Chabot Mathematics
§2.2 Graphs
of Functions
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
Review §
2.1
MTH 55
 Any QUESTIONS About
• §2.1 → Intro to Functions
 Any QUESTIONS About HomeWork
• §2.1 → HW-03
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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.
? 7
13
 An ordered pair
 4  5
3 3
that satisfies an
13 ? 7  4 5  3 28 15




equation is called a
3
3
3
3 3
solution of the equation 13 13
13
check (4, ) for
3
Chabot College Mathematics
3
7
y  x 5
3
3

3

Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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
Chabot College Mathematics
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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)
Chabot College Mathematics
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
Graph Fcn: f(x) = x2 – 2x – 6
 Let y = f(x)
x
3
2
0
1
2
3
4
5
f(x)
9
2
6
7
6
3
2
9
Chabot College Mathematics
6
 Plot-Pts & Connect-Dots
(x, y)
(3, 9)
(2, 2)
(0, 6)
(1, 7)
(2, 6)
(3, 3)
(4, 2)
(5, 9)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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.
Chabot College Mathematics
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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
Chabot College Mathematics
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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
Chabot College Mathematics
9
TRIPLE
Valued
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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
Chabot College Mathematics
10
SINGLE
SINGLE
Valued
Valued
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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
Chabot College Mathematics
11
SINGLE
Valued
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
Example  Analyze Fcn Graph
 Let: y  f x   x  2x  3.
2
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.
Chabot College Mathematics
12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
Example  Analyze Fcn Graph
 SOLUTION: a. Is the point (1, −3) on
the graph of f ?
f x   y  x 2  2x  3
2
?
3  1  2 1  3  4 No!
• (1, −3) is not on the graph of f
– That is, (1, −3) does NOT Make this equation
TRUE
2
f x   y  x  2x  3
3  1  2 1  3  4 No!
Chabot College Mathematics
13
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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.
5  x  2x  3
2
0  x  2x  8
2
0  x  4 x  2 
x  2 or x  4
• (−2, 5) and (4, 5) are on the graph of f
Chabot College Mathematics
14
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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.
y  x  2x  3
2
y  0  2 0   3
2
y  3
• The only y-intercept is (0, −3)
Chabot College Mathematics
15
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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.
y  x 2  2x  3
0  x  2x  3
2
0  x  1x  3
x  1 or x  3
• The x-intercepts of
the graph of f are (−1, 0) and (3, 0)
Chabot College Mathematics
16
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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
Chabot College Mathematics
17
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
Example  Graph Domain/Range
 Graph this function.
Then estimate the
domain and range.
f ( x)  x  1
 Graphing
f ( x)  x  1
 Domain = [1, )
• Covers X-axis from 1
to infinity
 Range = [0, )
• Covers the
NON-negative portion
of the Y-axis
Chabot College Mathematics
18
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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
Chabot College Mathematics
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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
1
previous day's maximum concentration   6
2
Chabot College Mathematics
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
Example  Statin Drugs
 Make T-Table using
Eqn:
MaxConcen =
½(Previous Max) + 6
Chabot College Mathematics
22
Day Max Concentration
1
6.000
2
9.000
3
4
5
6
10.500
11.250
11.625
11.813
7
8
11.906
11.953
9
10
11.977
11.988
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
Example  Statin Drugs
 Graph T-table
 Find Answer by
Analyzing Graph
The Maximum Statin
Concentration
Approaches 12 mg.
Chabot College Mathematics
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
WhiteBoard Work
 Problems From
§2.2 Exercise Set
• PPT → 50, 82
• 18, 52

Alabama Auto
Accident Rates
by County
Chabot College Mathematics
24
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
Prob 2.2-50
ID two
ages for
which
drivers
have the
same
number of
Accidents
Chabot College Mathematics
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
P2.2-50
 Draw Horizontal Line at GUESSED, or
Estimated, location then find ages
22
Chabot College Mathematics
26
68
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
P2.2-50
 Check Graph Estimates in Eqn
• A  22 years
f 22  0.422  3622  1000
2
f 22  193.6  792  1000  401.6
 Check Graph Estimates in Eqn
• A  68 years
f 68  0.468  3668  1000
2

f 22  1849.6  2448  1000  401.6
Chabot College Mathematics
27
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
P2.2-54  Find x for f(x) = 2.5
ReCall for Graphing: use y = f(x)
 Horizontal
Line at
y = 2.5%
(12, 2.5)
12
Chabot College Mathematics
28
(94, 2.5)
94
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
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:
y  f 1912  2.5%  f 1994  y
Chabot College Mathematics
29
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
All Done for Today
Engineering
Degrees
in USA
Chabot College Mathematics
30
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
31
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55A_Lec-05_sec_2-2_Fcn_Graphs.ppt