1. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §1.2 Graphs Of Functions

2. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §1.1 → Introduction to Functions  Any QUESTIONS About HomeWork §1.1 → HW-01 1.1

3. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 3 Bruce Mayer, PE Chabot College Mathematics §1.2 Learning Goals  Review the rectangular coordinate system  Graph several functions  Study intersections of graphs, the vertical line test, and intercepts  Sketch and use graphs of quadratic functions in applications

4. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 4 Bruce Mayer, PE Chabot College Mathematics Points and Ordered-Pairs  To graph, or plot, points we use two perpendicular number lines called axes. The point at which the axes cross is called the origin. Arrows on the axes indicate the positive directions  Consider the pair (2, 3). The numbers in such a pair are called the CoOrdinates. The first coordinate, x, in this case is 2 and the second, y, coordinate is 3.

5. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 5 Bruce Mayer, PE Chabot College Mathematics Plot-Pt using Ordered Pair  To plot the point (2, 3) we start at the origin, move horizontally to the 2, move up vertically 3 units, and then make a “dot” x = 2 y = 3 (2, 3)

6. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 6 Bruce Mayer, PE Chabot College Mathematics Example  Plot the point (–4,3)  Starting at the origin, we move 4 units in the negative horizontal direction. The second number, 3, is positive, so we move 3 units in the positive vertical direction (up) x = –4;y = 3 4 units left 3 units up

7. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 7 Bruce Mayer, PE Chabot College Mathematics Example  Read XY-Plot  Find the coordinates of pts A, B, C, D, E, F, G A B C D E F G Solution: Point A is 5 units to the right of the origin and 3 units above the origin. Its coordinates are (5, 3). The other coordinates are as follows: –B: (−2,4) –C: (−3,−4) –D: (3,−2) –E: (2, 3) –F: (−3,0) –G: (0, 2)

8. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 8 Bruce Mayer, PE Chabot College Mathematics Tool For XY Graphing  Called “ Engineering Computation Pad” Light Green Backgound Tremendous Help with Graphing and Sketching Available in Chabot College Book Store I use it for ALL my Hand-Work Graph on this side!

9. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 9 Bruce Mayer, PE Chabot College Mathematics XY Quadrants  The horizontal and vertical axes divide the plotting plane into four regions, or quadrants Note the Ordinate & Abscissa (Ordinate) (Abscissa)

10. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 10 Bruce Mayer, PE Chabot College Mathematics The Distance Formula  The distance between the points (x 1, y 1 ) and (x 2, y 1 ) on a horizontal line is |x 2 – x 1 |.  Similarly, the distance between the points (x 2, y 1 ) and (x 2, y 2 ) on a vertical line is |y 2 – y 1 |.

11. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 11 Bruce Mayer, PE Chabot College Mathematics Pythagorean Distance  Now consider any two points (x 1, y 1 ) and (x 2, y 2 ).  These points, along with (x 2, y 1 ), describe a right triangle. The lengths of the legs are |x 2 – x 1 | and |y 2 – y 1 |.

12. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 12 Bruce Mayer, PE Chabot College Mathematics Pythagorean Distance  Find d, the length of the hypotenuse, by using the Pythagorean theorem: d 2 = |x 2 – x 1 | 2 + |y 2 – y 1 | 2  Since the square of a number is the same as the square of its opposite, we can replace the absolute-value signs with parentheses: d 2 = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2

13. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 13 Bruce Mayer, PE Chabot College Mathematics Distance Formula Formally  The distance d between any two points (x 1, y 1 ) and (x 2, y 2 ) is given by

14. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 14 Bruce Mayer, PE Chabot College Mathematics Example Find Distance  Find Distance Between Pt1 & Pt2  Use Dist Formula Pt-1 Pt-2

15. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 15 Bruce Mayer, PE Chabot College Mathematics Graphing by Dot Connection  “Connecting the Dots” ALWAYS works for plotting any y = f(x) from an eqn  The procedure Use Fcn Eqn to make a “T-Table” Properly Construct and Label Graph Plot Ordered-Pairs in T-Table Connect Dots with Straight or Curved Lines T-Table for

16. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 16 Bruce Mayer, PE Chabot College Mathematics Making Complete Plots 1.Arrows in POSITIVE Direction Only 2.Label x & y axes on POSITIVE ends 3.Mark and label at least one unit on each axis 4.Use a ruler for Axes & Straight-Lines 5.Label significant points or quantities           

17. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Graph f(x) = 2x 2  Solution: Make T-Table and Connect-Dots xy(x, y) 0 1 –1 2 –2 0228802288 (0, 0) (1, 2) (–1, 2) (2, 8) (–2, 8)  x = 0 is Axis of Symm  (0,0) is Vertex

18. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 18 Bruce Mayer, PE Chabot College Mathematics Plot PieceWise Function: f(x) f(x) f(x) f(x) = |x||x||x||x|  ReCalling the Absolute Value Definition can State Function in PieceWise Form  Make T-Table from Above Fcn Def What will be the SHAPE of the the Graph of this Function?  Class Question: What will be the SHAPE of the the Graph of this Function?

19. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Graph f(x) = |x|  Make T-table  Plot Points, and Connect Dots

20. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 20 Bruce Mayer, PE Chabot College Mathematics Graph Intersections  How To Find Solutions to the Equality of Functions? Graph Both Functions and Find Intersections –At Intersections x & y are the SAME for both functions, and ANY point on the graph is a “Solution” to Fcn  Thus at Intersections BOTH Fcns are Simultaneously Solved

21. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 21 Bruce Mayer, PE Chabot College Mathematics Graph InterSection Example  Consider two Functions:  Want to Find solution(s), x s, such that  Note that this Equation can NOT Solved exactly; The solutions are irrational Numbers Such “NonAlgebraic” Eqns are Called “Transcendental”  Find Solution by Graph Intersection(s)

22. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 22 Bruce Mayer, PE Chabot College Mathematics Graph InterSection Example  Plot Both Functions on Same Graph  Find Intersection(s)  Read x s from intersection points ≈1.44≈4.97≈7.54

23. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 23 Bruce Mayer, PE Chabot College Mathematics MSExcel vs Transcendental  The “Goal Seek” Command in MicroSoft Excel to Find x s with greater Accuracy  Use Excel to Solve the Transcendental Equation  Collect Terms on One Side, and use “Goal Seek” to find x that satisfies eqn  For the Eqn Above the solutions, x s, are called the “zeros” or “roots” of the “zeroed” eqn

24. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 24 Bruce Mayer, PE Chabot College Mathematics MSExcel vs Transcendental  Use The “Goal Seek” Command in MicroSoft Excel to Find x s with greater Accuracy

25. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 25 Bruce Mayer, PE Chabot College Mathematics Goal Seek (on Data Tab)

26. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 26 Bruce Mayer, PE Chabot College Mathematics Goal Seek Results (2 Roots)

27. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 27 Bruce Mayer, PE Chabot College Mathematics Zeros Graphed by MATLAB >> u = linspace(0, 2.5*pi, 300); >> v = cos_ln(u); >> xZ = [0,8]; yZ = [0, 0]; >> plot(u,v, xZ,yZ, 'LineWidth',3), grid, xlabel('u'), ylabel('v'); >> Z1 = fzero(cos_ln,2) Z1 = 1.4429 >> Z2 = fzero(cos_ln,5) Z2 = 4.9705 >> Z3 = fzero(cos_ln,8) Z3 = 7.5425

28. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 28 Bruce Mayer, PE Chabot College Mathematics Power Function  f(x) = Kx n  In the Power Function “n” can be ANY number, positive, negative, rational or Irrational. Some Examples M15PwrFcnGraphs_1306.m

29. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 29 Bruce Mayer, PE Chabot College Mathematics PolyNomial Function  The General PolyNomial Function  Where n ≡ a positive integer constant a k ≡ any real number constant  n (the largest exponent) is called the DEGREE of the Polynomial

30. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 30 Bruce Mayer, PE Chabot College Mathematics PolyNomial Function  The plot of p(x) is continuous and crosses the X-axis no more than n-times  Some Examples M15PloyNomialFcnGraphs_1306.m

31. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 31 Bruce Mayer, PE Chabot College Mathematics Rational Function  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.

32. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 32 Bruce Mayer, PE Chabot College Mathematics Rational Fcn Examples  Note the Asymptotic Behavior

33. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 33 Bruce Mayer, PE Chabot College Mathematics Graphing & Vertical-Line-Test  Test a Reln-Graph to see if the Relation represents a Fcn  If no VERTICAL line intersects the graph of a relation at more than one point, then the graph is the graph of a function.

34. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 34 Bruce Mayer, PE Chabot College Mathematics Example  Vertical-Line-Test  Use the Vertical Line Test to determine if the graph represents a function  SOLUTION NOT a function as the Graph Does not pass the vertical line test

35. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 35 Bruce Mayer, PE Chabot College Mathematics Example  Vertical-Line-Test  Use the Vertical Line Test to determine if the graph represents a function  SOLUTION NOT a function as the Graph Does not pass the vertical line test TRIPLE Valued

36. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 36 Bruce Mayer, PE Chabot College Mathematics Example  Vertical-Line-Test  Use the Vertical Line Test to determine if the graph represents a function  SOLUTION IS a function as the Graph Does pass the vertical line test SINGLE Valued SINGLE Valued

37. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 37 Bruce Mayer, PE Chabot College Mathematics Example  Vertical-Line-Test  Use the Vertical Line Test to determine if the graph represents a function  SOLUTION IS a function as the Graph Does pass the vertical line test SINGLE Valued

38. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 38 Bruce Mayer, PE Chabot College Mathematics Quadratic Functions  All quadratic functions have graphs similar to y = x 2. Such curves are called parabolas. They are U-shaped and symmetric with respect to a vertical line known as the parabola’s line of symmetry or axis of symmetry.  For the graph of f(x) = x 2, the y-axis is the axis of symmetry. The point (0, 0) is known as the vertex of this parabola.

39. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 39 Bruce Mayer, PE Chabot College Mathematics The Vertex of a Parabola  The FORMULA for the vertex of a parabola given by f(x) = ax 2 + bx + c: The x-coordinate of the vertex is −b/(2a). The axis of symmetry is x = −b/(2a). The second coordinate of the vertex is most commonly found by computing f(−b/[2a])

40. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 40 Bruce Mayer, PE Chabot College Mathematics Graphing f(x) = ax 2 + bx + c 1.The graph is a parabola. Identify a, b, and c 2.Determine how the parabola opens If a > 0, the parabola opens up. If a < 0, the parabola opens down 3.Find the vertex (h, k). Use the formula

41. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 41 Bruce Mayer, PE Chabot College Mathematics Graphing f(x) = ax 2 + bx + c 4.Find the x-intercepts Let y = f(x) = 0. Find x by solving the equation ax 2 + bx + c = 0. If the solutions are real numbers, they are the x-intercepts. If not, the parabola either lies –above the x–axis when a > 0 –below the x–axis when a < 0

42. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 42 Bruce Mayer, PE Chabot College Mathematics Graphing f(x) = ax 2 + bx + c 5.Find the y-intercept. Let x = 0. The result f(0) = c is the y-intercept. 6.The parabola is symmetric with respect to its axis, x = −b/(2a) Use this symmetry to find additional points. 7.Draw a parabola through the points found in Steps 3-6.

43. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 43 Bruce Mayer, PE Chabot College Mathematics Example  Graph  SOLUTION Step 1 a = –2, b = 8, and c = –5 Step 2 a = –2, a < 0, the parabola opens down. Step 3 Find (h, k). Maximum value of y = 3 at x = 2

44. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 44 Bruce Mayer, PE Chabot College Mathematics Example  Graph  SOLUTION Step 4 Let f (x) = 0. Step 5 Let x = 0.

45. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 45 Bruce Mayer, PE Chabot College Mathematics Example  Graph  SOLUTION Step 6 Axis of symmetry is x = 2. Let x = 1, then the point (1, 1) is on the graph, the symmetric image of (1, 1) with respect to the axis x = 2 is (3, 1). The symmetric image of the y–intercept (0, –5) with respect to the axis x = 2 is (4, –5). Step 7 The parabola passing through the points found in Steps 3–6 is sketched on the next slide.

46. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 46 Bruce Mayer, PE Chabot College Mathematics Example  Graph  SOLUTION cont. Sketch Graph Using the points Just Determined

47. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 47 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems §1.2-44 Supply & Demand

48. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 48 Bruce Mayer, PE Chabot College Mathematics All Done for Today AutoMobile Stopping Distance

49. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 49 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

50. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 50 Bruce Mayer, PE Chabot College Mathematics

51. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 51 Bruce Mayer, PE Chabot College Mathematics

52. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 52 Bruce Mayer, PE Chabot College Mathematics

53. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 53 Bruce Mayer, PE Chabot College Mathematics (120, 0)

54. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 54 Bruce Mayer, PE Chabot College Mathematics

55. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 55 Bruce Mayer, PE Chabot College Mathematics

56. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 56 Bruce Mayer, PE Chabot College Mathematics

57. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 57 Bruce Mayer, PE Chabot College Mathematics Graph by MATLAB

58. BMayer@ChabotCollege.edu MTH15_Lec-02_Fa13_sec_1-2_Fcn_Graphs.pptx 58 Bruce Mayer, PE Chabot College Mathematics MATLAB Code % Bruce Mayer, PE % MTH-15 23Jun13 % M15P12441306.m % % The FUNCTION p = linspace(0,120,500); E = -p.^2/5 + 24*p; % % the Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(p,E, 'LineWidth', 3),axis([0 120 0 800]),... grid, xlabel('\fontsize{14}p ($/Unit)'), ylabel('\fontsize{14}E ($k/Month)'),... title(['\fontsize{16}MTH15 P1.2-44 Bruce Mayer, PE',]),... annotation('textbox',[.55.055.0.1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'M15P12441306.m','FontSize',9)