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Learn how to write quadratic functions in vertex form, graph equations, find the discriminant, and solve using the quadratic formula. Practice transforming graphs and solving real-world problems.

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 4–6) CCSS Then/Now New Vocabulary Example 1: Write Functions in Vertex Form Example 2: Standardized Test Example: Write an Equation Given a Graph Concept Summary: Transformations of Quadratic Functions Example 3: Graph Equations in Vertex Form Lesson Menu

  3. Find the value of the discriminant for the equation 5x2 – x – 4 = 0. A. –81 B. –21 C. 21 D. 81 5-Minute Check 1

  4. Find the value of the discriminant for the equation 5x2 – x – 4 = 0. A. –81 B. –21 C. 21 D. 81 5-Minute Check 1

  5. Describe the number and type of roots for the equation 5x2 – x – 4 = 0. A. 2 irrational roots B. 2 real, rational roots C. 1 real, rational root D. 2 complex roots 5-Minute Check 2

  6. Describe the number and type of roots for the equation 5x2 – x – 4 = 0. A. 2 irrational roots B. 2 real, rational roots C. 1 real, rational root D. 2 complex roots 5-Minute Check 2

  7. A.1, – B. C.–1, D. Find the exact solutions of 5x2 – x – 4 = 0 by using the Quadratic Formula. 5-Minute Check 3

  8. A.1, – B. C.–1, D. Find the exact solutions of 5x2 – x – 4 = 0 by using the Quadratic Formula. 5-Minute Check 3

  9. A. B. C. D. Solve x2 – 9x + 21 = 0 by using the Quadratic Formula. 5-Minute Check 4

  10. A. B. C. D. Solve x2 – 9x + 21 = 0 by using the Quadratic Formula. 5-Minute Check 4

  11. A. B. C. D. Solve 2x2 + 4x = 10 by using the Quadratic Formula. 5-Minute Check 5

  12. A. B. C. D. Solve 2x2 + 4x = 10 by using the Quadratic Formula. 5-Minute Check 5

  13. The function h(t) = –16t2 + 70t + 6 models the height h in feet of an arrow shot into the air at time t in seconds. How long, to the nearest tenth, does it take for the arrow to hit the ground? A. about 3.2 s B. about 4.5 s C. about 5.1 s D. about 5.5 s 5-Minute Check 6

  14. The function h(t) = –16t2 + 70t + 6 models the height h in feet of an arrow shot into the air at time t in seconds. How long, to the nearest tenth, does it take for the arrow to hit the ground? A. about 3.2 s B. about 4.5 s C. about 5.1 s D. about 5.5 s 5-Minute Check 6

  15. Content Standards F.IF.8.a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f (kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Mathematical Practices 7 Look for and make use of structure. CCSS

  16. You transformed graphs of functions. • Write a quadratic function in the form y = a(x – h)2 + k. • Transform graphs of quadratic functions of the form y = a(x – h)2 + k. Then/Now

  17. vertex form Vocabulary

  18. Write Functions in Vertex Form A.Write y = x2 – 2x + 4 in vertex form. y = x2 – 2x + 4 Notice that x2 – 2x + 4 is not a perfect square. y = (x2 – 2x+ 1) + 4 – 1 Balance this addition by subtracting 1. Example 1

  19. Write Functions in Vertex Form y = (x – 1)2 + 3 Write x2 – 2x + 1 as a perfect square. Answer: Example 1

  20. Write Functions in Vertex Form y = (x – 1)2 + 3 Write x2 – 2x + 1 as a perfect square. Answer:y = (x – 1)2 + 3 Example 1

  21. Write Functions in Vertex Form B.Write y = –3x2 – 18x + 10 in vertex form. Then analyze the function. y = –3x2 – 18x + 10 Original equation y = –3(x2 + 6x) + 10 Group ax2 + bx and factor, dividing by a. y = –3(x2 + 6x + 9) + 10 – (–3)(9) Complete the square by adding 9 inside the parentheses. Balance this addition by subtracting –3(9). Example 1

  22. Write Functions in Vertex Form y = –3(x + 3)2 + 37 Write x2 + 6x + 9 as a perfect square. Answer: Example 1

  23. Write Functions in Vertex Form y = –3(x + 3)2 + 37 Write x2 + 6x + 9 as a perfect square. Answer:y = –3(x + 3)2 + 37 Example 1

  24. A. Write y = x2 + 6x + 5 in vertex form. A.y = (x + 6)2 – 31 B.y = (x – 3)2 + 14 C.y = (x + 3)2 + 14 D.y = (x + 3)2 – 4 Example 1

  25. A. Write y = x2 + 6x + 5 in vertex form. A.y = (x + 6)2 – 31 B.y = (x – 3)2 + 14 C.y = (x + 3)2 + 14 D.y = (x + 3)2 – 4 Example 1

  26. B. Write y = –3x2 – 18x + 4 in vertex form. A.y = –3(x + 3)2 – 23 B.y = –3(x + 3)2 + 31 C.y = –3(x – 3)2 – 23 D.y = –3(x – 3)2 + 31 Example 1

  27. B. Write y = –3x2 – 18x + 4 in vertex form. A.y = –3(x + 3)2 – 23 B.y = –3(x + 3)2 + 31 C.y = –3(x – 3)2 – 23 D.y = –3(x – 3)2 + 31 Example 1

  28. Write an Equation Given a Graph Which is an equation of the function shown in the graph? Example 2

  29. Write an Equation Given a Graph Read the Test ItemYou are given a graph of a parabola. You need to find an equation of the parabola. Solve the Test ItemThe vertex of the parabola is at (2, –3), so h = 2 and k = –3. Since the graph passes through (0, –1), let x = 0 and y = –1. Substitute these values into the vertex form of the equation and solve for a. Example 2

  30. Vertex form Substitute –1 for y, 0 for x, 2 for h, and –3 for k. Simplify. Add 3 to each side. Divide each side by 4. Write an Equation Given a Graph Example 2

  31. The equation of the parabola in vertex form is Write an Equation Given a Graph Answer: Example 2

  32. The equation of the parabola in vertex form is Write an Equation Given a Graph Answer: The answer is B. Example 2

  33. A. B. C. D. Which is an equation of the function shown in the graph? Example 2

  34. A. B. C. D. Which is an equation of the function shown in the graph? Example 2

  35. Concept

  36. Graph Equations in Vertex Form Graph y = –2x2 + 4x + 1. Step 1Rewrite the equation in vertex form. y = –2x2 + 4x + 1 Original equation y = –2(x2 – 2x) + 1 Distributive Property y = –2(x2 – 2x + 1) + 1 – (–2)(1) Complete the square. y = –2(x – 1)2 + 3 Simplify. Example 3

  37. Graph Equations in Vertex Form Step 2The vertex is at (1, 3). The axis of symmetry is x = 1. Because a = –2, the graph opens down and is narrower than the graph of y = –x2. Step 3 Plot additional points to help you complete the graph. Answer: Example 3

  38. Graph Equations in Vertex Form Step 2The vertex is at (1, 3). The axis of symmetry is x = 1. Because a = –2, the graph opens down and is narrower than the graph of y = –x2. Step 3 Plot additional points to help you complete the graph. Answer: Example 3

  39. A. B. C.D. Graph y = 3x2 + 12x + 8. Example 3

  40. A. B. C.D. Graph y = 3x2 + 12x + 8. Example 3

  41. End of the Lesson

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