Algebra IIA Unit II: Quadratic Functions Foundational Material o Graphing and transforming linear functions o Solving linear equations and inequalities o Fit data using linear models Goal o Graph and transform quadratic functions o Use and perform operations with real numbers o Solve quadratic equations and inequalities o Fit data to quadratic models Why? o Build a foundation for higher level mathematics o These skills can be used in Chemistry, Physics and Economics o Observe and understand patterns and relationships in science, social studies, and model data Key Vocabulary o Complex conjugate o Complex number o Axis of symmetry o Standard form o Imaginary number o Zero of a function o Minimum value o Parabola o Use and perform operations with imaginary and other complex numbers o Maximum value o Vertex form o Quadratic function o Discriminant Lesson 1: Using Transformations to Graph Quadratic Functions and Properties of Functions in Standard Form Apply transformations to quadratic functions (CC.9-12.F.BF.3) Interpret transformations of real-world data (CC.9-12.A.CED.3) Define, identify, and graph quadratic functions (CC.9-12.F.IF.7a) Identify and use maximums & minimums of quadratic functions to solve problems (CC.9-12.A.CED.2-3) Form of a linear function: Form of a quadratic function: Warm-Up 1. For each translation of the point (- 2, 5), give the coordinates of the translated point. a. 6 units down b. 3 units right 2. Given!! =!! + 2! + 6, find! 2 =,! 0 =,! 3 = 3. Given!! = 2!! 5! + 6, find! 2 =,! 0 =,! 3 = Graph the functions that follow in the grid provided (be sure to draw and label the x-axis and y-axis). 4.!! =! 5.!! =!! The graph of a linear function is a The graph of a quadratic function is a
Graph of a Quadratic Function A Parabola Vertex: Axis of Symmetry: X intercept: Translating Quadratic Functions Use the graph of!! =!! to describe the transformations and then graph each function. Method 1: Table Method a. Create a table for 5 values. b. Calculate the x-value of the vertex (! =! ). Place this x-value as the!! 3rd value of the table. c. Write two x-values less than and more than the value calculated in part b into the table. d. Complete the table, plot the points and draw a parabola.!! =!! 6! + 8!! =!! + 2! + 6 Vertex: Domain: Vertex: Domain: Line of Symmetry: Range: Line of Symmetry: Range: Maximum or Minimum? Maximum or Minimum? Vertex Form of a quadratic function is Method 2: Vertex Form Method 1. Calculate the x-value of the vertex by finding! =!!!. 2. Substitute the x-value into the equation to get the y-value. This coordinate is the vertex (h, k). 3. Using a, h, and k substitute into vertex form to write the equation.
Describe the translation and then graph the quadratic functions that follow using!! =!! as a guide.!! =! 2!!! =!! + 3!! =! + 2! 1!! = 2!! Translation: Translation: Translation: Translation: Vertex: Line of Symmetry: Maximum or Minimum? Domain: Range: Vertex: Line of Symmetry: Maximum or Minimum? Domain: Range: Vertex: Line of Symmetry: Maximum or Minimum? Domain: Range: Vertex: Line of Symmetry: Maximum or Minimum? Domain: Range:
Writing Transformed Quadratic Equations (Working Backwards) Use the description to write the quadratic function in vertex form. 1. The parent function!! =!! is vertically stretched by a factor of 4/3 and then translated 2 units left and 5 units down to create g. 2. The parent function!! =!! is reflected through the x-axis, is vertically compressed by a factor of 1/3, and then translated 3 units right and 2 units up to create g. 3. Can a quadratic function have both a maximum and a minimum? Why? 4. How do you know whether a change in the value of a represents a vertical stretch or a vertical compression? 5. On Earth, the distance d in meters that a dropped object falls in t seconds is approximated by!! = 4.9!!. On the moon, the corresponding function is!!! = 0.8!!. What kind of transformation describes this change from!! = 4.9!!, and what does the transformation mean? 6. Explain how the values of a, h, and k in the vertex form of a quadratic function affect the function's graph. 7. Complete the graphic organizer below. In each row, write and equation that represents the indicated transformation of the quadratic parent function. Transformation Equation Graph Transformation Equation Graph Vertical Translation Horizontal Translation Reflection Vertical Stretch Assignment: Page 64, 17-37 odd, 39-44, 46-50 and Page 72, 13 37 odd, 42-46
Algebra IIA Unit II: Quadratic Functions Foundational Material o Graphing and transforming linear functions o Solving linear equations and inequalities o Fit data using linear models Goal Graph and transform quadratic functions Solve quadratic equations and inequalities Fit data to quadratic models o Use and perform operations with real numbers Use and perform operations with imaginary and other complex numbers Why? o Build a foundation for higher level mathematics o These skills can be used in Chemistry, Physics and Economics o Observe and understand patterns and relationships in science, social studies, and model data Key Vocabulary o Complex conjugate o Complex number o Axis of symmetry o Standard form o Imaginary number o Zero of a function o Minimum value o Parabola o Maximum value o Vertex form o Quadratic function o Discriminant Lesson 2: Solving Quadratic Equations by Graphing and Factoring Solve quadratic equations by graphing or factoring (CC.9-12.F.IF.8a), (CC.9-12.F.IF.7a) Determine a quadratic equation from its roots (CC.9-12.A.CED.1, and CC.9-12.A.REI.11) In your words What is a quadratic function? What is vertex form? What is a maximum value? What is a minimum value? What is a zero of a function? What is a parabola? Warm-Up Find the x-intercepts of each function. 1.!! = 3! + 9 2.!! = 6! + 4 The directions could be Solve the quadratic equation or Find the zeros or Find the solutions or Factor each expression. 3. 3!! 12! 4.!! 49 Find the x-intercepts or Find the roots. The method used could be Factoring Quadratic Formula Graphing
Find the zeros of!! =!! + 2! 3 by graphing. Find the zeros of!! =!! 6! + 8 by graphing. Vertex: Line of Symmetry: Maximum or Minimum? Domain: Range: Vertex: Line of Symmetry: Maximum or Minimum? Domain: Range: Find the solutions to the quadratic equations that follow by factoring. 5.!! =!! 8! + 12 6.!! = 3!! + 12! 7.!! =!! 4! 12 Physics and Projectile Functions Example: A soccer ball is kicked from ground level with an initial vertical velocity of 32 ft/s. 8. When will the ball hit the ground? 9. What is the maximum height of the ball? 10. When does the ball reach its max height? 11. When does the ball have a velocity of 0 ft/s? A golf ball is hit from ground level with an initial vertical velocity of 80 ft/s. 12. When will the ball hit the ground? 13. What is the maximum height of the ball? 14. When does the ball reach its max height? 15. When does the ball have a velocity of 0 ft/s?
Ms. Woods throws a water balloon from an initial height of 20 feet with an initial velocity of 158 ft/s. 16. When will the ball hit the ground? 17. What is the maximum height of the ball? 18. When does the ball reach its maximum height? 19. When does the ball have a velocity of 0 ft/s? A Review Finding Roots by Using Special Factors (still a review) Find the roots of the equation by factoring. Make a graph of each equation. 20. 9!! = 1 21. 40! = 8!! + 50 A double-root is On a graph, at the location of a double root, the graph 22. Write a quadratic function in standard form with the zeros 2 and 1. 23. Write a quadratic function in standard form with the zeros 5 and 5. 24. Write a quadratic function in standard form with the zeros 4 and 7. Page 82 and 83, 19-57 odd, 67-70
Algebra IIA Unit II: Quadratic Functions Foundational Material o Graphing and transforming linear functions o Solving linear equations and inequalities o Fit data using linear models Goal Graph and transform quadratic functions Solve quadratic equations and inequalities Fit data to quadratic models o Use and perform operations with real numbers Use and perform operations with imaginary and other complex numbers Why? o Build a foundation for higher level mathematics o These skills can be used in Chemistry, Physics and Economics o Observe and understand patterns and relationships in science, social studies, and model data Key Vocabulary o Complex conjugate o Complex number o Axis of symmetry o Standard form o Imaginary number o Zero of a function o Minimum value o Parabola o Maximum value o Vertex form o Quadratic function o Discriminant Lesson 3: Solving Quadratics by Completing the Square Solve quadratic equations by graphing or factoring (CC.9-12.F.IF.8a), (CC.9-12.F.IF.7a) Determine a quadratic equation from its roots (CC.9-12.A.CED.1, and CC.9-12.A.REI.11) In your words Write the two forms for a quadratic function? What is a line of symmetry? What is a maximum value? What is a minimum value? What is a zero of a function? What is a parabola? Write the formula used to solve projectile motion problems: Write the methods used to solve a quadratic equation. Review: Solve the equation by using the square root property. 3!! 4 = 68 Solve the equation by factoring. 16!! + 24! = 9
Completing the Square another method to solve quadratic equations. Use the method of completing the square to solve the quadratic equations that follow.!! = 27 6!! 2 2 = 9! 2!! + 8! = 12 3!! 24! 27 = 0
Write each function in vertex form and identify its vertex and axis of symmetry. Then discuss the shifts and translations for the given function based on the quadratic parent function. Determine the domain and range of the function and whether the function has a minimum or maximum value.!! =!! + 10! 13!! = 2!! 8! + 3!! =!! + 24! + 145!! = 5!! 50! + 128 Assignment: Page 89, 20-22, 23-39 odd, 41-59 odd, 63, 73-77
Algebra IIA Unit II: Quadratic Functions Foundational Material o Graphing and transforming linear functions o Solving linear equations and inequalities o Fit data using linear models Goal Graph and transform quadratic functions Solve quadratic equations and inequalities Fit data to quadratic models o Use and perform operations with real numbers Use and perform operations with imaginary and other complex numbers Why? o Build a foundation for higher level mathematics o These skills can be used in Chemistry, Physics and Economics o Observe and understand patterns and relationships in science, social studies, and model data Key Vocabulary o Complex conjugate o Complex number o Axis of symmetry o Standard form o Imaginary number o Zero of a function o Minimum value o Parabola Lesson 4: Complex Numbers and Roots and the Quadratic Formula Define and use imaginary and complex numbers (CC.9-12.N.CN.1), (CC.9-12.N.CN.2) Solve quadratic equations with complex roots (CC.9-12.N.CN.7), (CC.9-12.A.CED.1) Solve quadratic equations using the quadratic formula (CC.9-12.N.CN.7) Classify roots using the discriminant (CC.9-12.A.CED.1) In your words o Maximum value o Vertex form o Quadratic function o Discriminant Write the two forms for a quadratic function? What is a vertex of a parabola? What is a zero of a function? How can a zero of a function be found? Warm-Up 1. Solve the equation.!! + 1 = 0 i. Vertex: ii. iii. iv. Line of Symmetry: Zeros: Domain: v. Range: vi. Graph the equation!! + 1 = 0
Simplify the expressions that follow. Solve the equations that follow. 8.! 63 11. 3!! + 75 = 0 6. 25 9.!! 9 = 0 12. 9!! + 25 = 0 7. 2 36 10.!! = 81 13. 3!! = 48 2. 108 5. 3. 10 15!" 4. 5-5!! 12! Complex Numbers and Roots Holt McDougal Algebra 2 Examples Find the values of x and y that makes each equation true. 14. 3! 5! = 6 10!! 16. 2! 6! = 8 + 20!! 17. 8 + 6!! = 5!! 6 Find the zeros of the quadratic equations that follow. 18.!! =!! 2! + 5 19.!! =!! + 10! + 35 20.!! =!! 4! + 12
Complex Conjugate: Find each complex conjugate. 18. 8 + 5! 19. 6! 20.!!!! 21. 4 3! 2! 22. In a graph, the zeros are the same as 23. When finding the zeros, how can your answers be checked? 24. Name the ways to solve quadratic equations. Another method the Quadratic Formula. Where does this formula come from? Solve using the quadratic formula.!! 4! + 3 = 0!! =!! + 2! + 1!! 2! = 2 2!! + 27 = 16! 8-9 The Quadratic Formula and the Discriminant Holt McDougal Algebra 1 Find the type and number of solutions for each equation.!! 4! = 4!! + 8 = 4!!! 4! = 2 Assignment: Page 97, 27-57 odd and 76-80 and Page 105, 19-37 odd and 61-64
Algebra IIA Unit II: Quadratic Functions Foundational Material o Graphing and transforming linear functions o Solving linear equations and inequalities o Fit data using linear models Goal Graph and transform quadratic functions Solve quadratic equations and inequalities Fit data to quadratic models o Use and perform operations with real numbers Use and perform operations with imaginary and other complex numbers Why? o Build a foundation for higher level mathematics o These skills can be used in Chemistry, Physics and Economics o Observe and understand patterns and relationships in science, social studies, and model data Key Vocabulary o Complex conjugate o Complex number o Axis of symmetry o Standard form o Imaginary number o Zero of a function o Minimum value o Parabola Lesson 5: Solving Quadratic Inequalities Solve quadratic inequalities by using tables and graphs (CC.9-12.A.CED.1) Solve quadratic inequalities by using algebra (CC.9-12.A.CED.1), (CC.9-12.A.CED.3) Warm-Up: o Maximum value o Vertex form o Quadratic function o Discriminant Determine whether each statement is always, sometimes, or never true. If you answer sometimes, give an example to support your answer. 1. A real number is an imaginary number. 2. An imaginary number is a complex number. 3. A rational number is a complex number. 4. A complex number is an imaginary number. 5. An integer is a complex number. 6. Quadratic equations have no real solutions. 7. Quadratic equations have roots that are real and complex. 8. Roots of quadratic equations are conjugate pairs. 9. How is an inequality different than an equation? 10. Explain the difference between the graph of an equation and the graph of an inequality
Graph the following: 1.!! < 2! + 1 3.! 2!! 5! 2 2.! <!! 2! + 3 4.! < 3!! 6! + 1 Use a calculator to assist in graphing the following: 1.!! 6! + 8 3 3.!! 6! + 8 > 3 2.!!! + 5 < 5 4. 2!! 5! + 1 1
5. How do you know where to shade? 6. How do you know if the line is dotted or solid? 7. How do you choose a test point to verify the solution? Solve the quadratic inequalities using algebra. 8.!! 4! + 1 < 6 9.!! 6! + 10 2 10. 2!! + 3! + 7 < 2 11. The monthly profit of a small business that sells bicycle helmets can be modeled by the function!! = 8!! + 600! 4200, where x is the average selling price of a helmet. What is the range of selling prices that will generate a monthly profit of at lease $6000? 12. A business offers tours to the Amazon. The profit that the company ears can be modeled by the equation!! = 25!! + 1000! 3000. In the equation, x represents the number of tourist and P represents the profit. How many people are needed for a profit of at least $5000? Assignment: Page 114, 19-47 odd, 48-50, 55, 57, 62-64 and Page 122, 15-37 odd and 45-48
Algebra IIA Unit II: Quadratic Functions Foundational Material o Graphing and transforming linear functions o Solving linear equations and inequalities o Fit data using linear models Goal Graph and transform quadratic functions Solve quadratic equations and inequalities Fit data to quadratic models o Use and perform operations with real numbers Use and perform operations with imaginary and other complex numbers Why? o Build a foundation for higher level mathematics o These skills can be used in Chemistry, Physics and Economics o Observe and understand patterns and relationships in science, social studies, and model data Key Vocabulary o Complex conjugate o Complex number o Axis of symmetry o Standard form o Imaginary number o Zero of a function o Minimum value o Parabola Lesson 6: Curve Fitting with Quadratic Models Use quadratic functions to model data (CC.9-12.A.CED.2) Use quadratic models to analyze and predict (CC.9-12.A.CED.2), (CC.9-12.A.CED.3) o Maximum value o Vertex form o Quadratic function o Discriminant Now let s look at curve fitting again. Let s analyze the pattern in the data provided in each table. For the set of data in each table, the x-values are equally spaced and the second difference in the y-values is constant. These tables represent quadratic regression tables. x - 2-1 0 1 2!! =!! 4 1 0 1 4 x 0 2 4 6 8 y 12 10 9 9 10 Determine if the data sets below could represent a quadratic function. x 3 4 5 6 7 y 11 21 35 53 75 x 10 9 8 7 6 y 6 8 10 12 14
Write a quadratic function from data. 13. (0, 5), (2, 1), and (3, 2) (x, y)!! =!!! +!"+c System in a, b, and c (0, 5) (2, 1) (3, 2) 14. (0, - 3), (1, 0), and (2, 1) (x, y)!! =!!! +!"+c System in a, b, and c Now let s use quadratic regression and the calculator. 15. The table below shows approximate run times for 16 mm films, given the diameter of the film on the reel. Find a quadratic model for the run time given the diameter. Use the model to estimate the run time for a reel of film with a diameter of 15 in. Film Run Times (16 mm) Diameter in inches Reel Length in feet Run Time in minutes 5 200 5.55 7 400 11.12 9.25 600 16.67 10.5 800 22.22 12.25 1200 33.33 13.75 1600 44.45
16. The table below shows approximate run times for 16 mm films, given the diameter of the film on the reel. Find a quadratic model for the reel length given the diameter of the film. Use the model to estimate the reel length for an 8-inch diameter film. Film Run Times (16 mm) Diameter in inches Reel Length in feet Run Time in minutes 5 200 5.55 7 400 11.12 9.25 600 16.67 10.5 800 22.22 12.25 1200 33.33 13.75 1600 44.45 Assignment: Page 114, 19-47 odd, 48-50, 55, 57, 62-64 and Page 122, 15-37 odd and 45-48
Algebra IIA Unit II: Quadratic Functions Foundational Material o Graphing and transforming linear functions o Solving linear equations and inequalities o Fit data using linear models o Use and perform operations with real numbers Goal Graph and transform quadratic functions Solve quadratic equations and inequalities Fit data to quadratic models Use and perform operations with imaginary and other complex numbers Why? o Build a foundation for higher level mathematics o These skills can be used in Chemistry, Physics and Economics o Observe and understand patterns and relationships in science, social studies, and model data Key Vocabulary o Complex conjugate o Complex number o Axis of symmetry o Standard form Lesson 7: Operations with Complex Numbers o Imaginary number o Zero of a function o Minimum value o Parabola Perform operations with complex numbers (CC.9-12.N.CN.2) o Maximum value o Vertex form o Quadratic function o Discriminant The form of a complex number is The form of an imaginary number is 1 =!! = Warm-Up Simplify each expression. 17. 81 18. 18 Find each complex conjugate. 19. 6! 3 20. 4! + 2 Find each product. 21. 2 + 3 3 2 22. 6 + 3 5 1 5
Operations with Complex Numbers When complex numbers are added, subtracted, multiplied or divided, the resulting number is also a complex number. Add or subtract. Write the final answer in complex number form. 23. 2 + 4i + (3 11i) 25. 6 2i + ( 6 + 2i) 24. (4 i) (5 + 8i) 26. (10 + 3i) (10 4i) Multiply. Write the final answer in complex number form. 27. 2 + 4i (3 11i) 29. 6 2i ( 6 + 2i) 28. (4 i)(5 + 8i) 30. (10 + 3i)(10 4i) Divide. Write the final answer in complex number form. 31.!!!!!! 32.!!!!!!! 33.!!!!!! 34.!!!!!! Evaluate powers of i. Write the final answer in imaginary number form. 35.!!!! 36.!!" 37.!!"!! 38. 3!!" Assignment: Page 131, 47-51 odd, 55-69 odd, 85-101 odd, 115, 116 Ready To Go On Problems Page 109 and 135