6.6 Analyzing Graphs of Quadratic Functions

1. 6.6 Analyzing Graphs of Quadratic Functions Write a Quadratic Equation in Vertex form

2. CCSS: A.SSE.3 • CHOOSE and PRODUCE an equivalent form of an expression to REVEAL and EXPLAIN properties of the quantity represented by the expression. • a. FACTOR a quadratic expression to reveal the zeros of the function it defines. • b. COMPLETE THE SQUARE iin a quadratic expression to REVEAL the maximum or minimum value of the function it defines.

3. CCSS: F.IF.7 • GRAPH functions expressed symbolically and SHOW key features of the graph, by hand in simple cases and using technology for more complicated cases.* • a. GRAPH linear and quadratic functions and show intercepts, maxima, and minima.

4. Standards for Mathematical Practice • 1. Make sense of problems and persevere in solving them. • 2.Reason abstractly and quantitatively. • 3. Construct viable arguments and critique the reasoning of others. • 4. Model with mathematics. • 5. Use appropriate tools strategically. • 6.Attend to precision. • 7. Look for and make use of structure. • 8.Look for and express regularity in repeated reasoning.

5. Essential Questions • How do I determine the domain, range, maximum, minimum, roots, and y-intercept of a quadratic function from its graph? • How do I use quadratic functions to model data?  • How do I solve a quadratic equation with non - real roots?

6. Vertex form of the Quadratic Equation So far the only way we seen the Quadratic Equation is ax2 + bx + c =0. This form works great for the Quadratic Equation. Vertex form works best for Graphing. We need to remember how to find the vertex. The x part of the vertex come from part of the quadratic equation.

7. Vertex form of the Quadratic Equation The x part of the vertex come from part of the quadratic equation. To find the y part, we put the x part of the vertex. The vertex as not (x, y), but (h, k)

8. Find the vertex of the Quadratic Equation

9. Find the vertex of the Quadratic Equation

10. The Vertex form of the Quadratic Equation

11. The Vertex form of the Quadratic Equation

12. The Vertex form of the Quadratic Equation

13. Write the Quadratic Equation in Vertex form Find a, h and k a= 1 h = -1 k = 3

14. Write the Quadratic Equation in Vertex form Find a, h and k a= 1 h = -1 k = 3

15. Vertex is better to use in graphing y = 2(x - 3)2 – 2 Vertex (3 , -2) Put in 4 for x, y = 2(3 - 4)2 – 2 (4, 0) Then (2, 0) is also a point

16. Let see what changes happen when you change “a”

17. Let see what changes happen when you change “a”

18. Let see what changes happen when you change “a” The larger the “a”, the skinner the graph What if “a” is a fraction?

19. Let see what changes happen when you change “a” What if “a” is a fraction?

20. What if we change “h” in the Vertex Let a = 1, k = 0 Changing the “h” moves the graph Left or Right.

21. What if we change “k” in the Vertex Let a = 1, h = 0 “k” moves the graph up or down.

22. Write an equation Given the vertex and a point on the graph. The vertex gives you “h” and “k”. We have to solve for “a” Given vertex (1, 2) and point on the graph passing through (3, 4) h =1; k = 2

23. Write an equation Given vertex (1, 2) and point on the graph passing through (3, 4) x=3, y=4 Solve for “a”

24. Write an equation a = ½ Solve for “a”

25. Write an equation a = ½ Final Answer

26. 6.7 Graphing and Solving Quadratic Inequalities

27. Solving by Graphing Find the Vertex and the zeros of the Quadratic Equation You can find the zero in anyway we used in this chapter. Making Table Factoring Completing the Square Quadratic Formula

28. Solving by Graphing Given: Zeros:

29. Solving by Graphing Given: Zeros: y = x 2 -3x +2

30. Solving by Graphing Which way do I shade?, inside or outside y = x 2 -3x +2

31. Solving by Graphing The answers are in the shaded area y > x2 -3x +2 Why is it a dotted line?

32. Solve x2 - 4x + 3 > 0 Find the zeros, (x - 3)(x – 1) = 0 x = 3 ; x = 1 Is the Graph up or Down?

33. Solve x2 - 4x + 3 > 0 Find the zeros, (x - 3)(x – 1) = 0 x = 3 ; x = 1 Where do we shade? Inside or Outside

34. Try a few points One lower then the lowest zero, one higher then the highest zero and one in the middle. Let x =0 Let x = 4 Let x = 2

35. Try a few points One lower then the lowest zero, one higher then the highest zero and one in the middle. Let x =0 02 - 4(0) + 3 > 0 True Let x = 4 42 – 4(4) + 3 > 0 True Let x = 2 22 – 4(2) + 3 > 0 False Only shade where it is true.

36. Solve x2 - 4x + 3 > 0

37. Solve Quadratic Inequalities Algebraically

38. Break the number line into three parts Test a number less then -2 x ≤ -2 Let x = - 3 (-3)2 + ( -3) ≤ 2 9 – 3 = 6 6 is not less then – 2 False

39. Break the number line into three parts Test a number between -2 and 1 -2 ≤ x ≤ 1 Let x = 0 (0)2 + (0) ≤ 2 0 + 0 = 0 0 is less then 2 True

40. Break the number line into three parts Test a number great then 1 x ≥ 1 Let x = 2 (2)2 + (2) ≤ 2 4 + 2 = 6 6 is not less then – 2 False

41. Break the number line into three parts So the answer is -2 ≤ x ≤ 1