Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §1.2 GraphsOf Functions Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 1.1 Review § • Any QUESTIONS About • §1.1 → Introduction to Functions • Any QUESTIONS About HomeWork • §1.1 → HW-01

3. §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. 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. (2, 3) 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

6. 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. Find the coordinates of pts A, B, C, D, E, F, G B A E G F D C Example Read XY-Plot • 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. 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 Tool For XY Graphing Graph on this side!

9. XY Quadrants (Abscissa) • The horizontal and vertical axes divide the plotting plane into four regions, or quadrants • Note the Ordinate & Abscissa (Ordinate)

10. The Distance Formula • The distance between the points (x1, y1) and (x2, y1) on a horizontal line is |x2 – x1|. • Similarly, the distance between the points (x2, y1) and (x2, y2) on a vertical line is |y2 – y1|.

11. Pythagorean Distance • Now consider any two points (x1, y1) and (x2, y2). • These points, along with (x2, y1), describe a right triangle. The lengths of the legsare |x2 – x1| and |y2 – y1|.

12. Pythagorean Distance • Find d, the length of the hypotenuse, by using the Pythagorean theorem: d2 = |x2 – x1|2 + |y2 – y1|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: d2 = (x2 – x1)2 + (y2 – y1)2

13. Distance Formula Formally • The distance d between any two points (x1, y1) and (x2, y2) is given by

14. Example  Find Distance • Find Distance Between Pt1 & Pt2 • Use Dist Formula Pt-2 Pt-1

15. Graphing by Dot Connection • “Connecting the Dots” ALWAYS works for plotting any y = f(x) from an eqn • The procedure • Use FcnEqn 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. Making Complete Plots   Arrows in POSITIVE Direction Only Label x & y axes on POSITIVE ends Mark and label at least one unit on each axis Use a ruler for Axes & Straight-Lines Label significant points or quantities         

17. Solution:Make T-Table andConnect-Dots y (-2,8) (2,8) 8 7 6 5 4 3 (-1,2 ) 2 (1,2) 1 -5 -4 -3 -2 -1 1 2 3 4 5 -1 x (0, 0) -2 Example  Graph f(x) = 2x2 • x = 0 is Axis of Symm • (0,0) is Vertex

18. Plot PieceWise Function: f(x) = |x| • ReCalling the Absolute Value Definition can State Function in PieceWise Form • Make T-Table from Above Fcn Def • Class Question: What will be the SHAPE of the the Graph of this Function?

19. Example  Graph f(x) = |x| • Make T-table • Plot Points, and Connect Dots

20. 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. Graph InterSection Example • Consider two Functions: • Want to Find solution(s), xs, 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. Graph InterSection Example • Plot Both Functions on Same Graph • Find Intersection(s) • Read xs from intersection points ≈1.44 ≈4.97 ≈7.54

23. MSExcelvs Transcendental • The “Goal Seek” Command in MicroSoft Excel to Find xs 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, xs, are called the “zeros” or “roots” of the “zeroed” eqn

24. MSExcelvs Transcendental • Use The “Goal Seek” Command in MicroSoft Excel to Find xs with greater Accuracy Time forLiveDemo

25. Goal Seek (on Data Tab)

26. Goal Seek Results (2 Roots)

27. 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. Power Function  f(x) = Kxn • In the Power Function “n” can be ANY number, positive, negative, rational or Irrational. Some Examples M15PwrFcnGraphs_1306.m

29. PolyNomial Function • The General PolyNomial Function • Where • n ≡ a positive integer constant • ak ≡ any real number constant • n (the largest exponent) is called the DEGREE of the Polynomial

30. 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. 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. Rational Fcn Examples • Note the Asymptotic Behavior

33. Graphing & Vertical-Line-Test • Test a Reln-Graph to see if the Relation represents a Fcn • If noVERTICAL lineintersects the graph of a relation at morethan one point, then the graph is the graph of a function. FAILS Test

34. 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. 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 TRIPLEValued

36. 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 SINGLEValued SINGLEValued

37. 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 SINGLEValued

38. Quadratic Functions • All quadratic functions have graphs similar to y = x2. 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) = x2, the y-axis is the axis of symmetry. The point (0, 0) is known as the vertex of this parabola.

39. The Vertex of a Parabola • The FORMULA for the vertex of a parabola given by f(x) = ax2 + 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. Graphing f(x) = ax2 + bx + c • The graph is a parabola. Identify a, b, and c • Determine how the parabola opens • If a > 0, the parabola opens up. • If a < 0, the parabola opens down • Find the vertex (h, k). Use the formula

41. Graphing f(x) = ax2 + bx + c • Find the x-interceptsLet y = f(x) = 0. Find x by solving the equation ax2 + 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. Graphing f(x) = ax2 + bx + c • Find the y-intercept. Let x = 0. The result f(0) = c is the y-intercept. • The parabola is symmetric with respect to its axis, x = −b/(2a) • Use this symmetry to find additional points. • Draw a parabola through the points found in Steps 3-6.

43. Example  Graph • SOLUTION Step 1a = –2, b = 8, and c = –5 Step 2a = –2, a < 0, the parabola opens down. Step 3 Find(h, k). Maximum value of y = 3 at x = 2

44. Example  Graph • SOLUTION Step 4Let f (x) = 0. Step 5Let x = 0.

45. Example  Graph • SOLUTION Step 6Axis 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 7The parabola passing through the points found in Steps 3–6 is sketched on the next slide.

46. Example  Graph • SOLUTION cont. • Sketch GraphUsing the pointsJust Determined

47. WhiteBoard Work • Problems §1.2-44 • Supply & Demand

48. All Done for Today AutoMobileStoppingDistance

49. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –