. Adam and Joanna both rode their bikes for five hours last weekend while training for a race. The graph shows their distance traveled over the five hours as a function of time. a. Describe Joanna’s speed during the 5-hour bike ride. b. Describe Adam’s speed during the 5-hour bike ride. c. Find each rider’s average speed over the 5-hour time interval. Who had the fastest average speed? d. Find each rider's average speed over the interval [0,1]. Who had the fastest average speed over this interval? e. Find Adam’s average speed over the interval [1, 2]. Was Adam traveling faster over the interval [0,1] or [1, 2]? How does the graph show this? 2. An object is dropped from a 256-foot bridge into the water below. The height of this object with respect to time can be modeled by the function ℎ(𝑡) = −16𝑡 2 + 256. a. Use the graph or symbolic representation to evaluate the function for the following values of t. ℎ(0) ℎ(1) ℎ(2) ℎ(3) ℎ(4) b. Find the average rate of change over the interval [0, 4]. Is the object traveling this speed at every point in its descent? Explain. c. Find the average rate of change over the interval [0, 1]. d. Find the average rate of change over the interval [1, 2]. e. Find the average rate of change over the interval [2, 3] f. Find the average rate of change over the interval [3, 4] g. Explain what is happening to the average rate of change of the object as t increases. Why is this happening? h. If the speed of the object is increasing as it falls, why is the average rate of change negative over the interval? 3. The following tables show the distance traveled by three different cars over five seconds. Time (s) 1 2 3 4 5 Car 1 Distance (ft) 4 7 10 13 16 Time (s) 1 2 3 4 5 Car 2 Distance (ft) 2 5 10 17 26 Time (s) 1 2 3 4 5 Car 3 Distance (ft) 3 5 9 17 33 a. Using the above tables, compare the three cars and their positions after t seconds. Which car is traveling the fastest? Justify your answer. b. What is the average rate of change for each car over the interval [0, 2]? c. What is the average rate of change for each car over the interval [3, 5]? d. Think about it again. Which car is traveling the fastest? 4. The following are graphs of functions 𝑓(𝑥) = 10𝑥, 𝑔(𝑥) = 𝑥 2 + 5, ℎ(𝑥) = 2𝑥 respectively. Question 4 continued…… Use the graphs from the previous page to answer the following questions. e. Find the average rate of change for the function 𝑓(𝑥) = 10𝑥 over the following intervals: [0,2]: [2,5]: [-4,-1]: f. What do you notice about the average rate of change of a linear function? g. Find the average rate of change for the function 𝑔(𝑥) = 𝑥 2 + 5 over the following intervals: [2, 5]: [8,10]: [-5,-2]: [-10, -8]: h. What do you notice about the average rate of change of a quadratic function? i. Use your answers from part g to complete the following statements: If a function is increasing, its average rate of change will be…… If a function is decreasing, its average rate of change will be…… j. Compare the average rate of change of the function 𝑔(𝑥) = 𝑥 2 + 5 over the interval [2, 5] vs. [-5,-2] 5. The following is the graph of the function 𝑓(𝑥) = |𝑥|. Use the graphs and symbolic representations (equations) to answer the questions to follow. a. Find the average rate of change for each function over the interval [0,2]. Which function is increasing at the fastest rate over this interval? b. Find the average rate of change for each function over the interval [4,6]. Which function is increasing at the fastest rate over this interval? c. Find the average rate of change for each function over the interval [8,10]. Which function is increasing at the fastest rate over this interval? d. Which function will have the greatest average rate of change as x gets larger and larger? Which will have the smallest average rate of change as x gets larger and larger? Make reference to your answers in parts a-c and the graphs to support your answer. Use the graphs and symbolic representations to answer the questions that follow. a. In what ways is an absolute value function similar to a linear function? b. Using what you learned in the previous problem, tell how the average rate of change of an absolute value function will be similar to the average rate of change of a linear function. In what was will it be different? c. In what ways is an absolute value function similar to a quadratic function? d. Using what you learned in the previous problem, tell how the average rate of change of an absolute value function will be similar to the average rate of change of a quadratic function. In what ways will it be different? e. If you know the average rate of change of this function 𝑓(𝑥) = |𝑥| over the interval [0,2] is 1, find the average rate of change over the following intervals without doing any calculations or using the graph. [2,4] [-2,0] [1,3] [-10,-8] [2,5] [-5,2] [99,100] [-100,-99] Summary Notes: Unit 6 Lesson 6 – Average Rate of Change Ready, Set, Go! - Assignment 6.6 Name____________________________________ Date__________ Hour________ Ready 1. An average rate of change may also be found when there is no formula given. For instance, suppose the temperature at 1:00 PM was 82°and at 9:00 PM the temperature was 70°. What was the average rate of change of the temperature over that interval? 2. Find the average rate of change of the function 𝑓(𝑥) = 𝑥 2 as x varies from 𝑥 = 1 𝑡𝑜 𝑥 = 3. 3. For the function 𝑓(𝑥) = 𝑥 2 , find the average rate of change of the function over the interval [−4, −1]. 4. On Monday, the price of a gallon of gas was $3.74 and on Saturday, the price had risen to $4.06. What is the average rate of change of the price of a gallon of gas from Monday to Sunday? 5. According to census figures, the population of Clovis was 31, 194 in 1980 and 32, 511 in 2001. What was the average rate of change of the population over that time interval? Set 6. Given the function 𝑓(𝑥) = 𝑥 2 − 11𝑥 + 24, find and interpret the average rate of change over each interval. a. [0,3] b. [4,7] c. [6, 8] d. [−1,1] 7. A potato is launched into the air. Use rates of change over different intervals to describe the flight of the potato. Note: for full credit more than 2 intervals must be used. M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com 11 y 10 9 8 7 6 5 4 3 2 1 -2 -1 -1 x 1 2 3 4 5 -2 -3 8. Consider the function f ( x ) 3 x 2 6 x 4 . (a) Find the average rate of change of the function from x 1 to x 4. (b) Sketch a graph of f (x ) and a graph of the line connecting the corresponding points at x 1 and x 4. 9. The graph of y f (x ) is shown below. A secant line is a line connecting two points on the graph. If a secant line is created to pass through the graph at the endpoints of a given interval, over which of the following intervals is the slope of the secant line the greatest? (a) [7, 3] (b) [5, 1] (c) [2, 1] (d) [1, 2 ] 10. Compare the equation 𝑦 = 9𝑥 − 4𝑥 2 , the graph below, and the table below. Which has the steepest rate of change from 𝑥 = 1 𝑡𝑜 𝑥 = 2, and what is its value? The equation: 𝑦 = 9𝑥 − 4𝑥 2 x -1 1 2 3 y 0 2 0 -4 Go! 11. The path of a pumpkin “chucked” into the air can be modeled by the function ℎ(𝑡) = −16𝑡 2 + 96𝑡 + 10 where h is the height of the pumpkin in feet and t is the time in seconds after being “chucked”. Find and interpret the vertical speed of the pumpkin over the following time interval. a. [0,2] b. [3,6] c. [2,4] 12. If, after 2.5 hours of driving at a constant speed, you have traveled 175 miles, what is the rate of change of your distance d over time? 13. Calculate the avergae rate of change of 𝑓(𝑥) = 4𝑥 2 + 3𝑥 + 5, 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑥 = 2 𝑎𝑛𝑑 𝑥 = 5, as a function of x. 14. 1 Calculate the average rate of change of 𝑔(𝑥) = 𝑥 − 𝑥 2 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑥 = −2 𝑎𝑛𝑑 𝑥 = 3, as a functin of x. Unit 6 Lesson 7 – Quadratic Models Task 6.7 Name____________________________________ Date__________ Hour_________ Be sure to answer all parts of the task and show your thoughts, process, and work. Label your answers. Task 1: Find the maximum height of the path of an arrow modeled by the function ℎ(𝑡) = −16𝑡 2 + 96𝑡. During what time interval is the arrow going up? Going down? When does it hit the ground? Task 2: Create a situation that could have produced the given data. Use appropriate vocabulary and key features to tell the story. time 𝑓 (𝑡 ) 0 300 5 777.5 10 1010 15 997.5 20 740 25 237.5 Unit 6 Lesson 7 – Quadratic Models Notes 6.7 Many quadratic model problems can be broken down into 3 types. 1. Maximum/minimum problems: Any time we are trying to find the highest or lowest value in a quadratic model, we are looking for the vertex. 2. Places where the value of the equation is “0”: These are when we have a height function, for example, and we want to know where the height is zero. For this, use the methods of solving quadratic equations that you have learned. 3. Finding values at certain points: For these, just plot in the “x” value, if that’s what we know, and solve for “𝑓(𝑥)” or “y”. If what we are given or what we know is the “𝑓(𝑥)” value, then we are going to have to get our equation in standard form and use #2 above. Example 1: The number of mosquitoes 𝑀(𝑥), in millions, in a certain area depends on the June rainfall x, in inches, according to the equation 𝑀(𝑥) = 10𝑥 − 2𝑥 2 . What rainfall produces the maximum number of mosquitoes? Example 2: The polynmial function 𝐼(𝑡) = −0.1𝑡 2 + 1.9𝑡 represents the yearly income (or loss) from a real estate investment, where t is time in years after 1970. During what year does the maximum income occur? Exmaple 3: Your company uses the quadratic model 𝑦 = −7𝑥 2 + 350𝑥 to represent how many units y of a new product will be sold x weeks after its release. How many units can you expect to sell in week 27? Example 4: A rectangular pen for a dog uses 80 feet of fencing. What dimensions will result in an area of 384 𝑓𝑡 2 . Unit 6 Lesson 7 – Quadratic Models Ready, Set, Go! - Assignment 6.7 Name____________________________________ http://goo.gl/wtY5G Date__________ Hour________ Ready 1. A bottle rocket is fired from the ground upwards at 64 feet per second. a) What is the maximum height the bottle rocket reaches? b) How long does it take for the bottle rocket to hit the ground? 2. Your company uses the quadratic model 𝑦 = −4.5𝑥 2 + 150𝑥 to represent the average number of new customers who will be signed on x weeks after the release of your new service. How many new customers can you expect to gain in week 8? 3. 1 Suppose the cost of producting x crates of pencils is given by 𝐶(𝑥) = 2 𝑥 2 − 10𝑥 + 1000. a) How much does it cost to produce 100 crates of pencils? b) How many crates of pencils will minimize the cost of production? 4. The length of a room is 5 feet more than its width. If the area of the room is 266 𝑓𝑡 2 , what is the width of the room? 5. The length of a rectangular frame is 4 cm more than its width. The area inside the frame is 60 squre cm. Find the width of the fram. Set 6. The profit for a company is given by 𝑃(𝑥) = −0.0002𝑥 2 + 140𝑥 − 250,000, where x is the number of units produced. What production level will yeild a maximum profit? 7. The perimeter of a rectangle is 400 feet. Let x represent the width of the rectangle and write a quadratic function that expresses th area of the rectangle in terms of width. Of all possible rectangles with perimeters of 400 feet, what are the measurements of the one that has the greatest area? 8. The billboard is 10 feet longer than it is high. The billboard has 336 square feet of advertising space. What are the measurements of the billboard? 9. An astornaut standing on the surface of the moon throws a rock into space. The height of the rock is given by the equation 𝑠(𝑡) = −2.7𝑡 2 + 27𝑡 + 6. How much time will ellapse before the rock strikes the lunar surface? 10. The demand equation for a certain product is 𝑝 = 50 − 0.0005𝑥 where p is the price per unit and x is the number of units sold. The total revenue for selling x units is given by 𝑅 = 𝑥𝑝. How many units must be sold to produce a revenue of $250,000. 11. If an M-16 is fired straight upward, then the height h(t) of the bullet in feet at time t in seconds is given by ℎ(𝑡) = −16𝑡 2 + 325𝑡. a) What is the height of the bullet 5 seconds after it is fired? b) How long does it take for the bullet to return to the earth? 12. A boy tosses a ball upward at 32 feet per second from a window that is 48 feet above ground. The height of the ball above ground (in feet) at time t (in seconds) is given by ℎ(𝑡) = −16𝑡 2 + 32𝑡 + 48. Find the time at which the ball strikes the ground. Go! 13. The height h( x) (in feet) of a ball thrown by a child is h( x) 1 2 x x 2 where x is the horizontal 12 distance (in feet) from where the ball is thrown. a) How high is the ball when it is at its maximum height? b) How high is the ball when it leaves the child’s hand? (Hint: find y when x=0) c) How far from the child does the ball strike the ground? 14. If Robert kicks a football straight up into the air with an initial velocity of 100 feet per second, the function h t 16t 2 100t gives the height, in feet, of the ball after t seconds. What is the maximum height reached by the football (to the nearest foot)? How long is the football in the air before it hits the ground (to the nearest hundredth of a second)? 15. Parker throws a ball off the top of a building. The building is 350 feet high and the initial velocity of the ball is 30 feet per second. The position of the ball at any time t is represented by p t 16t 2 30t 350 . How long will it take the ball to hit the ground (to the nearest hundredth of the second)? Unit 6 Lesson 8 – Quadratic Practice Understanding Task Task 6.8 Use the table to identify the vertex, the equation for the axis of symmetry, and state the number of x intercept(s) the parabola will have, if any. Will the vertex be a maximum or minimum? 1. 2. 3. x y x y x y -4 10 -2 49 -7 -9 -3 3 -1 28 -6 3 -2 -2 0 13 -5 7 -1 -5 1 4 -4 3 0 -6 2 1 -3 -9 1 -5 3 4 -2 -29 2 -2 4 13 -1 -57 Vertex _____________ Vertex _____________ Vertex _____________ A.O.S. _____________ A.O.S. _____________ A.O.S. _____________ x-inter______________ x-inter______________ x-inter______________ max or min? max or min? Write max or min? the equation for each problem below. Use a second representation to check your equation. 4. The area of a square with side length x, where the side length is decreased by 3, the area is multiplied by 2 and then 4 units are added to the area. 5. x -4 f(x) 7 -3 -2 -1 0 1 2 3 4 2 -1 -2 -1 2 7 14 23 6. 7. For each problem below, you are given a piece of information that tells you a lot. Use what you know about that information to fill in the rest. 8. You get this: Fill in this: Factored form on the equation: 𝑦 = 𝑥 2 − 𝑥 − 12 Graph of the equation: 9. You get this: Fill in this: Vertex form of the equation: Standard form of the equation: 10. You get this: . Fill in this: Either form of the equation other than standard form. y= −x 2 – 6x + 16 Vertex of the parabola x-intercepts and y-intercept Additioanl Notes: Unit 6 Lesson 8 – Quadratic Practice Understanding Ready, Set, Go! - Assignment 6.8 Name____________________________________ Date__________ Hour________ Ready 1. Prove that a. 4(𝑥 − 2)(𝑥 + 6) = 4(𝑥 +)2 − 64 b. −3(𝑥 + 2)(𝑥 − 6) = −3(𝑥 − 2)2 + 48 c. (𝑥 + 5)(𝑥 + 7) = (𝑥 + 6)2 − 1 2. Multiply and write each product in the form 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. Then, identify a, b, and c. 𝑎. 𝑦 = 𝑥(𝑥 − 4) 𝑎= 𝑏= 𝑐= 𝑏. 𝑦 = (𝑥 − 1)(2𝑥 − 1) 𝑎= 𝑏= 𝑐= 𝑐. 𝑦 = −(𝑥 + 5)2 𝑎= 𝑏= 𝑐= 3. Write three different equations for a parabola that has x-intercepts (−3, 0)𝑎𝑛𝑑 (1, 0) Set For each problem below, you are given a piece of information that tells you a lot. Use what you know about that information to fill in the rest. 4. You get this: Fill in this: Vertex form of the equation: y = x 2 – 6x + 3 Graph of the equation: 5. You get this: Fill in this: Factored form of the equation: Standard form of the equation: 6. You get this: Fill in this: Either form of the equation other than standard form. y = 2x 2 + 12x + 13 Vertex of the parabola x-intercepts and y-intercept Go! A golf-pro practices his swing by driving golf balls of the edge of a cliff into a lake. The height of the ball (measured in meters) as a function of time (mesured in seconds and represented by variable t) from the instand of impact with the golf club is 𝑓(𝑡) = 58.8 + 19.6𝑡 − 4.9𝑡 2 . The expressions below are equivalent: a. 𝑓(𝑡) = −4.9𝑡 2 + 19.6𝑡 + 58.8 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑓𝑜𝑟𝑚 b. 𝑓(𝑡) = −4.9(𝑡 − 6)(𝑡 + 2) 𝑓𝑎𝑐𝑡𝑜𝑟𝑒𝑑 𝑓𝑜𝑟𝑚 c. 𝑓(𝑡) = −4.9(𝑡 − 2)2 + 78.4 𝑣𝑒𝑟𝑡𝑒𝑥 𝑓𝑜𝑟𝑚 7. Whick equation is most useful for finding how many seconds it takes for the ball to hit the water? Justify your answer. 8. Which equation is the most useful for finding the maximum height of the ball? Justify your asnwer. 9. If you wanted to know the height of the ball at exactly 3.5 seconds, which equation would you use to find your answer? Explain why. 10. If you wanted to know the height of the cliff above the lake, which equation would you use? Explain why. One form of a quadratic function is given. Fill in the missing form. 11. Standard form Vertex form Factored form 𝑦= ( 𝑥+ 5 )( 𝑥− 3) Table (Show the vertex and at least 2 points on each side of the vertex.) 12. Standard form Graph Vertex form 𝒚= −𝟑(𝒙− 𝟏)𝟐+ 𝟒 Table (Show the vertex and at least 2 points on each side of the vertex.) Graph Factored form Unit 6 Lesson 9 – Comparing Linear, Quadratic, and Exponential Functions Task 6.9 Name_____________________________________ Date________________ Hour___________ 1. Determine whether each chart below represents a linear or exponential or quadratic model; then create a function for each chart. (recursive or explicit) f(n) = x y 0 -10 1 x y x y -6 -4 22 0 5 2 -2 -3 17 -1 10 3 2 -2 12 -2 20 4 6 -1 7 -3 40 5 10 0 2 -4 80 6 14 Difference Linear or exponential? x y x y 0 9 0 10 -1 12 -1 12 -2 16 -2 14 -3 25 -3 16 -4 34 -4 18 2. How can you tell if a chart represents a linear function? 3. How can you tell if a chart represents an exponential function? 4. How can you tell if a chart represents a quadratic function? The table below shows the values of 2𝑥 and 2𝑥 3 + 1. Fill in the chart. 𝒙 𝟐𝒙 𝟐𝒙𝟑 + 𝟏 1 2 3 4 5 6 5. The numbers in the third column are larger than the numbers in the second column. Does this remain true if the table is extended to the number 10? 6. At what value of x does the exponential function surpass the cubic function? 7. Find a quadratic and exponential function that: (a graphing calculator may be a good tool to use) a. Do not intersect b. Intersect once c. Intersect twice d. Intersect more than once. Unit 6 Lesson 9 – Comparing Linear, Quadratic, and Exponential Functions Notes 6.9 Decide if the function is linear, exponential, quadratic, or neither. Please give a detailed reason for your answer. 1. M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com 6 5 4 3 2 1 y x 1 2 3 4 5 6 7 8 2. −3𝑥 = 4𝑦 + 7 3. 𝑦 = 7𝑥 2 4. Each term in a sequence is exactly 1/3 of the pervious term. 5. You should be able to prove that eventually, as long as the functions are headed in the same direction, a quantity increasing exponentially will "beat" linear, quadratic, and polynomial functions. It's probably obvious that the function y = 3x will eventually surpass y = 3x + 3. We can see this via a table of values or a graph. Somewhere down the line, when x gets closer and closer to infinity, the y value of the exponential function will be larger than the y value of the linear function. We can see that this happens at x = 2 whether we graph it or look at the table of values. x 3x 3x + 3 13 6 29 9 3 27 12 4 81 15 5 243 18 6. Which of the following x values proves that 𝑓(𝑥) = 2𝑥 will suprpass the function 𝑦 = 𝑥 2 + 𝑥 + 3? Then, graph both on the same axis and be sure to label both your axes with your chosen scale. A. B. C. D. 20 30 40 None of the above, it would be a larger value of x. 7. A locan newspaper has projected that its online revenues are growing at a rate modeled by 𝑦 = 1.5𝑥 while its print media only increases at a rate of 𝑦 = 36𝑥 + 35. At which point (in terms of x) do online revenues surpass that of print revenues? 9. At what point will the function 𝑦 = 16𝑥 surpass 𝑦 = 16𝑥? Unit 6 Lesson 9 – Comparing Linear, Quadratic, and Exponential Functions Ready, Set, Go! - Assignment 6.9 Name____________________________________ Date__________ Hour________ Ready 1. Calculate the differences between each chart to determine if the function is linear, exponential, quadratic, or neither. x y Diff x y 0 200 0 6 1 170 1 7.8 2 140 2 10.14 3 110 3 13.18 4 80 4 17.13 5 50 5 22.7 6 20 6 28.95 Type of function: Type of function: x y Diff x y 0 425 0 2 1 387 1 3 2 331 2 18 3 257 3 83 4 165 4 258 5 55 5 627 6 -73 6 1298 Type of function: Diff Type of function: Diff Set Determine whether each function is linear, exponential, quadratic, or none. Be sure to explain. 2. 3. A newborn that doubles its weight in the first 4 months. 4. 5. 𝑦 = 𝑥 2 + 2𝑥 3 + 5 6. 7. f(0) = 2 f(1) = 3 f(n) = f(n-1) +f(n-2) *may be helpful to make a table* 8. The function has a constant 2nd difference. 9. The function has a constant increasing ratio. Go! Identify the graph as linear, quadratic, exponential, or neither. 10. 11. x -3 -2 -1 0 1 2 3 x -3 -2 -1 0 1 2 3 y 14 10 6 2 -2 -6 -10 y .5 1 2 4 8 16 32 12. 13. x -3 -2 -1 0 1 2 3 x -3 -2 -1 0 1 2 3 y 21 12 5 0 -3 -4 -3 y -16 -13 -10 -7 -4 -1 2 14. Given the following graph, which of the lines/curves most likely represents an exponential funtion? (Assuming there is only one)? 15. By simply looking at the equations below, which one will eventually surpass the others? (A) 𝑦 = 𝑥100 (B) 𝑦 = 100𝑥 (C) 𝑦 = 100𝑥 (D) 𝑦 = 𝑥100 + 100𝑥 Explain your choice. Quadratic Modeling Additional Practice Task/Worksheet Name___________________________________________ Date_________ Hour_________ 1. A rock is thrown upward so that its distance, in feet, above the ground after t seconds is h(t) = –14t2 +336t. a) Find the zeros of the function and explain the meaning in the context of the problem. b) Find the vertex of the function and explain the meaning in the context of the problem. 2. A firework is shot upward so that its distance, in feet, above the ground after t seconds is h(t) = –13t2 +312t. a) Find the zeros of the function and explain the meaning in the context of the problem. b) Find the vertex of the function and explain the meaning in the context of the problem. 3. John owns a hotdog stand. He has found that his profit is represented by the equation P(x) = –x2 + 68x + 77, with P being the profit in dollars, and x the number of hotdogs sold. How many hotdogs must he sell to earn the most profit? 4. Bob owns a watch repair shop. He has found that the cost of operating his shop is given by C(x) = 4x2 – 264x + 85, where c is the cost in dollars, and x is the number of watches repaired. How many watches must he repair to have the lowest cost? 5. A projectile is thrown upward so that its distance above the ground after t seconds is h(t) = –11t2 + 286t. When does it reach its maximum height? 6. The number of mosquitoes M(x), in millions, in a certain area depends on the June rainfall x, in inches: M(x) = 10x – x2. What rainfall produces the maximum number of mosquitoes? 7. The manufacturer of a CD player has found that the revenue R (in dollars) is R(p) = –4p2 + 1280p, when the unit price is p dollars. If the manufacturer sets the price p to maximize revenue, what is the maximum revenue to the nearest whole dollar? 8. A projectile is thrown upward so that its distance above the ground after t seconds is h(t) = –12t2 + 504t. When does it reach its maximum height? 9. The owner of a video store has determined that the cost C, in dollars, of operating the store is approximately given by C(x) = 2x2 – 30x + 700, where x is the number of videos rented daily. Find the lowest cost to the nearest dollar. *10. The quadratic function f(x) = 0.0042x2 – 0.42x + 36.23 models the average age at which U.S. men were first married x years after 1900. a) In which year was this average age at a minimum? (Round to the nearest year) b) What was the average age at first marriage for that year? (Round to the nearest tenth). Unit 6 Lesson 10 – Fundamental Theorem of Algebra Task/Notes 6.10 Name______________________________________ Date_______ Hour________ The Fundamental Theorem of Algebra states that every polynomial of degree n with complex coefficients has exactly n roots in the complex number system. Polynomial Identities 1. (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2 2. (𝑎 + 𝑏)(𝑐 + 𝑑) = 𝑎𝑐 + 𝑎𝑑 + 𝑏𝑐 + 𝑏𝑑 3. 𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏) 4. 𝑥 2 + (𝑎 + 𝑏)𝑥 + 𝑎𝑏 = (𝑥 + 𝑎)(𝑥 + 𝑏) 5. If 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 then 𝑥 = −𝑏±√𝑏 2 −4𝑎𝑐 2𝑎 State the degree of the polynomial and how mayn complex roots it has 1. 𝑓(𝑥) = 𝑥 2 + 𝑥 + 3 2. 𝑓(𝑥) = 5𝑥 3 + 2𝑥 2 − 5𝑥 + 4 For each example, find the degree, the number of roots, and the complex roots. Write your answer in factored form. 3. (𝑥 + 5)(2𝑥 − 6) = 0 4. 𝑥 2 + 64 = 0 5. 2𝑥 2 + 12𝑥 = −20 Additional Notes/Examples: 6. 𝑓(𝑥) = 𝑥 4 + 10𝑥 2 + 24 Unit 6 Lesson 10 – Fundamental Theorem of Algebra Assignment - 6.10 Name______________________________________ Date_______ Hour________ Find the degree and number of roots. Then find the complex roots and write them in factored form. 1. 𝑥 2 + 9 2. 𝑥 2 + 𝑥 + 1 3. 𝑥 2 − 2𝑥 + 2 4. 𝑥 2 − 6𝑥 + 10 5. 𝑥 2 − 4𝑥 + 5 6. 𝑥 2 − 2𝑥 + 5 7. 𝑥 4 + 5𝑥 2 + 4 8. 𝑥 4 + 13𝑥 2 + 36 9. 𝑥 4 − 1 Find the degree and number of roots. State the roots aand write the polynomial in factored form. 10. 𝑥 2 + 16 11. 𝑥 2 + 8𝑥 − 4 12. 2𝑥 2 + 4𝑥 − 10 13. 4𝑥 2 − 4𝑥 − 3 14. 𝑥 2 − 6𝑥 + 11 15. 𝑥 2 + 3𝑥 + 5 16. 𝑥 2 + 10𝑥 + 16 17. 𝑥 2 − 11𝑥 + 18 18. 𝑡 2 − 7𝑡 − 44 Unit 6 Lesson 11 – Exponentials Take All Task/Notes 6.11 1. The graph show three different investment plans. 𝐿(𝑡) = 200𝑡 + 300 𝑄(𝑡) = 5𝑡 2 + 300 𝐸(𝑡) = 300(1.095𝑡 ) a. Which investment has the greatest average rate of change between 0 and 10 years? Show the work that leads to your answer. Write your answer in the form of a setence. b. Which investment has the greatest average rate of change between 30 to 40 years? Show the work that leads to your answer. Write your answer in sentence form. 2. Compare rates of change using graphs. a. Using a graphing calculator, graph the following functions on the same coordinate plane. 𝐿(𝑥) = 3 𝑥−1 2 Q(x) = 2(x − 3)2 − 5 𝐸(𝑥) = 3𝑥−2 − 7 b. Find the average rate of change for the functions 𝐿(𝑥), 𝐸(𝑥), 𝑎𝑛𝑑 𝑄(𝑥) for the specified intervals. Determine which of the three functions is increasing the fastest on the interval. [−4, 2] [3, 5] c. Using the functions 𝐿(𝑥), 𝐸(𝑥), 𝑎𝑛𝑑 𝑄(𝑥), determine which function has the greatest average rate of change on the interval [0, ∞). In genteral, what type of function (linear, exponential, or quadratic) will increase faster as x approaches infinity? Explain your answer. Matching Parabolas to Quadratic Equations Sorting Activity Teacher Notes There is a matching parabolas activity available at www.cachemath2.wordpress.com It is not included in this book as it is many pages of graphs and equations in PDF format. It is filed under additional teacher resources in the “TEACHER” tab. Additional Quadratic Story Problems Teacher Notes There is an additional quadratics worksheet/task with story problems. You can use them as a review, or type them up for a test or test review. You can find this activity at www.cachemath2.wordpress.com It is filed under additional teacher resources in the “TEACHER” tab.