1. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
§2.2 Graphs of Functions
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. 2.1 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
MTH 55
Review §
Any QUESTIONS About
§2.1 → Intro to Functions
Any QUESTIONS About HomeWork
§2.1 → HW-03

3. 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. 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. 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
Form Fcn-Graph by Letting y = f(x)

6. Graph Fcn: f(x) = x2 – 2x – 6 Let y = f(x) Plot-Pts & Connect-Dots x
(x, y) 3 9
(3, 9) 2 2
(2, 2) 6
(0, 6) 1 7
(1, 7)
(2, 6) 3
(3, 3) 4
(4, 2) 5
(5, 9)

7. 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.
FAILS Test

8. 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

9. 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

10. 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

11. 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

12. Example  Analyze Fcn Graph
Let:
Is the point (1, −3) on the graph of f ?
Find all values of x such that (x, 5) is on the graph of f.
Find all y-intercepts of the graph of f.
Find all x-intercepts of the graph of f.

13. 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

14. 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

15. 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)

16. 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)

17. 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

18. 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

19. 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

20. 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
30% is “Absorbed” by the body. The rest passes thru withOUT absorbtion

21. 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

22. Example  Statin Drugs Make T-Table using Eqn:
MaxConcen = ½(Previous Max) + 6
Day
Max Concentration 1
6.000 2
9.000 3
10.500 4
11.250 5
11.625 6
11.813 7
11.906 8
11.953 9
11.977 10
11.988

23. The Maximum Statin Concentration Approaches 12 mg.
Example  Statin Drugs
Graph T-table
Find Answer by Analyzing Graph
The Maximum Statin Concentration Approaches 12 mg.

24. WhiteBoard Work Problems From §2.2 Exercise Set
PPT → 50, 82
18, 52
Alabama Auto Accident Rates by County

25. Prob
ID two ages for which drivers have the same number of Accidents

26. P2.2-50
Draw Horizontal Line at GUESSED, or Estimated, location then find ages
22
68

27.  P2.2-50 Check Graph Estimates in Eqn Check Graph Estimates in Eqn
A  22 years 
Check Graph Estimates in Eqn
A  68 years

28. ReCall for Graphing: use y = f(x)
P  Find x for f(x) = 2.5
ReCall for Graphing: use y = f(x)
Horizontal Line at y = 2.5%
(12, 2.5)
(94, 2.5)
12
94

29. P  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:

30. Engineering Degrees in USA
All Done for Today
Engineering Degrees in USA

31. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer –