Transcript INTERMEDIATE ALGEBRA Class Notes Sections 7.1 and 7.2

INTERMEDIATE ALGEBRA
Class Notes
Sections 7.1 and 7.2
Chapter 7:
Introduction to Relations and
Functions
Section 7.1
Chapter 7:
Introduction to Relations and
Functions
Section 7.1
Exercise #7
Write the slope-intercept form of the line.
Then identify the slope and y-intercept.
4x + 2y = 8
2y = – 4x + 8
y = – 2x + 4
slope: – 2
y-intercept: 0, 4 
Chapter 7:
Introduction to Relations and
Functions
Section 7.1
Exercise #31
Use the point-slope formula (if possible) to write
an equation of the line given the following
information. Write the final answer in
slope-intercept form if possible.
1
The slope is and the line passes
4
through the point  –8, 6 .

Point-slope formula: y – y1 = m x – x1
1
m=
4
point: – 8, 6 

1
The slope is and the line passes
4
through the point  –8, 6 .

Point-slope formula: y – y1 = m x – x1
1   
y –6 =
x – –8
4
1
y –6 =
x + 8
4
1
y –6 = x +2
4
1
y = x +8
4

Chapter 7:
Introduction to Relations and
Functions
Section 7.1
Exercise #39
Use the point-slope formula (if possible) to write
an equation of the line given the following
information. Write the final answer in
slope-intercept form if possible.
The line passes through the points
1, – 3 and –7, 2
Point-slope formula: y – y1 = m x – x1
2 –  –3 
5
5
m=
=
=–
–7 – 1
–8
8
5
m = – , use the point 1, – 3 
8



Point-slope formula: y – y1 = m x – x1
5
m = – , use the point 1, – 3 
8
5


y – –3 = – x – 1
8
5
5
y +3= – x +
8
8
5
5
y = – x + –3
8
8
5
5 24
y=– x+ –
8
8 8
5
19
y= – x–
8
8

Chapter 7:
Introduction to Relations and
Functions
Section 7.1
Exercise #45
Use the point-slope formula (if possible) to write
an equation of the line given the following
information.
The line passes through the point 2,6 
and is perpendicular to the line y = 1.
(Hint: Sketch the line first.)
Write the final answer in
slope-intercept form
if possible.
The line passes through the point 2,6 
and is perpendicular to the line y = 1.
y
7
x =2
–4
6
–3
x
y =1
Chapter 7:
Introduction to Relations and
Functions
Section 7.1
Exercise #55
The following table represents the median
selling price, y, of new privately-owned
one-family houses sold in the Midwest
from 1980 to 2000. Let x represent the
number of years since 1980. Let y
represent price in thousands of dollars.
Year
1980 x = 0
1985 x = 5
1990 x = 10
1995 x = 15
2000 x = 20
Price in $1000
67
84
108
142
167
Price (in $1000)
200
 20,167 
150
100
 0,67 
50
0
0
5
10
15
20
Year (x = 0 corresponds to 1980)
a. Find the slope of the line between
the points (0,67) and (20,167).
Price (in $1000)
200
 20,167 
150
100
 0,67 
50
0
0
5
10
15
20
Year (x = 0 corresponds to 1980)
167 – 67 100
m=
=
20 – 0
20
=5
Price (in $1000)
200
 20,167 
150
100
 0,67 
50
0
0
5
10
15
20
Year (x = 0 corresponds to 1980)
b. Find the equation of the line between
the points (0,67) and (20,167). Write
the answer in slope-intercept form.
Price (in $1000)
200
y = 5x + 67
150
100
 0,67 
50
0
 20,167 
0
5
10
15
20
Year (x = 0 corresponds to 1980)
m = 5, use (0, 67)
y = mx + b
y = 5x + 67
Price (in $1000)
200
y = 5x + 67
150
100
 0,67 
50
0
 20,167 
0
5
10
15
20
Year (x = 0 corresponds to 1980)
c. Use the equation from part b
to estimate the median price
of a one-family house sold in
the Midwest in the year 2005.
2005 is 25 years after 1980.
x = 25
y = 5x + 67
y = 5  25  + 67
y = 125 + 67
y = 192
The median price in 2005 will be
$192,000.
Chapter 7:
Introduction to Relations and
Functions
Section 7.2
Applications of Linear
Equations
Chapter 7:
Introduction to Relations and
Functions
Section 7.2
Exercise #5
The average daily temperature in January
for cities along the Eastern seaboard of the
United States and Canada generally
decreases for cities farther north.
A city’s latitude in the Northern
Hemisphere is a measure of
how far north it is on the globe.
The average temperature, y, can be described by
the equation y = – 2.333x + 124.0 where x is the
latitude of the city.
City
Jacksonville, FL
Miami, FL
Atlanta, GA
Baltimore, MD
Boston, MA
Atlantic City, NJ
New York, NY
Portland, ME
Charlotte, NC
Norfolk, VA
x
Latitude
(N)
y
Average Daily
Temperature(F)
30.3
25.8
33.8
39.3
42.3
39.4
40.7
43.7
35.2
36.9
52.4
67.2
41.0
31.8
28.6
30.9
31.5
20.8
39.3
39.1
The average temperature, y, can be described by
the equation y = – 2.333x + 124.0 where x is the
latitude of the city.
Temperature
100
80
60
40
20
0
20
30
Latitude
40
50
y = – 2.333x + 124.0
a. Which variable is the dependent
variable?
y, the temperature
b. Which variable is the independent
variable?
x, the latitude
y = – 2.333x + 124.0
c. Use the equation to predict the average
daily temperature in January for
Philadelphia, PA, whose latitude
is 40.0 N. Round to
one decimal place.
x = 40.0
y = – 2.333  40.0  + 124.0
= – 93.32 + 124.0
= 30.68
- 30.7°
Chapter 7:
Introduction to Relations and
Functions
Section 7.2
Applications of Linear
Equations
Chapter 7:
Introduction to Relations and
Functions
Section 7.2
Exercise #9
This figure depicts a relationship between a
person’s height, y (in inches), and the length
of the person’s arm, x (measured in inches
from shoulder to wrist).
100
90
80
70
60
50
40
30
20
10
0
 24, 82.25 
17, 57.75 
3
6
9 12 15 18 21 24 28
a. Use the points 17, 57.75 and 24, 82.25 
to find a linear equation relating height to
arm length.
82.25 – 57.75
24.5
= 3.5
m=
=
24 – 17
7
m = 3.5, use (17, 57.75)

y – y1 = m x – x1

y – 57.75 = 3.5  x – 17 
y – 57.75 = 3.5x – 59.5
y = 3.5x – 1.75
b. What is the slope of the line? Interpret
the slope in the context of the problem.
The slope = 3.5
For each additional inch in
length of a person’s arm, the
person’s height increases by
3.5 inches.
c. Use the equation from part a to estimate the
height of a person whose arm length
is 21.5 in.
y = 3.5x – 1.75
y = 3.5  21.5  – 1.75
y = 75.25 – 1.75
y = 73.5
73.5 inches
Chapter 7:
Introduction to Relations and
Functions
Section 7.2
Exercise #11
The cost to rent a car, y, for 1 day is
$20 plus $0.25 per mile.
a. Write a linear equation to
compute the cost, y, of driving
a car x miles for 1 day.
y = 20 + 0.25x
or
y = 0.25x + 20
The cost to rent a car, y, for 1 day is
$20 plus $0.25 per mile.
b. Use the linear equation to
compute the cost of driving
258 miles in the rental car.
y = 0.25x + 20
x = 258
y = 0.25  258  + 20
= 64.5 + 20
= 84.5
Cost = $84.50
Chapter 7:
Introduction to Relations and
Functions
Section 7.3
Introduction to Relations
Chapter 7:
Introduction to Relations and
Functions
Section 7.3
Exercise #7
Write the relation as a set of
ordered pairs.
A,1, A,2 , B,2 , C,3 , D,5 , E,4 
A
1
B
2
C
3
D
4
E
5
Chapter 7:
Introduction to Relations and
Functions
Section 7.3
Exercise #11
List the domain and range.
A
1
B
2
C
3
D
4
E
5
Domain: A, B, C, D, E
Range: 1, 2, 3, 4, 5 
Chapter 7:
Introduction to Relations and
Functions
Section 7.3
Exercise #23
Find the domain and range of the relations.
Use interval notation where appropriate.
y
x

Domain: – ,0 
Range: – , 
Chapter 7:
Introduction to Relations and
Functions
Section 7.3
Exercise #35
The percentage of male high school
students, y, who participated in an
organized physical activity for
the year 1995 is approximated by
y = –12.64x + 195.22. For this
model, x represents the grade
level 9 Š x Š 12.
(Source: Centers for Disease Control)
The percentage of male high school
students, y, who participated in an
organized physical activity for
the year 1995 is approximated by
y = –12.64x + 195.22. For this
model, x represents the grade
level 9 Š x Š 12.
a. Approximate the percentage
of males who participated in
organized physical activity
for grades 9, 10, 11, and 12,
respectively.
y = –12.64x + 195.22
Grade 9, x = 9:
y = –12.64  9  + 195.22
= –113.76 + 195.22
= 81.46
Grade 9: 81.46%
y = –12.64x + 195.22
Grade 10, x = 10:
y = –12.64 10  + 195.22
= –126.4 + 195.22
= 68.82
Grade 10: 68.82%
y = –12.64x + 195.22
Grade 11, x = 11:
y = –12.64 11 + 195.22
= –139.04 + 195.22
= 56.18
Grade 11: 56.18%
y = –12.64x + 195.22
Grade 12, x = 12:
y = –12.64 12  + 195.22
= –151.68 + 195.22
= 43.54
Grade 12: 43.54%
The percentage of male high school
students, y, who participated in an
organized physical activity for
the year 1995 is approximated by
y = –12.64x + 195.22. For this
model, x represents the grade
level 9 Š x Š 12.
b. Can we use this model to
predict seventh-grade
participation? Explain your
answer.
No. This model is for
9 Š x Š 12.