1. Splash Screen

2. Lesson Menu 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

3. Over Lesson 4–1 5-Minute Check 1 A.maximum B.minimum Does the function f(x) = 3x 2 + 6x have a maximum or a minimum value?

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

5. Over Lesson 4–1 5-Minute Check 3 A.x = y + 1 B.x = 2 C.x = 0 D.x = –1 Find the equation of the axis of symmetry for f(x) = 3x 2 + 6x.

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

7. Over Lesson 4–1 5-Minute Check 5 Graph f(x) = 3x 2 + 6x. A.ansB. C.ansD.ans

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

9. CCSS 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.

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

11. Vocabulary quadratic equation standard form root zero

12. Example 1 Two Real Solutions Solve x 2 + 6x + 8 = 0 by graphing. Graph the related quadratic function f(x) = x 2 + 6x + 8. The equation of the axis of symmetry is x = –3. Make a table using x-values around –3. Then graph each point.

13. Example 1 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. CheckCheck the solutions by substituting each solution into the original equation to see if it is satisfied. x 2 + 6x + 8 = 0 0 = 0 (–4) 2 + 6(–4) + 8 = 0 (–2) 2 + 6(–2) + 8 = 0 ??

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

15. Concept

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

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

18. Example 2 Solve x 2 – 6x = –9 by graphing. A.B. C.D. –3 33

19. Example 3 No Real Solution NUMBER THEORY Use a quadratic equation to find two numbers with a sum of 4 and a product of 5. UnderstandLet x = one of the numbers. Then 4 – x = the other number. Plan x(4 – x)=5The product is 5. 4x – x 2 =5Distributive Property x 2 – 4x + 5=0Add x 2 and subtract 4x from each side. SolveGraph the related function.

20. Example 3 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.

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

22. Example 4 Estimate Roots Solve –x 2 + 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.

23. Example 4 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.

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

25. End of the Lesson

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