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Splash Screen. Five-Minute Check (over Lesson 4–1) CCSS Then/Now New Vocabulary Example 1: Two Real Solutions Key Concept: Solutions of a Quadratic Equation Example 2: One Real Solution Example 3: No Real Solution Example 4: Estimate Roots Example 5: Solve by Using a Table

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Splash Screen

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 4–1) CCSS Then/Now New Vocabulary Example 1: Two Real Solutions Key Concept: Solutions of a Quadratic Equation Example 2: One Real Solution Example 3: No Real Solution Example 4: Estimate Roots Example 5: Solve by Using a Table Example 6: Real-World Example: Solve by Using a Calculator Lesson Menu

  3. Does the function f(x) = 3x2 + 6x have a maximum or a minimum value? A. maximum B. minimum 5-Minute Check 1

  4. Find the y-intercept of f(x) = 3x2 + 6x. A. –1 B. 0 C. 1 D. 2 5-Minute Check 2

  5. Find the equation of the axis of symmetry for f(x) = 3x2 + 6x. A.x = y + 1 B.x = 2 C.x = 0 D.x = –1 5-Minute Check 3

  6. Find the x-coordinate of the vertex of the graph of the functionf(x) = 3x2 + 6x. A. 1 B. 0 C. –1 D. –2 5-Minute Check 4

  7. A. ans B. C. ans D. ans Graph f(x) = 3x2 + 6x. 5-Minute Check 5

  8. A.y = 2x2 – 4x + 3 B.y = –x2 + 2x – 1 C.y = x2 + x + 1 D.y = 3x2 – 6x 1 __ 2 Which parabola has its vertex at (1, 0)? 5-Minute Check 6

  9. Content Standards A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. CCSS

  10. You solved systems of equations by graphing. • Solve quadratic equations by graphing. • Estimate solutions of quadratic equations by graphing. Then/Now

  11. quadratic equation • standard form • root • zero Vocabulary

  12. Two Real Solutions Solve x2 + 6x + 8 = 0 by graphing. Graph the related quadratic function f(x) = x2 + 6x + 8. The equation of the axis of symmetry is x = –3. Make a table using x-values around –3. Then graph each point. Example 1

  13. ? ? (–4)2 + 6(–4) + 8 = 0 (–2)2 + 6(–2) + 8 = 0 Two Real Solutions We can see that the zeros of the function are –4 and –2. Answer: The solutions of the equation are –4 and –2. Check Check the solutions by substituting each solution into the original equation to see if it is satisfied. x2 + 6x + 8 = 0 x2 + 6x + 8 = 0 0 = 0 0 = 0   Example 1

  14. A.B. C.D. –3, 1 –1, 3 –3, 1 –1, 3 Solve x2 + 2x – 3 = 0 by graphing. Example 1

  15. Concept

  16. x2 – 4x = –4 x2 – 4x + 4 = 0 Add 4 to each side. One Real Solution Solve x2 – 4x = –4 by graphing. Write the equation in ax2 + bx + c = 0 form. Graph the related quadratic function f(x) = x2 – 4x + 4. Example 2

  17. One Real Solution Notice that the graph has only one x-intercept, 2. Answer: The only solution is 2. Example 2

  18. A.B. C. D. 3 3 –3 –3 Solve x2 – 6x = –9 by graphing. Example 2

  19. No Real Solution NUMBER THEORYUse a quadratic equation to find two numbers with a sum of 4 and a product of 5. Understand Let x = one of the numbers. Then 4 – x = the other number. Plan x(4 – x) = 5 The product is 5. 4x – x2 = 5 Distributive Property x2 – 4x + 5 = 0 Add x2 and subtract 4x from each side. SolveGraph the related function. Example 3

  20. No Real Solution The graph has no x-intercepts. This means that the original equation has no real solution. Answer: It is not possible for two real numbers to have a sum of 4 and a product of 5. CheckTry finding the product of several numbers whose sum is 4. Example 3

  21. NUMBER THEORYUse a quadratic equation to find two numbers with a sum of 7 and a product of 14. A. 7, 2 B. –7, –2 C. 5, 2 D. no such numbers exist Example 3

  22. Estimate Roots Solve –x2 + 4x – 1 = 0 by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. Make a table of values and graph the related function. Example 4

  23. Estimate Roots The x-intercepts of the graph are between 0 and 1 and between 3 and 4. Answer: One solution is between 0 and 1 and the other is between 3 and 4. Example 4

  24. Solve x2 – 4x + 2 = 0 by graphing. What are the consecutive integers between which the roots are located? A. 0 and 1, 3 and 4 B. 0 and 1 C. 3 and 4 D. –1 and 0, 2 and 3 Example 4

  25. End of the Lesson

  26. Page 234 – 236 #41 – 48, 61 – 63, 67 – 73

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