1. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §2.2 Graphs of Functions

2. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §2.1 → Intro to Functions  Any QUESTIONS About HomeWork §2.1 → HW-03 2.1 MTH 55

3. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 3 Bruce Mayer, PE Chabot College Mathematics Recall Ordered Pairs  An ordered pair (a, b) is said to satisfy an equation with variables a and b if, when a is substituted for x and b is substituted for y in the equation, the resulting statement is true.  An ordered pair that satisfies an equation is called a solution of the equation 

4. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 4 Bruce Mayer, PE Chabot College Mathematics Ordered Pair Dependency  Frequently, the numerical values of the variable y can be determined by assigning appropriate values to the variable x. For this reason, y is sometimes referred to as the dependent variable and x as the independent variable. i.e., if we KNOW x, we can CALCULATE y

5. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 5 Bruce Mayer, PE Chabot College Mathematics Mathematical RELATION  Any set of ordered pairs is called a relation. The set of all first (x) components is called the domain of the relation, and the set of all SECOND (y) components is called the RANGE of the relation  Any (x, y) Relation can be plotted on a “Cartesian” GRAPH

6. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 6 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.

7. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 7 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

8. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 8 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

9. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 9 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

10. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 10 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

11. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Analyze Fcn Graph  Let: a.Is the point (1, − 3) on the graph of f ? b.Find all values of x such that (x, 5) is on the graph of f. c.Find all y-intercepts of the graph of f. d.Find all x-intercepts of the graph of f.

12. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Analyze Fcn Graph  SOLUTION: a. Is the point (1, −3) on the graph of f ? ? (1, − 3) is not on the graph of f –That is, (1, − 3) does NOT Make this equation TRUE

13. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Analyze Fcn Graph  SOLUTION: b. Find all values of x such that (x, 5) is on the graph of f Substitute 5 for y and solve for x. ( − 2, 5) and (4, 5) are on the graph of f

14. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Analyze Fcn Graph  SOLUTION: c. Find all y-intercepts (y when x = 0) of the graph of f. Substitute 0 for x and solve for y. The only y-intercept is (0, − 3)

15. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Analyze Fcn Graph  SOLUTION: d. Find all x-intercepts (x when y = 0) of the graph of f. Substitute 0 for y and solve for x. The x-intercepts of the graph of f are ( − 1, 0) and (3, 0)

16. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 16 Bruce Mayer, PE Chabot College Mathematics Domain & Range from Graph  Regarding the graph of a function:  Domain = the set of a function’s inputs, found on the horizontal axis That is, the Fcn’s X-axis Coverage  Range = the set of a function’s outputs, found on the vertical axis That is, the Fcn’s Y-axis Coverage

17. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Graph Domain/Range  Graph this function. Then estimate the domain and range.  Graphing  Domain = [1,  ) Covers X-axis from 1 to infinity  Range = [0,  ) Covers the NON- negative portion of the Y-axis

18. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Statin Drugs  Many pharmaceuticals used to lower high blood cholesterol levels are called statins and are very popular and widely prescribed. These drugs, along with proper diet and exercise, help prevent heart attacks and strokes.  BioChemists define bioavailability is the amount of a drug you have ingested that makes it into your bloodstream

19. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Statin Drugs  A statin with a BioAvailability of 30% has been prescribed for Fernando to treat his cholesterol levels. Fernando takes 20 milligrams (20 mg) of this statin every day. During the same day, one-half of the statin is filtered OUT of the BloodStream.  Find the maximum concentration of the statin in the bloodstream on each of the first ten days of using the drug, and graph the result

20. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 20 Bruce Mayer, PE Chabot College Mathematics Example  Statin Drugs  SOLUTION Since the statin has 30% bioavailability and Fernando takes 20 milligrams per day, the maximum concentration in the bloodstream is 30% of 20 mg, or 20(0.3) = 6 mg from each day’s prescription. Because one-half of the statin is filtered out of the blood each day, the daily maximum concentration is

21. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Statin Drugs  Make T-Table using Eqn: MaxConcen = ½(Previous Max) + 6 DayMax Concentration 16.000 29.000 310.500 411.250 511.625 611.813 711.906 811.953 911.977 1011.988

22. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Statin Drugs  Graph T-table  Find Answer by Analyzing Graph The Maximum Statin Concentration Approaches 12 mg.

23. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 23 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §2.2 Exercise Set PPT → 50, 82 18, 52  Alabama Auto Accident Rates by County

24. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 24 Bruce Mayer, PE Chabot College Mathematics Prob 2.2-50 ID two ages for which drivers have the same number of Accidents

25. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 25 Bruce Mayer, PE Chabot College Mathematics P2.2-50  Draw Horizontal Line at GUESSED, or Estimated location, then find ages  22  68

26. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 26 Bruce Mayer, PE Chabot College Mathematics P2.2-50  Check Graph Estimates in Eqn A  22 years  Check Graph Estimates in Eqn A  68 years 

27. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 27 Bruce Mayer, PE Chabot College Mathematics P2.2-54  Find x for f(x) = 2.5  Horizontal Line at y = 2.5%  12  94 (12, 2.5)(94, 2.5)

28. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 28 Bruce Mayer, PE Chabot College Mathematics P2.2-54  Find x for f(x) = 2.5  What do These Results mean in terms of the variables in this situation?  The Functional Interpretation is that TWO values for the INdependent variable, x, produce the SAME value for the DEpendent variable y:

29. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 29 Bruce Mayer, PE Chabot College Mathematics All Done for Today Engineering Degrees in USA

30. BMayer@ChabotCollege.edu MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt 30 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –