09/01/2013 Linear Relations and Functions Chapter 2 .7 Pg 109

1. 09/01/2013Linear Relations and FunctionsChapter 2 .7 Pg 109 Parent Functions and Transformations

2. Terminology/Keywords • Terminology : • parent graph, • parent function, • constant function, • Transformation: translation, reflection, • identity function, quadratic function, translation, reflection, • line of reflection , • dilation

3. Parent Functions Identify the followingFunctions

4. Parent Functions

5. Learning Objectives 1. Describe and graph translations. 2. Describe and graph Reflections. 3.Identify Transformations. 4.Real Life Problems.

6. Draw the graph of :y = / x + 4/

7. PRACTICE • Homework Page 116 • Exercises: 50.51 • Practice Worksheet • See website and Email • Classwork Pages 113 – 116 Exercises 1-8,9,10 – 13 • 33 to 38 • Bonus: 44 ,49 to 51

8. f(x) = x

11. We will now see how certain transformations (operations) of a function change its graph. This will give us a better idea of how to quickly sketch the graph of certain functions. The transformations are (1) translations, (2) reflections, and (3) stretching/dilation

12. Vertical Translations • f(x) = x2 f(x) + y y 0 = x2 0 1 = x2 +1 3 = x2 + 3 2 = x2+2 x

13. Vertical Translations • f(x) = x2 f(x) + y y 0 = x2 -1 = x2 -1 0 -3 = x2-3 -2 =x2 - 2 x Adding c to f(x) moves the graph up by c units if c is positive, down if c is negative

14. Horizontal Translations • f(x) = x2 y y f(x + 0) = (x+0)2 f(x+1)=(x+1)2 f(x+2) =(x+2)2 f(x+3) = (x+3)2 x

15. Horizontal Translations • f(x) = x2 y y f(x – 0) = (x-0)2 f(x-1)=(x-1)2 f(x-2) =(x-2)2 f(x-3) = (x-3)2 x • Changing a function from f(x) to f(x-d) will move the graph d units to the right. • Changing a function from f(x) to f(x+d) will move the graph d units to the left.

16. Combining Translations • If f(x) = x2, graph f(x-2) +3: y y f(x) = x2 f(x-2)=(x-2)2 f(x-2) +3 =(x-2)2 +3 x

17. Parent Functions

18. Example 2 : Describe and graph Translations : • Describe the translation in y = /x/ + 2 The graph of y = /x/ + 2 is a translation of the graph of y = /x/ up 2 units. Example 3 : Describe and graph Reflections: Describe the reflection in y = -x2.Then graph the function.

19. QUIZ 2 • PG 110 • 1. 2A and 2B • 2.Describe the Term Translation. • 3.Pg 113 – Question 1 - 4

20. Example 3 : Describe and graph Reflections: Describe the reflection in y = .Then graph the function. The graph of y = is a reflection of the graph y = in the x- axis.

21. QUIZ 2 • PG 110 • 1. Describe what is a reflection and what is a reflection line. • 2.Pg 113 Question 5 and 6.

22. Example 4 : Describe and graph Dilations Describe the dilation in y = 4x .Then graph the function. The graph of y = 4x is a dilation of y = x .The slope of the graph of y = 4x is steeper than that of the graph of y = x

23. Example 5 : Real – World Question - Identify Transformations Lanscaping: Etham is going to add a brick walkway around the perimeter of his vegetable garden. The area of the walkway can be represented by the function f(x) =4 (x+2.5)2 - 25.Describe the transformation in the function. Then graph the function. Determine how each transformation affects the parent graph. f(x) =4(x+2.5)2 - 25 +2.5 translates f(x) = x2 left 2.5 units -2.5 translates f(x) = x2 down 25 units 4 stretches f(x) = x2 vertically The graph of f(x) =4 (x+2.5)2 - 25 is a combination of transformations of the parent graph f(x) = x2.

24. Group work • In groups of 4 look at the magic Equation and design your own Equation and draw graphs for your function. • Identify and describe the function.

25. a = adjusting shape (compress, stretch or reflect) c = moving up/down d = moving left/right Note: a ,c ,d  R Remember f(x) means – function with variable x Recall “Transforming”

26. Lesson Quiz: Part I Identify the parent function for g from its function rule. Then graph gby using grapher on your laptop and describe what transformation of the function it represents. 1.g(x) = x + 7 linear; translation up 7 units

27. Lesson Quiz: Part II Identify the parent function for g from its function rule. Then graph g and describe what transformation of the parent function it represents. 2.g(x) = x2 – 7 quadratic; translation down 7 units

28. Check It Out! The cost of playing an online video game depends on the number of months for which the online service is used. Graph the relationship from number of months to cost, and identify which parent function best describes the data. Then use the graph to estimate the cost of 5 months of online service.

29. Check It Out! Example 3 Continued Step 1 Graph the relation. Graph the points given in the table. Draw a smooth line through them to help you see the shape. Step 2 Identify the parent function. The graph of the data set resembles the shape of a linear parent function ƒ(x) = x. Step 3 Estimate the cost for 5 months of online service. The linear graph indicates that the cost for 5 months of online service is $72.

30. Lesson Quiz: Part I Identify the parent function for g from its function rule. Then graph g on your laptop and describe what transformation of the parent function it represents. 1.g(x) = x + 7 linear; translation up 7 units

31. Lesson Quiz: Part II Identify the parent function for g from its function rule. Then graph g on your laptop and describe what transformation of the parent function it represents. 2.g(x) = x2 – 7 quadratic; translation down 6 units

32. Lesson Quiz: Part III 3. Stacy earns $7.50 per hour. Graph the relationship from hours to amount earned and identify which parent function best describes it. Then use the graph to estimate how many hours it would take Stacy to earn $60. linear: 8 hr