Answers (Anticipation Guide and Lesson 10-1)
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10
DATE
Before you begin Chapter 10
Radical Expressions and Triangles
Anticipation Guide
PERIOD
A1
A
A
D
A
A
6. When solving radical equations by squaring each side of the
equation, it is possible to obtain solutions that are not
solutions to the original equation.
7. The longest side of any triangle is called the hypotenuse.
8. Because 5 = 4 + 3 , a triangle whose sides have lengths
3, 4, and 5 will be a right triangle.
9. On a coordinate plane, the distance between any two points
can be found using the Pythagorean Theorem.
A
D
11. Two triangles are similar only if their corresponding angles
are congruent and the measures of their corresponding sides
are in proportion.
12. All right triangles are similar.
After you complete Chapter 10
D
10. The Distance Formula cannot be used to find the distance
between two points on the same vertical line.
2
D
5. Before multiplying two radical expressions with different
radicands the square roots must be evaluated.
2
A
? and 2 ¡Ì3
? will equal 5 ¡Ì3
?.
4. The sum of 3 ¡Ì3
2
D
¡Ì?
3
1
3. ?
is in simplest form because ¡Ì?
3 is not a whole number.
¡¤ ¡Ìy?.
2. It is always true that ¡Ìxy
? will equal
¡Ì?
x
A
Glencoe Algebra 1
Answers
3
Glencoe Algebra 1
? For those statements that you mark with a D, use a piece of paper to write an
example of why you disagree.
? Did any of your opinions about the statements change from the first column?
Chapter 10
STEP 2
A or D
1. An expression that contains a square root is called a
radical expression.
Statement
? Reread each statement and complete the last column by entering an A or a D.
Step 2
STEP 1
A, D, or NS
? Write A or D in the first column OR if you are not sure whether you agree or
disagree, write NS (Not Sure).
? Decide whether you Agree (A) or Disagree (D) with the statement.
? Read each statement.
Step 1
10
NAME
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
PERIOD
¡Ö 4.24
6
2
4
x
0
y
y=3 x
x
¡Ì?
x
x
Chapter 10
Dilation of y = ¡Ì"
x;
D = {x | x ¡Ý 0};
R = {y | y ¡Ý 0}
0
y
2
3
1. y = ?
x
5
Dilation of y = ¡Ìx";
D = {x | x ¡Ý 0};
R = {y | y ¡Ý 0}
0
y
2. y = 4 ¡Ìx?
¡Ì?
x
x
Glencoe Algebra 1
Dilation of y = ¡Ì"
x;
D = {x | x ¡Ý 0};
R = {y | y ¡Ý 0}
0
y
2
5
3. y = ?
Graph each function, and compare to the parent graph. State the domain and range.
Exercises
y= x
Step 2 Plot points and draw a
smooth curve.
The domain is {x|x ¡Ý 0} and the range is {y|y ¡Ý 0}.
¡Ö 7.35
3
1
6
¡Ö 2.12
0.5
y
0
0
x
0
y
Graph y = 3 ¡Ìx
?. State the domain and range.
Range: {y|y ¡Ý 0}
Domain: {x|x ¡Ý 0}
Type of graph: curve
Step 1 Make a table. Choose nonnegative
values for x.
Example
Square Root Function
x
Parent function: f(x) = ¡Ì?
A square root function contains the square root of
a variable. Square root functions are a type of radical function.
In order for a square root to be a real number, the radicand, or the expression under the
radical sign, cannot be negative. Values that make the radicant negative are not included in
the domain.
Square Root Functions
Study Guide and Intervention
Dilations of Radical Functions
10-1
NAME
Answers (Anticipation Guide and Lesson 10-1)
Lesson 10-1
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter Resources
A2
Glencoe Algebra 1
Square Root Functions
Study Guide and Intervention
DATE
0
-1
0
-1
1
-1.41
3
-2
8
-3
0
y
y
¡Ì"
x
+ 3
x
Chapter 10
translation of y = ¡Ì"
x
up 3 units;
D = {x | x ¡Ý 0};
R = {y | y ¡Ý 3}
0
1. y =
x
6
translation of y = ¡Ì"
x
right 1 unit;
D = {x | x ¡Ý 1};
R = {y | y ¡Ý 0}
0
y
2. y = ¡Ì"""
x- 1
Glencoe Algebra 1
translation of y = ¡Ì"
x
right 1 unit and reflected
across the x-axis;
D = {x | x ¡Ý 1}
R = {y | y ¡Ü 0}
0
y
3. y = - ¡Ì"""
x- 1
x
y =- x+ 1
y= x
Graph each function, and compare to the parent graph. State the domain
and range.
Exercises
Step 2 This is a horizontal translation 1 unit to the left of
the parent function and reflected across the x-axis.
The domain is {x | x ¡Ý 0 } and the range is {y | y ¡Ü 0 }.
y
x
Step 1 Make a table of values.
x
Step 3 Translate the graph ?h? units left if h is positive and right
if h is negative.
Step 2 Translate the graph ?c? units up if c is positive and down
if c is negative.
Example
x + 1 and compare to the parent graph. State the
Graph y = - ¡Ì"""
domain and range.
Graphs of Square Root
Functions
Step 1 Draw the graph of y = +c ¡Ì??
x . The graph starts at the
origin and passes through the point at (1, a). If a > 0, the
graph is in the 1st quadrant. If a < 0, the graph is reflected
across the x-axis and is in the 4th quadrant.
Radical functions, like
quadratic functions, can be translated horizontally and vertically, as well as reflected across
the x-axis. To draw the graph of y = a ¡Ì"""
x + h , follow these steps.
(continued)
PERIOD
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10
Reflections and Translations of Radical Functions
10-1
NAME
Square Root Functions
Skills Practice
DATE
PERIOD
¡Ìx
"
??;
¡Ìx
x
y
¡Ì"
x
+1
x
x
Chapter 10
translation of y = ¡Ì"
x;
right 3 units reflected across
the x-axis; D = {x | x ¡Ý 3},
R = {y | y ¡Ü 0}
0
y
7. y = - ¡Ì"""
x-3
D = {x | x ¡Ý 0}, R = { y | y ¡Ý 1}
translation of
y = ¡Ì"
x up 1 unit;
0
4. y =
D = {x | x ¡Ý 0}, R = { y | y ¡Ý 0}
dilation of y =
0
y
1. y = 2
¡Ìx
"
¡Ì??
x;
x
0
y
6. y = ¡Ì"""
x-1
x
x
7
translation of y = ¡Ì??
x
right 2 units and up
3 units; D = {x | x ¡Ý 2},
R = {y | y ¡Ý 3}
0
y
8. y = ¡Ì"""
x - 2+ 3
2
x
Glencoe Algebra 1
dialation of y = ¡Ìx" reflected
across the x-axis
translated right 4 units up
1 units; D = {x | x ¡Ý 4},
R = {y | y ¡Ü 1}
0
y
1 ¡Ì"""
9. y = - ?
x-4 + 1
D = {x | x ¡Ý 1}, R = { y | y ¡Ý 0}
x
x
D = {x | x ¡Ý 0}, R = { y | y ¡Ý 0}
??;
¡Ìx
4
D = {x | x ¡Ý 0}, R = { y | y ¡Ý -4}
-4
2
translation of
y = ¡Ì??
x right 1 unit;
y
¡Ì"
x
0
y
dilation of y =
?2
4
8
12
3. y = 5 ¡Ìx"
translation of
y = ¡Ìx?? down 4 units;
0
5. y =
D = {x | x ¡Ý 0}, R = { y | y ¡Ý 0}
dilation of y =
0
y
2
1
2. y = ?
Graph each function, and compare to the parent graph. State the domain
and range.
10-1
NAME
Answers (Lesson 10-1)
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 10-1
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A3
Square Root Functions
Practice
DATE
PERIOD
¡Ìx
#
x
x
y
¡Ìx
#
+2
x
x
dilation of y = ¡Ì"
x
translated up 1 unit
and right 1 unit;
D = {x | x ¡Ý 1},
R = {y | y ¡Ý 1}
0
y
?????
5. y = 2 ¡Ìx - 1 + 1
translation of y = ¡Ìx"
up 2 unit;
D = {x | x ¡Ý 0},
R = {y | y ¡Ý 2}
0
2. y =
Chapter 10
Glencoe Algebra 1
Answers
8
amperes, P is the power in watts, and R is the resistance of
the circuit in ohms. Graph this function for a circuit with a
resistance of 4 ohms.
¡ÌR
x
x
0
1
2
3
4
5
Glencoe Algebra 1
Power (watts)
20 40 60 80 100
translation of y = ¡Ì"
x;
up 2 units and right
2 units, reflected
in the x-axis;
D = {x | x ¡Ý 2},
R = {y | y ¡Ü 2}
0
y
?????
6. y = - ¡Ìx - 2 + 2
translation of y = ¡Ì"
x
left 3 units;
D = {x | x ¡Ý -3},
R = {y | y ¡Ý 0}
0
y
3. y = ¡Ì###
x-3
7. OHM¡¯S LAW In electrical engineering, the resistance of a circuit
??
P
can be found by the equation I = ?
, where I is the current in
translation of y = ¡Ì"
x
up 1 unit reflected
in the x-axis;
D = {x | x ¡Ý 0},
R = {y | y ¡Ü 1}
0
y
4. y = - ¡Ìx# + 1
dilation of y = ¡Ì"
x;
D = {x | x ¡Ý 0},
R = {y | y ¡Ý 0}
0
y
3
4
1. y = ?
Graph each function, and compare to the parent graph. State the domain
and range.
10-1
NAME
Current (amperes)
8
12
16
Pendulum Length (ft)
4
20
Chapter 10
The domain is actually {y % y ¡Ü 0}
because the graph has been
reflected across the x-axis.
3. ERROR ANALYSIS Gregory is drawing
the graph of y = -5 ¡Ì###
x + 1 . He describes
the range and domain as {x % x ¡Ý -1},
{ y % y ¡Ý 0}. Explain and correct the
mistake that Gregory made.
approximately 284 ft/s
2. EMPIRE STATE BUILDING The roof of
the Empire State Building is 1250 feet
above the ground. The velocity of an
object dropped from a height of h meters
is given by the function V = ¡Ì##
2 gh ,
where g is the gravitational constant,
32.2 feet per second squared. If an object
is dropped from the roof of the building,
how fast is it traveling when it hits the
street below?
0
1
2
3
4
5
9
Square Root Functions
DATE
¡Ì
PERIOD
2
4
6
Energy (joules)
8
10
Glencoe Algebra 1
translation of V = 100 ¡Ì"
E
one unit down
d. How would the graph differ if you
applied a voltage of V + 1 volts
instead?
translation of V = 100 ¡Ì"
E
one unit to the left
c. How would the graph differ if you
wished to store E + 1 joules of energy
in the capacitor instead?
0
50
100
150
200
250
300
350
b. Graph the equation you found in part a.
??
V = 100 ¡ÌE
a. Rewrite and simplify the equation for
the case of a 0.0002 farad capacitor.
4. CAPACITORS A capacitor is a set of
plates that can store energy in an
electric field. The voltage V required to
store E joules of energy in a capacitor
with a capacitance of C farads is given
???
2E
by V = ?
C .
Word Problem Practice
1. PENDULUM MOTION The period T of
a pendulum in seconds, which is the time
for the pendulum to return to the point
of release, is given by the equation
T = 1.11 ¡Ì#
L . The length of the
pendulum in feet is given by L. Graph
this function.
10-1
NAME
Period (sec)
Chapter 10
Voltage (volts)
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 10-1)
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 10-1
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A4
Glencoe Algebra 1
Cubic Root Functions
Enrichment
DATE
PERIOD
Graph y =
?1.44
?1
0
1
1.44
1.71
?3
?1
0
1
3
5
3
¡Ìx
".
y
x
x
3
Chapter 10
translation of y = ¡Ì"
x
up 2 units and
right 1 unit
0
y
4. y = ¡Ì"""
x- 1 + 2
3
3
dilation of y = ¡Ìx"
0
3
1. y = 2 ¡Ì
x
"
x
0
y
x
10
dilation of
3
y = ¡Ì"
x translated
right 2 units
3
5. y = 3 ¡Ì"""
x- 2
translation of
3
y = ¡Ì"
x up 1 unit
0
y
3
2. y = ¡Ì
x" + 1
x
x
x
3
Glencoe Algebra 1
reflection of y = ¡Ì"
x
across the x-axis
translated up 3 units
0
y
3
6. y = - ¡Ì
x" + 3
translation of
3
y = ¡Ì"
x left 1 unit
0
y
3
3. y = ¡Ì"""
x + 1
0
y
Step 2 Plot points and draw a
smooth curve.
Graph each function, and compare to the parent graph.
Exercises
y
?1.71
x
?5
Step 1 Make a table.
Example
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10
A cubic root function contains the cubic root of a variable. The cubic root of a number
3
x are the numbers y that satisfy the equation y ¡¤ y ¡¤ y = x (or, alternatively, y = ¡Ì
x" ).
Unlike square root functions, cubic root functions return real numbers when the radicand
is negative.
10-1
NAME
DATE
Simplifying Radical Expressions
Study Guide and Intervention
PERIOD
Simplify ¡Ì""
180 .
Simplify.
Simplify.
3 ¡Ì"
2
10. ¡Ì""
9x4
9 ¡Ì"
2
9. ¡Ì""
4a2
Chapter 10
6?a3bc? ¡Ì""
2b
20. ¡Ì""""
72a6b3c2
2a2 b ¡Ì"
6
17. ¡Ì"""
24a4b2
24 ¡Ì""
15
"
13. 4 ¡Ì""
10 # 3 ¡Ì6
3 x y2z4 ¡Ì""
5y
2 5 8
""""
21. ¡Ì45x
yz
9x2 y
18. ¡Ì"""
81x4y2
9?x3?
14. ¡Ì""
3x2 # 3 ¡Ì""
3x4
3x2
6. ¡Ì"
3 ¡¤ ¡Ì"
6
5. ¡Ì""
162
2a
2 ¡Ì""
17
2. ¡Ì""
68
2 ¡Ì"
7
1. ¡Ì""
28
Simplify each expression.
Exercises
30b
= 2?a?b2c2 ¡Ì""
11
" # ¡Ì"
= ¡Ì"
22 # ¡Ì"
2 # ¡Ì3
5 # ¡Ì"
a2 # ¡Ì"""
b4 # b # ¡Ì"
c4
" # c2
" # ¡Ì3
" # ¡Ì5
" # ?a? # b2 # ¡Ìb
= 2 # ¡Ì2
7x2?y3z? ¡Ì"
2
4 6 2
""""
22. ¡Ì98x
yz
5?ab? ¡Ì""
6c
19. ¡Ì""""
150a2b2c
2 a b2 ¡Ì"
5
15. ¡Ì"""
20a2b4
10a2 ¡Ì"
3
11. ¡Ì"""
300a4
Glencoe Algebra 1
10 x ¡Ì"
xy
16. ¡Ì"""
100x3y
"
8?c3? ¡Ì2
12. ¡Ì"""
128c6
5 ¡Ì"
2
" ¡¤ ¡Ì""
8. ¡Ì5
10
¡Ì""
10
5 ¡Ì"
3
"
7. ¡Ì"
2 ¡¤ ¡Ì5
4. ¡Ì""
75
2 ¡Ì""
15
3. ¡Ì""
60
Product Property of Square Roots
Prime factorization of 180
ab = ¡Ì"
a # ¡Ì"
b.
For any numbers a and b, where a ¡Ý 0 and b ¡Ý 0, ¡Ì""
120a2 ¡¤ b5 ¡¤ c4 .
Simplify ¡Ì""""""
¡Ì""""""
120a2 # b5 # c4
3
""""""""
# 3 # 5 # a2 # b5 # c4
= ¡Ì2
Example 2
¡Ì""
180 = ¡Ì""""""
2#2#3#3#5
= ¡Ì"
22 # ¡Ì"
32 # ¡Ì"
5
5
= 2 # 3 # ¡Ì"
= 6 ¡Ì"
5
Example 1
Product Property of Square Roots
The Product Property of Square Roots and
prime factorization can be used to simplify expressions involving irrational square roots.
When you simplify radical expressions with variables, use absolute value to ensure
nonnegative results.
Product Property of Square Roots
10-2
NAME
Answers (Lesson 10-1 and Lesson 10-2)
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 10-2
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 10
DATE
Simplifying Radical Expressions
Study Guide and Intervention
(continued)
PERIOD
Simplify
A5
?
¡Ì2
2
10
11
¡Ì100
??
¡Ì121
??
10|b |
?
3
??
|a| ¡Ì30
??
¡Ì30
4
?
12
Glencoe Algebra 1
Answers
? + 4 ¡Ì10
?
2 ¡Ì7
Chapter 10
¡Ì8
?
¡Ì8
?
?
2 + ¡Ì3
a
??
5?bc3? ¡Ì3b
?
?a?
??
¡Ì14
7
?
?
2 ¡Ì3
5
?
??
? - ¡Ì14
4 ¡Ì5
?
33
? - 2 ¡Ì6
?
14. ? 4 ¡Ì2
¡Ì
x
y
?2
3
???
75b3c6
?
2
4
¡Ì??yx
6
¡Ì??75 " ¡Ì??52
¡Ì??52 " ¡Ì??56
16. ?
?
3 + ¡Ì5
?
2
6b
12.
10.
8.
6.
¡Ì3
?
¡Ì75
?
?
¡Ì3
3
4. ? 5
¡Ì8
?
¡Ì24
?
?
15. ? 5 - 2 ¡Ì5
¡Ì5
?
?
5 + ¡Ì5
¡Ì4
?
?
3 - ¡Ì5
¡Ì144b
??
100a4 5a2
?4
?
8
6
13. ?
11.
??
3a
¡Ì?
10b
9.
2
¡Ì??43 " ¡Ì??25
7.
?
2 ¡Ì8
5. ? 2
?
8 ¡Ì2
3. ? ?
1. ? ?
¡Ì9
?
¡Ì18
?
Simplify each expression.
Exercises
70
2 ¡Ì??
15
2. ? ?
Product Property of Square Roots
Simplify the numerator and denominator.
=?
=?
¡Ì?
5
Multiply by ? to rationalize the denominator.
¡Ì?
5
56
?
.
¡Ì??
45
For any numbers a and b, where a ¡Ý 0 and b > 0,
14
2 " ¡Ì??
??
3 " ¡Ì15
5
2 ¡Ì??
14 ¡Ì?
=?"?
¡Ì?
?
3 ¡Ì5
5
56
4 " 14
?
?
= ¡Ì???
¡Ì??
45
9"5
Example
Quotient Property of Square Roots
Glencoe Algebra 1
?
.
¡Ì??ba = ?
¡Ìb
?
¡Ìa
A fraction containing radicals is in simplest
form if no radicals are left in the denominator. The Quotient Property of Square Roots
and rationalizing the denominator can be used to simplify radical expressions that
involve division. When you rationalize the denominator, you multiply the numerator and
denominator by a radical expression that gives a rational number in the denominator.
Quotient Property of Square Roots
10-2
NAME
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
?
¡Ì6
6
Chapter 10
7 + ¡Ì?
7
5
23. ?
35 - 5 ¡Ì?
7
42
?
8 - 2 ¡Ì?
5
11
?
2 ¡Ì?
3
|b|
?
6
3q
¡Ì??
5m2| P | ¡Ì??
3m
?
?
4 + ¡Ì?
5
2
12
?
¡Ì??
b
q
??
¡Ì?
12
¡Ì??16
2
21. ?
19.
17.
15.
5 2
???
13. ¡Ì75m
P
¡Ì??
15
3
?
4
13
3 - ¡Ì?
2
4
24. ?
2
12 + 4 ¡Ì?
7
?
6 + 3 ¡Ì?
3
3 ¡Ì?
5
2m
?2
2 ¡Ì??
5h
5
?
2 - ¡Ì?
3
45
?
¡Ì??
4m
4h
?
¡Ì??
5
¡Ì??
14
7
?
2x2?y3? ¡Ì??
10
4b2
¡Ì??67 ¡¤ ¡Ì??13
¡Ì??53
3
22. ?
20.
18.
16.
14.
4 6
???
12. ¡Ì40x
y
2 4
???
11. ¡Ì81a
d
9| a |d2
10. ¡Ì??
16b4
9. 2 ¡Ì?
3 ¡¤ 3 ¡Ì??
15 18 ¡Ì?
5
48
8. ¡Ì?
6 ¡¤ 4 ¡Ì??
24
15
7. 3 ¡Ì?
5 ¡¤ ¡Ì?
5
3 ¡Ì??
11
6. ¡Ì?
5 ¡¤ ¡Ì??
60 10 ¡Ì?
3
4. ¡Ì??
99
6 ¡Ì?
2
3. ¡Ì??
72
5. ¡Ì?
2 ¡¤ ¡Ì??
10 2 ¡Ì?
5
2. ¡Ì??
40
2 ¡Ì?
7
??
1. ¡Ì28
2 ¡Ì??
10
Simplifying Radical Expressions
Skills Practice
Simplify each expression.
10-2
NAME
PERIOD
Glencoe Algebra 1
Answers (Lesson 10-2)
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 10-2
Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
................
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