NAME
6-1
DATE
Relations (Pages 238–243)
16.0,
S
T
17.0
A
N
D
A
R
D
S
A relation is a set of ordered pairs. The first coordinate in an ordered pair is
the x-coordinate. The second coordinate is the y-coordinate. The domain
of a relation is the set of all x-coordinates, and the range is the set of all
y-coordinates.
EXAMPLE
Express the relation {(4, 1), (1, 2), (1, 4), (2, 3), (4, 3)} as a
table and as a graph. Then determine the domain and range.
x
4
1
1
2
4
The domain is {4, 1, 1, 2, 4) and
y
y
1
2
4
3
3
(4, 3)
the range is {1, 2, 4, 3, 3}.
(–1, 2)
x
O
(–4, –1)
(2, –3)
(1, –4)
PRACTICE
Express each relation as a table and as a graph. Then determine the domain
and the range.
1. {(2, 1), (1, 0), (1, 2), (2, 4), (4, 3)}
4)}
2. {(3, 3), (1, 1), (0, 0.5), (2.5, 2), (3,
Express each relation as a set of ordered pairs and in a table. Then
determine the domain and the range.
3.
4.
y
O
5.
y
y
x
x
O
O
x
Express each relation as a set of ordered pairs.
6.
B
3.
y
3
4
5
6
7.
x
4
3
2
1
8.
y
3
1
1
3
x
0
2
4
6
y
1
1
3
5
C
C
A
B
5.
C
B
6.
A
7.
B
A
8.
9. Standardized Test Practice What is the domain of the relation {(2, 7), (3, 5), (2, 8)}?
Answers: 1–2. See Answer Key for tables and graphs. 1. D {2, 1, 1, 2, 4}; R {1, 0, 2, 4, 3} 2. D {3, 1, 0, 2.5, 3};
R {3, 1, 0.5, 2, 4} 3–5. See Answer Key for tables. 3. {(4, 4), (3, 2), (1, 2), (2, 1)}; D {4, 3, 1, 2},
R {4, 2, 2, 1} 4. {(1, 2), (1, 2), (2, 2), (3, 3)}; D {1, 1, 2, 3}; R {2, 2, 3} 5. {(3, 2), (3, 1), (1, 2), (0, 3);
D {3, 1, 0}; R {2, 1, 2, 3} 6. {(3, 3), (2, 4), (1, 5), (0, 6)} 7. {(1, 3), (2, 1), (3, 1), (4, 3)} 8. {(6, 5), (4, 3),
(2, 1), (0, 1)} 9. D
4.
x
3
2
1
0
© Glencoe/McGraw-Hill
40
CA Parent and Student Study Guide
Algebra: Concepts and Applications
NAME
6-2
DATE
A {2, 3, 5, 7, 8}
C {2, 3, 8}
B {5, 7, 8}
D {2, 3}
25.0
S
T
A
N
D
A
R
D
S
Equations as Relations
(Pages 244–249)
An equation in two variables has solutions that are ordered pairs in the
form (x, y). The set of solutions to a problem is called the solution set.
Solution of an
Equation in Two
If a true statement results when the numbers in an ordered pair are substituted into
an equation in two variables, then the ordered pair is a solution of the equation.
Variables
2(1) 1
1
0
2(0) 1
1
(0, 1)
2(1) 1
3
(1, 3)
1
EXAMPLES
B Which of the ordered pairs (3, 5), (0, 1),
or (1, 1) are solutions of y 2x 1?
Make a table and substitute each value of x into the
equation to determine the corresponding value of y.
2x 1
range
y
(1, 1)
solution set: {(1, 1), (0, 1), (1, 3)}
A Solve y 2x 1 if the domain is {1, 0,
1}.
domain
x
1
Substitute the values for x and y into the equation to
see if they make a true statement.
ordered pair
(x, y)
Does 5 2(3) 1? Yes, 5 6 1.
Does 1 2(0) 1? No, 1 0 1.
Does 1 2(1) 1? No, 1 2 1.
(3, 5) is a solution of y 2x 1.
PRACTICE
Which ordered pairs are solutions of each equation?
1. y 2x 7
a. (4, 1)
b. (8, 9)
c. (1, 5)
d. (0, 7)
2. y 9x
a. (2, 11)
b. (1, 9)
c. (1, 9)
d. (3, 12)
3. 2x y 18
a. (1, 15)
b. (0, 18)
c. (2, 14)
d. (1, 20)
4. y 3x 10
a. (7, 31)
b. (0, 0)
c. (0, 10)
d. (2, 16)
5. 5x 3y 24
a. (1, 5)
b. (4, 2)
c. (3, 1)
d. (0, 8)
Solve each equation if the domain is {2, 1, 0, 2}. Graph the solution set.
6. y 3x 1
9. 2x y 2
B
3.
C
C
A
B
5.
C
B
6.
A
7.
B
A
10. 3x y 1
8. x y 4
11. 2x 2y 8
Find the domain of each equation if the range is {3, 1, 1, 3}.
8.
12. y 2x 1
13. y x 2
14. 4y 2x
Answers: 1. a, b 2. c 3. b, d 4. a, c 5. d 6–11. See Answer Key for graphs. 6. {(2, 5), (1, 2), (0, 1), (2, 7)}
7. {(2, 7), (1, 5), (0, 3), (2, 1)} 8. {(2, 6), (1, 5), (0, 4), (2, 2)} 9. {(2, 6), (1, 4), (0, 2), (2, 2)}
10. {(2, 7), (1, 4), (0, 1), (2, 5)} 11. {(2,6), (1, 5), (0, 4), (2, 2)} 12. {2, 1, 0, 1} 13. {1, 1, 3, 5}
14. {6, 2, 2, 6} 15. D
4.
7. y 2x 3
© Glencoe/McGraw-Hill
41
CA Parent and Student Study Guide
Algebra: Concepts and Applications
NAME
6-3
DATE
15. Standardized Test Practice Which of the following is a
solution of the equation 2x y 10?
A (2, 6)
B (2, 6)
C (2, 6)
An equation whose graph is a straight line is called a linear equation. A
linear equation may contain one or two variables with no variable having an
exponent other than 1. A linear equation can be written in the form Ax By C,
where A, B, and C are any real numbers, and A and B are not both zero. To
graph a linear equation, make a table of ordered pairs that are solutions. Then
graph the ordered pairs and draw a straight line through them.
EXAMPLES
B Graph the equation y 2.
A Determine whether the equation
y 2x 1 is a linear equation. If it is,
identify A, B, and C.
Select five values for the domain and make a table.
This is a linear equation, since the equation contains
only two variables and the power of each variable is 1.
First, rewrite the equation so that both variables are
on the same side of the equation.
y 2x 1
2x y 1
Subtract 2x from each side.
x
y
(x, y)
2
1
0
1
2
2
2
2
2
2
(2, 2)
(1, 2)
(0, 2)
(1, 2)
(2, 2)
Note that because the
equation does not
contain the variable x, x
can be any value and the
y-value will still be 2.
Then graph the ordered pairs and
connect them to draw the line.
Note that the graph of y 2 is a
horizontal line through (0, 2).
The equation is now in the form Ax By C, where
A 2, B 1, and C 1.
y
O
x
Try These Together
1. If the equation x 3 is a linear
equation, identify A, B, and C.
2. Graph the equation 3x y 5.
HINT: To find values for y more easily, solve the
equation for y. Subtract 3x from each side and then
divide each side by 1.
HINT: Since there is no variable y in this equation,
use the placeholder 0y.
PRACTICE
Determine whether each equation is a linear equation. Explain. If an
equation is linear, identify A, B, and C.
3. y 2x2 3
4. x 2y 8
5. y 1
6. y 4x 1
7. 3x 5y 7
8. 8 y x
Graph each equation.
B
4.
C
B
8.
12. y 3 0
13. y 5 0
14. x 2 0
15. x y 6
16. x y 15
17. 2x y 4
C
B
A
7.
11. y 3 2x
C
A
5.
6.
10. y 3x 1
B
A
18. Standardized Test Practice Write the equation y 2x 8 in the standard
form Ax By C.
A y 2x 8
B y 2x 8
C 2x y 8
D 2x y 8
Answers: 1. A 1, B 0, C 3 2. See Answer Key. 3. no 4. yes; A 1, B 2, C 8 5. yes; A 0, B 1, C 1
6. yes; A 4, B 1, C 1 7. yes; A 3, B 5, C 7 8. yes; A 1, B 1, C 8 9–17. See Answer Key. 18. C
3.
9. y x 4
© Glencoe/McGraw-Hill
42
CA Parent and Student Study Guide
Algebra: Concepts and Applications
NAME
6-4
DATE
Functions (Pages 256–261)
16.0,
S
T
18.0,
A
25.0
N
D
A
R
D
S
A function is a relation in which each element of the domain is paired with
exactly one element of the range. Equations that are functions can be written
in a form called functional notation, f(x) (read “f of x”). In a function, x is
an element of the domain and f(x) is the corresponding element in the range.
In f (3), f (3) is the functional value of f for x 3.
Vertical
Line Test
If each vertical line passes through no more than one point of the graph of a relation, then
the relation is a function.
EXAMPLES
B If f(x) 3x 1 and g(x) 2x, find
f(1) and g(3).
A Is {(1, 2), (1, 3)} a function? Is
{(1, 4), (3, 2), (5, 4)} a function?
1st relation: not a function
This relation has 1 paired with both 2 and 3.
f(x) 3x 1
f(1) 3(1) 1 or 2
Replace x with 1.
2nd relation: a function
In this relation, each x-value is paired with no more
than one y-value. A function can have a y-value
paired with more than one x-value.
g(x) 2x
g(3) 2(3) or 6
Replace x with 3.
PRACTICE
Determine whether each relation is a function.
1.
x
1
2
3
2.
y
10
13
16
x
2
2
3
3.
y
0
1
4
5. {(15, 0), (15, 2)}
y
8.
4. {(7, 4), (6, 3), (5, 2)}
y
7.
x
O
O
x
33
35
36
y
10
8
10
6. {(0, 1), (2, 1), (0, 3)}
y
9.
O
x
x
If f(x) 3x and g(x) x 5, find each value.
B
4.
14. f(a)
15. g(m)
16. g(9)
17. f(2)
C
B
8.
13. f(1)
C
B
A
7.
12. g(8)
C
A
5.
6.
11. g(7)
B
A
18. Standardized Test Practice Martha pays a flat $50 a month for the use of her
cell phone. She also pays $0.30 for each minute that she talks over 6 hours.
The cost of her phone bill can be represented by f(x) 50 0.30x, where x is
the number of minutes past 6 hours that she uses the phone. Evaluate f(60) to
find the amount of her phone bill if she uses the phone for 7 hours.
A $68.30
B $68.00
C $50.30
D $18.00
Answers: 1. yes 2. no 3. yes 4. yes 5. no 6. no 7. no 8. yes 9. no 10. 21 11. 2 12. 13 13. 3 14. 3a
15. m 5 16. 4 17. 6 18. B
3.
10. f(7)
© Glencoe/McGraw-Hill
43
CA Parent and Student Study Guide
Algebra: Concepts and Applications
NAME
6-5
DATE
Direct Variation (Pages 264–269)
15.0
S
T
A
N
D
A
R
D
S
A direct variation is a linear function that can be written in the form y kx,
where k 0. In this equation, k is called the constant of variation. We say
that y varies directly as x. Since y depends on x, y is the dependent variable
and x is the independent variable. The graph of a direct variation passes
through the origin. Direct variations can be used to solve rate problems.
You can also use 3 of the general forms for proportions to solve direct
y
x
x
x
y
y
1
1
1
1
2
variation problems. These include , 2 , and .
y2
x2 y2
y1
x1
x2
EXAMPLES
A Determine whether y 2x is a direct
variation.
B Suppose y varies directly as x and y 4
when x 6. Find y when x 9.
Graph the equation.
y
O
y1
y2
1
x
x
Direct proportion
4
y2
6
9
y1 4, x1 6, and x2 9
2
Find the cross products.
36 6y2
6 y2
Divide each side by 6.
So, y 6 when x 9.
x
Since the graph passes through the origin, the
equation is a direct variation.
PRACTICE
Determine whether each equation is a direct variation. Verify the answer
with a graph.
1. y x 2
2. y 5x
3. y x
4. y 2x 3
Solve. Assume that y varies directly as x.
5. If y 8 when x 5,
find x when y 64.
6. If y 14 when x 84,
find x when y 2.
7. Find y when x 9,
if y 15 when x 27.
8. Find y when x 52,
if y 3 when x 4.
Solve by using direct variation.
9. If there are 4 quarts in a gallon, how many quarts are in 7.5 gallons?
10. How many inches are in 2.5 yards if there are 36 inches in a yard?
B
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
11.
Standardized Test Practice The amount an employee earns varies directly
as the number of hours she works. If she gets paid $58.80 for 8 hours of
work, how much will she get paid for 15 hours of work?
A $110.25
B $112.50
C $117.60
D $120.00
Answers: 1–4. See Answer Key for graphs. 1. no 2. yes 3. yes 4. no 5. 40 6. 12 7. 5 8. 39 9. 30 10. 90 11. A
3.
© Glencoe/McGraw-Hill
44
CA Parent and Student Study Guide
Algebra: Concepts and Applications
NAME
6-6
DATE
Inverse Variation (Pages 270–275)
15.0
S
T
A
N
D
A
R
D
S
An inverse variation is described by an equation of the form xy k, where
k 0. We say that y varies inversely as x. You can use a proportion, such as
x1
x2
y
2
, to solve problems involving inverse variation.
y
1
EXAMPLES
A Suppose y varies inversely as x and y 9
when x 4. Find y when x 12.
y
x1
x2
2
y
4
12
2
9
1
y
Inverse variation proportion
x1 4, x2 12, and y1 9
36 12y2
Find the cross products.
36
12
Divide each side by 12.
12y2
12
B Suppose y varies inversely as x, and
y 3 when x 4. Find the constant of
variation and write an equation for the
statement.
xy k
(4)(3) k
12 k
Inverse variation
x 4, y 3
Multiply.
The constant of variation is 12. An equation for the
statement is xy 12.
3 y2
So, when x 12, y 3.
PRACTICE
Solve. Assume that y varies inversely as x.
1. Find x when y 7 if y 3 when x 14.
2. If y 8 when x 5, find x when y 10.
3. If y 9 when x 6, find y when x 2.
4. Suppose y 21 when x 4. Find y when x 28.
Find the constant of variation. Then write an equation for each statement.
5. y varies inversely as x, and y 2 when x 8.
6. y varies inversely as x, and y 5 when x 4.
1
7. y varies inversely as x, and y 4 when x .
2
8. y varies inversely as x, and y 3 when x 2.4.
9. Electronics The amount of current in a circuit varies inversely as the
resistance in the circuit. Suppose there is 100 milliamps of current when
the resistance is 60 ohms. What would the current be if the resistance
were increased to 150 ohms?
B
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
10. Standardized Test Practice Suppose y varies inversely as x and y 5 when
x 8. Find x when y 10.
A 3
10. B
4.
B 4
C 7
D 16
Answers: 1. 6 2. 4 3. 27 4. 3 5. 16; xy 16 6. 20; xy 20 7. 2; xy 2 8. 7.2; xy 7.2 9. 40 milliamps
3.
© Glencoe/McGraw-Hill
45
CA Parent and Student Study Guide
Algebra: Concepts and Applications
NAME
6
DATE
Chapter 6 Review
16.0,
S
T
17.0,
A
18.0,
N
D
25.0
A
R
D
S
Make a Map
See if you and a parent can find Captain Graphsalot’s fleet of ships. Use each
clue to graph points that show the locations of his ships. Three or more points in
a row indicate the location of a single ship.
y
x
O
Clue 1: Graph {(0, 1), (0, 3), (5, 1)}. State the domain of this relation.
Clue 2: Graph {(5, 0), (4, 2), (3, 2)}. State the range of this relation.
Clue 3: Solve y x 1 for the domain {2, 1, 2}. Plot the points in your
graph.
Clue 4: Determine whether each of the following relations is a function. If the
relation is a function, graph the given points. If it is not a function, do
not graph it.
a.
x
0
0
4
y
1
1
2
b.
x
3
2
1
y
2
2
3
c.
x
1
5
y
3
2
Clue 5: Given g(x) 2x 6, find g(1). This is the number of ships that
you should have found in the fleet.
Answers are located on page 115.
© Glencoe/McGraw-Hill
46
CA Parent and Student Study Guide
Algebra: Concepts and Applications