1. Quadratic Functions and Transformations EQ: What is a quadratic function?

2. SYMMETRY EQ: What are the different types of symmetry?

3. ACTIVATION Do the following graphs display symmetry What does symmetry mean?

4. Lesson Symmetry to a line— X-axis—if their y coordinates are additive inverses Y-axis—if their x coordinates are additive inverses Also called even functions Test if f(-x) = f(x) ie you get the original equation Symmetry to a point– If two points are equidistant from the point of symmetry and all three points are on the same line To the origin if both their x and y coordinates are additive inverses Also called odd functions Test if f(-x) = -f(x) ie ALL the signs change from the original equation

5. Determine the symmetry of each if it exists: 3y = x 2 + 4 3x 2 – 2y 2 = 3 Y-axis 3y = (-x) 2 + 4 3(-x) 2 – 2y 2 =3 3y = x 2 + 4 3x 2 – 2y 2 = 3 both are symmetric to the y-axis X-axis 3(-y) = x 2 + 4 3x 2 – 2(-y) 2 =3 -3y = x 2 + 4 3x 2 – 2y 2 = 3 the second is symmetric to the x-axis Lesson

6. Determine if the function is even, odd or neither f(x) = 2x 2 + 4xf(x) = 3x 4 – 4x 2 f(-x)= 2(-x) 2 + 4(-x) f(-x) = 3(-x) 4 – 4(-x) 2 f(x) = 2x 2 - 4x f(x) = 3x 4 – 4x 2 Mixed signssame as the original Neithereven Lesson

7. Do the following graphs display symmetry Lesson

8. HOMEWORK PAGE(S): 389 NUMBERS: 2 -40 even

9. TRANSFORMATIONS 9-2 EQ: How do you graph a function of the form f(x)=(x - h) 2 + k??

10. ACTIVATION How are the parabolas alike and how are they different?

11. Lesson Transformation— An alteration of the width/scale or direction of a graph Translation— the ability to move a graph left/right and/or up/down

12. Parent function– the most basic shape of any function These often go through the origin y = x y = x 2 y = x 3 y = |x| y = Lesson

13. y = a func(bx – h) + k Work with a partner to determine what the h and k do to a function using the following examples y = (x – 2) 2 y = (x + 3) 2 y = x 2 + 2 y = x 2 – 4 Lesson Do they do the same thing regardless of the equation? y = |x – 2| y = |x + 3| y = |x| + 2 y = |x| – 4

14. y = a func(bx – h) + k h moves it left/right +/ – k moves it up/down +/ – Lesson

15. HOMEWORK PAGE(S): 393 NUMBERS: 2 - 22 even

16. STRETCHING AND SHRINKING 9-3 EQ: How do you graph a function of the form f(x) = a(x) 2 ?

17. ACTIVATION How are the parabolas alike and how are they different?

18. y = a func(bx – h) + k Work with a partner to determine what the a does to a function using the following examples y = 2(x ) 2 y = -2(x) 2 y = ½ x 2 y = - ½ x 2 Lesson Do they do the same thing regardless of the equation? y = 2|x| y = -2|x| y = ½ |x| y = - ½ |x|

19. y = a func(bx – h) + k “a” Determines the width and reflection If ‘a” is positive it is standard (opens up) ie it has a minimum point at the vertex If “a” is negative it is reflected over the y-axis (inverted—opens down) ie it has a minimum point at the vertex If |a|>1 the graph narrows ie 2 makes it goes up twice as fast If |a|<1 the graph is wider ie 1/2 makes it goes up half as fast Lesson

20. If the graph is f(x) Sketch 3f(x) - ½ f(x) Hint: Pick 0’s and maxs and mins Lesson

21. HOMEWORK PAGE(S): 398 to 399 NUMBERS: 2 to 8 even

22. THE ENTIRE TRANSFORMATION 9-4&5 EQ: How do you graph a function of the form f(x) = a(x - h) 2 ? How do you graph a function of the form f(x) = a(x - h) 2 + k?

23. Lesson Find the vertex, line of symmetry y = 3 (x – 2) 2 + 1 Vertex (2, -1) Symmetric to the line x = 2 Opens up so minimum value of y = -1

24. Lesson Write the equation that is a transformation of f(x) = 2x 2 Given maximum at (3, -1) f(x) = -2(x – 3) 2 – 1

25. Lesson Given f(x) = the graph to the right Work with a partner to sketch each of the following f(x+1) -2 2f(x) -3f(x – 2) + 3

26. HOMEWORK PAGE(S): 399 402 406 NUMBERS: 38 2 to 16 even 4 to 20 by 4’s

27. STANDARD FORM FOR QUADRATIC EQUATIONS 9-6 EQ: How do you use standard form of a quadratic equation to solve word problems involving minimum and maximum?

28. Activation USE ROPES: which means???? R O P E S Read the problem Organize your thoughts in a chart Plan the equations that will work Evaluate the Solution Summarize your findings

29. Lesson Place each equation in standard form then find the vertex, line of symmetry and max or min value. f(x) = x 2 – 2x – 3 f(x) + 3 + = x 2 – 2x + f(x) + 4 = (x – 1) 2 f(x) = (x – 1) 2 – 4 11

30. Lesson Place each equation in standard form then find the vertex, line of symmetry and max or min value. f(x) = 3x 2 – 24x + 50 f(x) – 50 + = 3 (x 2 – 8x + ) f(x) – 2 = 3 (x – 4) 2 f(x) = 3 (x – 4) 2 + 2 4816

31. A carpenter is building a room with a perimeter of 68 feet. What dimensions would yield the room with maximum area? P = 2x + 2y 68 = 2(x + y) 34 = x + y 34 – x = y Lesson A = lw A = xy A = x (34 – x) A = - x 2 + 34x A + = -1(x 2 – 34x + ) A – 289 = -1(x – 17) 2 A = -1 (x – 17) 2 + 289 Vertex (17, 289) Max area 289 sq ft.

32. HOMEWORK PAGE(S): 410 - 411 NUMBERS: 4, 8, 10, 12, 16, 29

33. GRAPHS AND X-INTERCEPTS 9-7 EQ: How are the x-intercepts of a quadratic function found?

34. ACTIVATION What is true of an x-intercept?

35. Lesson x-intercept—the place where a graph crosses the x-axis Means y =0 Find the x-intercept of f(x) = x 2 – 4x + 1 0 = x 2 – 4x + 1 what methods could be used? factoring, completing the square or the quadratic formula

36. Find the x-intercepts and graph the function If you have the vertex and the x-intercepts you have a good approximation of the graph f(x) = 2x 2 – 4x - 1 Lesson

37. HOMEWORK PAGE(S): 413 NUMBERS: 2 – 14 even