1. For Educational Use Only © 2010 12.1 Functions Involving Square Roots Brian Preston Algebra 1 2009-2010

2. For Educational Use Only © 2010 Real World Application What is the biggest pendulum you have seen?

3. For Educational Use Only © 2010 Lesson Objectives 1) Evaluate and graph a function involving square roots. 2) Use functions involving square roots to model real-life problems, such as the length of a cycle of a pendulum.

4. For Educational Use Only © 2010 Graph y = 2x + 1 -3 -2 -4 -5 -3 -2 -4 -5 1 2 3 4 5 23 4 5 1 +2 +1 (0,1) slope = 2 y-int = 1 1 1 Regular line Review

5. For Educational Use Only © 2010 -3 -5 3 5 1 3 1 7 5 9 New line y = ax 2 + bx + c (0,0) (1,1) (2,4) (3,9) (-1,1) (-2,4) (-3,9) y = x 2 1 Parabola line Review

6. For Educational Use Only © 2010 2 2 y =+ 3 x–(–2) Center -3 -3 1 3 5 1 3 -5 5 ( h, k ) –2 2 + Review

7. For Educational Use Only © 2010 y =+ k x – h a Graphing a Square Root a = The bigger a is the steeper the curved line is. The smaller a is the flatter the curve line is. (h,k) = Starting point (Center) k = Translates (shifts) up or down. a + k h k Definition

8. For Educational Use Only © 2010 y =+ k Can you take the square root of a negative number? Graphing a Square Root Definition x – h a -3 -2 -4 -5 -3 -2 -4 -5 1 2 3 4 5 23 4 5 1 X is negative Y is negative No

9. For Educational Use Only © 2010 y =+ k We only use the positive quadrants. Graphing a Square Root Definition x – h a 2 6 8 4 4 2 8 6

10. For Educational Use Only © 2010 y =+ k Square Root Graph. Graphing a Square Root Starting point (h,k) Definition x – h a

11. For Educational Use Only © 2010 y = Sketch the graph. Find the domain & range. Example 2 6 8 4 4 2 8 6 1) x Not the right form

12. For Educational Use Only © 2010 0 0 y =+ Sketch the graph. Find the domain & range. Example x – 0 2 6 8 4 4 2 8 6 Center ( h, k ) 0 0 0 1) 1 Normal

13. For Educational Use Only © 2010 y =+ 0 Sketch the graph. Find the domain & range. Example x – 0 Domain 1) How can you get an undefined answer or error? 1 0 or – 2 No negative radicals

14. For Educational Use Only © 2010 y =+ 0 Sketch the graph. Find the domain & range. Example x – 0 Domain x – 0 1) x – 0 ≥ 0 x ≥ 0 The smallest number the radical can become is 0. Range

15. For Educational Use Only © 2010 y =+ 0 Sketch the graph. Find the domain & range. Example x – 0 Domain 1) x – 0 ≥ 0 x ≥ 0 y =+ 0 0 This is the lowest y value. Range

16. For Educational Use Only © 2010 0 0 y =+ Sketch the graph. Find the domain & range. Example x – 0 Domain 1) x – 0 ≥ 0 x ≥ 0 Range y ≥ 0 y ≥ 0

17. For Educational Use Only © 2010 2) The period T (in seconds) of a pendulum is the time it takes for the pendulum to swing back and forth. The period is related to the length L (in inches) of the pendulum by the model T = 2π. Find the period of a pendulum with a length of eight inches. Give your answer to the nearest tenth. period eight period Real World Application 8 2π2π L 384 L = T

18. For Educational Use Only © 2010 8 How long is the pendulum period? Real World Application 2) 2π2π 384 = T 2π2π = T 0.20… 2π2π = T 0.144…  0.9 = T seconds

19. For Educational Use Only © 2010 Real World Application How long does it take a cycle of the pendulum to occur? 0.9 seconds

20. For Educational Use Only © 2010 y =– 1 Sketch the graph. Find the domain & range. Example 2 6 8 4 4 2 8 6 3) x Not the right form

21. For Educational Use Only © 2010 y =– 1 Sketch the graph. Find the domain & range. Example x – 0 2 6 8 4 4 2 8 6 – 1 Center ( h, k ) 0 0 – 1 3) 1 Normal

22. For Educational Use Only © 2010 y =– 1 Sketch the graph. Find the domain & range. Example x – 0 Domain x – 0 3) x – 0 ≥ 0 x ≥ 0 The smallest number the radical can become is 0. Range

23. For Educational Use Only © 2010 y =– 1 Sketch the graph. Find the domain & range. Example x – 0 Domain 3) x – 0 ≥ 0 x ≥ 0 y =– 1 0 This is the lowest y value. Range

24. For Educational Use Only © 2010 y =– 1 Sketch the graph. Find the domain & range. Example x – 0 Domain 3) x – 0 ≥ 0 x ≥ 0 Range y ≥ – 1 y ≥

25. For Educational Use Only © 2010 y =+ 4 Sketch the graph. Find the domain & range. Example 2 6 8 4 4 2 8 6 4) x Not the right form

26. For Educational Use Only © 2010 4 4 y =+ Sketch the graph. Find the domain & range. Example x – 0 2 6 8 4 4 2 8 6 Center ( h, k ) 0 0 4 4) 1 Normal

27. For Educational Use Only © 2010 y =+ 4 Sketch the graph. Find the domain & range. Example x – 0 Domain x – 0 4) x – 0 ≥ 0 x ≥ 0 The smallest number the radical can become is 0. Range

28. For Educational Use Only © 2010 y =+ 4 Sketch the graph. Find the domain & range. Example x – 0 Domain 4) x – 0 ≥ 0 x ≥ 0 y =+ 4 0 This is the lowest y value. Range

29. For Educational Use Only © 2010 4 4 y =+ Sketch the graph. Find the domain & range. Example x – 0 Domain 4) x – 0 ≥ 0 x ≥ 0 Range y ≥ 4 y ≥ 4

30. For Educational Use Only © 2010 0 0 y =+ Sketch the graph. Find the domain & range. Example x – 3 2 6 8 4 4 2 8 6 Center ( h, k ) 3 3 0 5) 1 Normal

31. For Educational Use Only © 2010 + 3 x – 3 y =+ 0 Sketch the graph. Find the domain & range. Example Domain 5) x – 3 ≥ 0 x ≥ 3 The smallest number the radical can become is 0. Range + 3

32. For Educational Use Only © 2010 0 0 y =+ Sketch the graph. Find the domain & range. Example x – 3 Domain 5) Range y ≥ 0 y ≥ 0 x – 3 ≥ 0 + 3 x ≥ 3

33. For Educational Use Only © 2010 1 y =+ Sketch the graph. Find the domain & range. Example x + 1 2 6 8 4 4 2 8 6 6) Wrong form or think the opposite sign.

34. For Educational Use Only © 2010 1 1 y =+ Sketch the graph. Find the domain & range. Example x–(–1) 2 6 8 4 4 2 8 6 Center ( h, k ) –1 1 6) 1 Normal

35. For Educational Use Only © 2010 – 1 x–(–1) y =+ 1 Sketch the graph. Find the domain & range. Example Domain 6) x–(–1) ≥ 0 x ≥ – 1 The smallest number the radical can become is 0. Range – 1

36. For Educational Use Only © 2010 1 1 y =+ Sketch the graph. Find the domain & range. Example Domain 6) Range y ≥ 1 y ≥ 1 – 1 x–(–1) ≥ 0 x ≥ – 1 x–(–1)

37. For Educational Use Only © 2010 y =+ 0 Sketch the graph. Find the domain & range. Example 2 6 8 4 4 2 8 6 7) x Not the right form 2

38. For Educational Use Only © 2010 0 0 y =+ Sketch the graph. Find the domain & range. Example x – 0 2 6 8 4 4 2 8 6 Center ( h, k ) 0 0 0 7) 2 Steeper curved line

39. For Educational Use Only © 2010 y =+ 0 Sketch the graph. Find the domain & range. Example x – 0 Domain x – 0 7) x – 0 ≥ 0 x ≥ 0 The smallest number the radical can become is 0. Range 2

40. For Educational Use Only © 2010 y =+ 0 Sketch the graph. Find the domain & range. Example x – 0 Domain 7) x – 0 ≥ 0 x ≥ 0 y =+ 0 0 This is the lowest y value. Range 2

41. For Educational Use Only © 2010 0 0 y =+ Sketch the graph. Find the domain & range. Example x – 0 Domain 7) x – 0 ≥ 0 x ≥ 0 Range y ≥ 0 y ≥ 0 2

42. For Educational Use Only © 2010 3 3 y =+ Sketch the graph. Find the domain & range. Example x – 4 2 6 8 4 4 2 8 6 Center ( h, k ) 4 4 3 8) 2 Steeper curved line

43. For Educational Use Only © 2010 + 4 x – 4 y =+ 3 Sketch the graph. Find the domain & range. Example Domain 8) The smallest number the radical can become is 0. Range 2 x – 4 ≥ 0 x ≥ 4 + 4

44. For Educational Use Only © 2010 3 3 y =+ Sketch the graph. Find the domain & range. Example Domain 8) Range y ≥ 3 y ≥ 3 x – 4 2 + 4 x – 4 ≥ 0 x ≥ 4 + 4

45. For Educational Use Only © 2010 9 9 y = Evaluate the function for the given value of x. Example 9) 3 9 x ; y = 3 x

46. For Educational Use Only © 2010 9 9 y = Evaluate the function for the given value of x. Example 9) 3 x ; y = 3 3 y = 3   y = 9

47. For Educational Use Only © 2010 – 2 x y = Evaluate the function for the given value of x. Example 10) (– 2) 21 – 2x ; y = 21 – 2 y = 21 y = + 4 = 5 25

48. For Educational Use Only © 2010 x y = Evaluate the function for the given value of x. Example 11) ( ) 36x – 2 ; y = 36 y = 18 y = – 2 = 4 16 – 2 1 2 1 2

49. For Educational Use Only © 2010 1) Don’t forget the negative signs. 2) Make sure you have the right center. Key Points & Don’t Forget y =+ k x – h a (h,k) not (– h,k) h

50. For Educational Use Only © 2010 pg. 496-497 #’s 18-42 even, 43-45, 52-64 even The Assignment

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