1. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §G Translate Rational Plots

2. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §G → Graphing Rational Functions  Any QUESTIONS About HomeWork §G → HW-22 G MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 3 Bruce Mayer, PE Chabot College Mathematics GRAPH BY PLOTTING POINTS  Step1. Make a representative T-table of solutions of the equation.  Step 2. Plot the solutions as ordered pairs in the Cartesian coordinate plane.  Step 3. Connect the solutions (dots) in Step 2 by a smooth curve

4. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 4 Bruce Mayer, PE Chabot College Mathematics Translation of Graphs  Graph y = f(x) = x 2. Make T-Table & Connect-Dots  Select integers for x, starting with −2 and ending with +2. The T-table:

5. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 5 Bruce Mayer, PE Chabot College Mathematics Translation of Graphs  Now Plot the Five Points and connect them with a smooth Curve (−2,4)(2,4) (−1,1)(1,1) (0,0)

6. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 6 Bruce Mayer, PE Chabot College Mathematics Axes Translation  Now Move UP the graph of y = x 2 by two units as shown (−2,4)(2,4) (−1,1)(1,1) (0,0) (−2,6) (2,6) (−1,3) (1,3) (0,2)  What is the Equation for the new Curve?

7. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 7 Bruce Mayer, PE Chabot College Mathematics Axes Translation  Compare ordered pairs on the graph of with the corresponding ordered pairs on the new curve:

8. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 8 Bruce Mayer, PE Chabot College Mathematics Axes Translation  Notice that the x-coordinates on the new curve are the same, but the y-coordinates are 2 units greater  So every point on the new curve makes the equation y = x 2 +2 true, and every point off the new curve makes the equation y = x 2 +2 false.  An equation for the new curve is thus y = x 2 +2

9. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 9 Bruce Mayer, PE Chabot College Mathematics Axes Translation  Similarly, if every point on the graph of y = x 2 were is moved 2 units down, an equation of the new curve is y = x 2 −2

10. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 10 Bruce Mayer, PE Chabot College Mathematics Axes Translation  When every point on a graph is moved up or down by a given number of units, the new graph is called a vertical translation of the original graph.

11. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 11 Bruce Mayer, PE Chabot College Mathematics Vertical Translation  Given the Graph of y = f(x), and c > 0 1.The graph of y = f(x) + c is a vertical translation c-units UP 2.The graph of y = f(x) − c is a vertical translation c-units DOWN

12. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 12 Bruce Mayer, PE Chabot College Mathematics Horizontal Translation  What if every point on the graph of y = x 2 were moved 5 units to the right as shown below.  What is the eqn of the new curve? (−2,4)(2,4)(3,4)(7,4)

13. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 13 Bruce Mayer, PE Chabot College Mathematics Horizontal Translation  Compare ordered pairs on the graph of with the corresponding ordered pairs on the new curve:

14. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 14 Bruce Mayer, PE Chabot College Mathematics Horizontal Translation  Notice that the y-coordinates on the new curve are the same, but the x-coordinates are 5 units greater.  Does every point on the new curve make the equation y = (x+5) 2 true? No; for example if we input (5,0) we get 0 = (5+5) 2, which is false. But if we input (5,0) into the equation y = (x−5) 2, we get 0 = (5−5) 2, which is TRUE.  In fact, every point on the new curve makes the equation y = (x−5) 2 true, and every point off the new curve makes the equation y = (x−5) 2 false. Thus an equation for the new curve is y = (x−5) 2

15. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 15 Bruce Mayer, PE Chabot College Mathematics Horizontal Translation  Given the Graph of y = f(x), and c > 0 1.The graph of y = f(x−c) is a horizontal translation c-units to the RIGHT 2.The graph of y = f(x+c) is a horizontal translation c-units to the LEFT

16. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  Plot by Translation  Use Translation to graph f(x) = (x−3) 2 −2  LET y = f(x) → y = (x−3) 2 −2  Notice that the graph of y = (x−3) 2 −2 has the same shape as y = x 2, but is translated 3-unit RIGHT and 2-units DOWN.  In the y = (x−3) 2 −2, call −3 and −2 translators

17. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Plot by Translation  The graphs of y=x 2 and y=(x−3) 2 −2 are different; although they have the Same shape they have different locations  Now remove the translators by a substitution of x’ (“x-prime”) for x, and y’ (“y-prime”) for y  Remove translators for an (x’,y’) eqn

18. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Plot by Translation  Since the graph of y=(x−3) 2 −2 has the same shape as the graph of y’ =(x’) 2 we can use ordered pairs of y’ =(x’) 2 to determine the shape  T-table for y’ =(x’) 2

19. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Plot by Translation  Next use the translation rules to find the origin of the x’y’-plane. Draw the x’-axis and y’-axis through the translated origin The origin of the x’y’-plane is 3 units right and 2 units down from the origin of the xy-plane. Through the translated origin, we use dashed lines to draw a new horizontal axis (the x’-axis) and a new vertical axis (the y’-axis).

20. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 20 Bruce Mayer, PE Chabot College Mathematics Example  Plot by Translation  Locate the Origin of the Translated Axes Set using the translator values Move: 3-Right, 2-Down

21. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Plot by Translation  Now Plot the ordered pairs of the x’y’ equation on the x’y’-plane, and use the points to draw an appropriate graph. Remember that this graph is smooth

22. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Plot by Translation  Perform a partial-check to determine correctness of the last graph. Pick any point on the graph and find its (x,y) CoOrds; e.g., (4, −1) is on the graph  The Ordered Pair (4, −1) should make the xy Eqn True

23. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Plot by Translation  Sub (4, −1) into y=(x−3) 2 −2 :  Thus (4, −1) does make y = (x−3) 2 −2 true. In fact, all the points on the translated graph make the original Eqn true, and all the points off the translated graph make the original Eqn false

24. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Plot by Translation  What are the Domain & Range of y=(x−3) 2 −2?  To find the domain & range of the xy-eqn, examine the xy-graph (not the x’y’ graph).  The xy graph showns Domain of f is {x|x is any real number} Range of f is {y|y ≥ −2}

25. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 25 Bruce Mayer, PE Chabot College Mathematics Graphing Using Translation 1.Let y = f(x) 2.Remove the x-value & y-value “translators” to form an x’y’ eqn. 3.Find ordered pair solutions of the x’y’ eqn 4.Use the translation rules to find the origin of the x’y’-plane. Draw dashed x’ and y’ axes through the translated origin. 5.Plot the ordered pairs of the x’y’ equation on the x’y’-plane, and use the points to draw an appropriate graph.

26. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  ReCall Graph y = |x|  Make T-table

27. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Graph y = |x+2|+3  Step-1  Step-2  Step-3 → T-table in x’y’

28. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example  Graph y = |x+2|+3  Step-4: the x’y’-plane origin is 2 units LEFT and 3 units UP from xy-plane Up 3 Left 2

29. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 29 Bruce Mayer, PE Chabot College Mathematics Example  Graph y = |x+2|+3  Step-5: Remember that the graph of y = |x| is V-Shaped: Up 3 Left 2

30. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 30 Bruce Mayer, PE Chabot College Mathematics Rational Function Translation  A rational function is a function f that is a quotient of two polynomials, that is,  Where where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial. The domain of f consists of all inputs x for which q(x) ≠ 0.

31. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 31 Bruce Mayer, PE Chabot College Mathematics Example   Find the DOMAIN and GRAPH for f(x)  SOLUTION When the denom x = 0, we have x = 0, so the only input that results in a denominator of 0 is 0. Thus the domain {x|x  0} or (– , 0)  (0,  )  Construct T-table  Next Plot points & connect Dots

32. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 32 Bruce Mayer, PE Chabot College Mathematics Plot  Note that the Plot approaches, but never touches, the y-axis (as x ≠ 0) –In other words the graph approaches the LINE x = 0 the x-axis (as 1/   0) –In other words the graph approaches the LINE y = 0  A line that is approached by a graph is called an ASYMPTOTE

33. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 33 Bruce Mayer, PE Chabot College Mathematics ReCall Asymptotic Behavior  The graph of a rational function never crosses a vertical asymptote  The graph of a rational function might cross a horizontal asymptote but does not necessarily do so

34. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 34 Bruce Mayer, PE Chabot College Mathematics Recall Vertical Translation  Given the graph of the equation y = f(x), and c > 0,  the graph of y = f(x) + c is the graph of y = f(x) shifted UP (vertically) c units;  the graph of y = f(x) – c is the graph of y = f(x) shifted DOWN (vertically) c units y = 3x 2 y = 3x 2 − 3 y = 3x 2 +2

35. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 35 Bruce Mayer, PE Chabot College Mathematics Recall Horizontal Translation  Given the graph of the equation y = f(x), and c > 0,  the graph of y = f(x– c) is the graph of y = f(x) shifted RIGHT (Horizontally) c units;  the graph of y = f(x + c) is the graph of y = f(x) shifted LEFT (Horizontally) c units. y = 3x 2 y = 3(x-2) 2 y = 3(x+2) 2

36. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 36 Bruce Mayer, PE Chabot College Mathematics ReCall Graphing by Translation 1.Let y = f(x) 2.Remove the translators to form an x’y’ eqn 3.Find ordered pair solutions of the x’y’ eqn 4.Use the translation rules to find the origin of the x’y’-plane. Draw the dashed x’ and y’ axes through the translated origin. 5.Plot the ordered pairs of the x’y’ equation on the x’y’-plane, and use the points to draw an appropriate graph.

37. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 37 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Step-1  Step-2  Step-3 → T-table in x’y’

38. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 38 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Step-4: The origin of the x’y’ -plane is 2 units left and 1 unit up from the origin of the xy-plane:

39. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 39 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Step-5: We know that the basic shape of this graph is Hyperbolic. Thus we can sketch the graph using Fewer Points on the translated axis using the T-Table

40. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 40 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Examination of the Graph reveals Domain → {x|x ≠ −2} Range → {y|y ≠ 1}

41. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 41 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Step-1  Step-2  Step-3 → T-table in (x’y’) by y’ = −2/x’

42. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 42 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Step-4: The origin of the x’y’ -plane is 3 units RIGHT and 1 unit DOWN from the origin of the xy-plane

43. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 43 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Step-5: We know that the basic shape of this graph is Hyperbolic. Thus we can sketch the graph using Fewer Points on the translated axis using the T-Table

44. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 44 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Notice for this Graph that the Hyperbola is “mirrored”, or rotated 90°, by the leading Negative sign “Spread out”, or expanded, by the 2 in the numerator

45. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 45 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Examination of the Graph reveals Domain → {x|x ≠ 3} Range → {y|y ≠ −1}

46. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 46 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Step-1  Step-2  Step-3 → T-table in x’y’ by y’ = −1/(2x’)

47. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 47 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Step-4: The origin of the x’y’ -plane is 1 unit RIGHT and 2 units UP from the origin of the xy-plane

48. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 48 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Step-5: We know that the basic shape of this graph is Hyperbolic. Thus we can sketch the graph using Fewer Points on the translated axis using the T-Table

49. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 49 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Notice for this Graph that the Hyperbola is “mirrored”, or rotated 90°, by the leading Negative sign “Pulled in”, or contracted, by the 2 in the Denominator

50. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 50 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Examination of the Graph reveals Domain → {x|x ≠ 1} Range → {y|y ≠ 2}

51. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 51 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §G1 Exercise Set G8, G10, G12  A Function with TWO Vertical Asymptotes

52. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 52 Bruce Mayer, PE Chabot College Mathematics All Done for Today Another Cool Design by Asymptote Architecture

53. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 53 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

54. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 54 Bruce Mayer, PE Chabot College Mathematics Translate Up or Down  Make Graphs for  Notice: Of the form of VERTICAL Translations

55. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 55 Bruce Mayer, PE Chabot College Mathematics Translate Left or Right  Make Graphs for  Notice: Of the form of HORIZONTAL Translations

56. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 56 Bruce Mayer, PE Chabot College Mathematics Example  Graph y = |x+2|+3  Step-4: The origin of the x’y’ -plane is 2 units LEFT and 3 units UP from the origin of the xy-plane:

57. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 57 Bruce Mayer, PE Chabot College Mathematics Example  Graph y = |x+2|+3  Step-5: Remember that the graph of y = |x| is V-Shaped:

58. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 58 Bruce Mayer, PE Chabot College Mathematics Up 3 Over 2

59. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 59 Bruce Mayer, PE Chabot College Mathematics Up 3 Over 2

60. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 60 Bruce Mayer, PE Chabot College Mathematics

61. BMayer@ChabotCollege.edu MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt 61 Bruce Mayer, PE Chabot College Mathematics