Answers (Anticipation Guide and Lesson 10-1)

Glencoe Algebra 1

A1

Chapter 10

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NAME

10

DATE

Anticipation Guide

Radical Expressions and Triangles

PERIOD

Step 1

Before you begin Chapter 10

? Read each statement.

? Decide whether you Agree (A) or Disagree (D) with the statement.

? Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).

STEP 1 A, D, or NS

Statement

1. An expression that contains a square root is called a radical expression.

2. It is always true that xy will equal x ? y.

3.

1 3

is in simplest form because

3

is not a whole number.

4. The sum of 3 3 and 2 3 will equal 5 3 .

5. Before multiplying two radical expressions with different radicands the square roots must be evaluated.

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 52 = 42 + 32, 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.

10. The Distance Formula cannot be used to find the distance between two points on the same vertical line.

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.

STEP 2 A or D

A A D A D

A

D A A D

A D

Step 2

After you complete Chapter 10

? Reread each statement and complete the last column by entering an A or a D.

? Did any of your opinions about the statements change from the first column?

? For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.

Chapter 10

3

Glencoe Algebra 1

Answers

Chapter Resources

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NAME

DATE

10-1 Study Guide and Intervention

PERIOD

Square Root Functions

Dilations of Radical Functions 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.

y Parent function: f(x) = x

Square Root Function

Type of graph: curve Domain: {x|x 0}

y= x

Range: {y|y 0}

0

x

Example Graph y = 3 x. State the domain and range.

Step 1 Make a table. Choose nonnegative values for x.

Step 2 Plot points and draw a smooth curve.

x

y

0

0

0.5

2.12

1

3

2

4.24

4

6

6

7.35

y y=3 x

0

x

The domain is {x|x 0} and the range is {y|y 0}.

Exercises

Graph each function, and compare to the parent graph. State the domain and range.

1. y =

3 2

x

y

2. y = 4 x

y

3. y =

5 2

x

y

0

x

Dilation of y = x; D = {x | x 0}; R = {y | y 0}

Chapter 10

0

x

Dilation of y = x; D = {x | x 0}; R = {y | y 0}

5

0

x

Dilation of y = x; D = {x | x 0}; R = {y | y 0}

Glencoe Algebra 1

Lesson 10-1

Answers (Anticipation Guide and Lesson 10-1)

Glencoe Algebra 1

A2

Chapter 10

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NAME

DATE

PERIOD

10-1 Study Guide and Intervention (continued)

Square Root Functions

Reflections and Translations of Radical Functions 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 x + h, follow these steps.

Graphs of Square Root Functions

Step 1

Draw the graph of y = +c 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.

Step 2 Translate the graph c units up if c is positive and down if c is negative.

Step 3 Translate the graph h units left if h is positive and right if h is negative.

Example Graph y = - x + 1 and compare to the parent graph. State the domain and range.

Step 1 Make a table of values.

x

-1

0

1

3

8

y

0

-1

-1.41

-2

-3

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 0} and the range is {y | y 0}.

Exercises

y y= x

0

x

y=- x+1

Graph each function, and compare to the parent graph. State the domain and range.

1. y = x + 3

2. y = x - 1

3. y = -x - 1

y

y

y

0

x

0

x

0

x

translation of y = x up 3 units; D = {x | x 0}; R = {y | y 3}

Chapter 10

translation of y = x right 1 unit; D = {x | x 1}; R = {y | y 0}

6

translation of y = x right 1 unit and reflected across the x-axis; D = {x | x 1} R = {y | y 0}

Glencoe Algebra 1

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Lesson 10-1

NAME

10-1 Skills Practice

DATE

PERIOD

Square Root Functions

Graph each function, and compare to the parent graph. State the domain and range.

1. y = 2 x

2. y =

1 2

x

3. y = 5 x

y

y

y 12

8

4

0

x

0

x

-2 0

2 4x

dilation of y = x; D = {x | x 0}, R = { y | y 0}

4. y = x + 1

y

dilation of y = x; D = {x | x 0}, R = { y | y 0}

5. y = x - 4

y

dilation of y = x; D = {x | x 0}, R = { y | y 0}

6. y = x - 1

y

0

x

0

x

0

x

translation of y = x up 1 unit; D = {x | x 0}, R = { y | y 1}

7. y = -x - 3

y

0

x

translation of y = x;

right 3 units reflected across the x-axis; D = {x | x 3}, R = {y | y 0}

translation of y = x down 4 units; D = {x | x 0}, R = { y | y -4}

8. y = x - 2+ 3

y

translation of y = x right 1 unit; D = {x | x 1}, R = { y | y 0}

9. y = - 1 x - 4 + 1 2

y

0

x

translation of y = x

right 2 units and up 3 units; D = {x | x 2}, R = {y | y 3}

0

x

dialation of y = x reflected

across the x-axis

translated right 4 units up 1 units; D = {x | x 4}, R = {y | y 1}

Chapter 10

7

Glencoe Algebra 1

Answers (Lesson 10-1)

Glencoe Algebra 1

A3

Chapter 10

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NAME

10-1 Practice

DATE

PERIOD

Square Root Functions

Graph each function, and compare to the parent graph. State the domain and range.

1. y = 4 x 3

2. y = x + 2

3. y = x - 3

y

y

y

0

x

dilation of y = x; D = {x | x 0}, R = {y | y 0}

4. y = - x + 1

y

0

x

0

x

0

x

translation of y = x up 2 unit; D = {x | x 0}, R = {y | y 2}

5. y = 2 x-1 + 1

y

translation of y = x left 3 units; D = {x | x -3}, R = {y | y 0}

6. y = - x-2 + 2

y x

0

0

x

translation of y = x up 1 unit reflected in the x-axis; D = {x | x 0}, R = {y | y 1}

dilation of y = x translated up 1 unit and right 1 unit; D = {x | x 1}, R = {y | y 1}

translation of y = x ; up 2 units and right 2 units, reflected in the x-axis; D = {x | x 2}, R = {y | y 2}

7. OcaHnMbe'SfoLuAnWd byIntheeleectqruicaatiloennIgi=neerRPin,gw, thheerereIsiissttahnececuorfraenctiricnuit

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.

Current (amperes)

5 4 3 2 1

0 20 40 60 80 100 Power (watts)

Chapter 10

8

Glencoe Algebra 1

Answers

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NAME

DATE

10-1 Word Problem Practice

PERIOD

Square Root Functions

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 L. The length of the pendulum in feet is given by L. Graph this function.

5

4

3

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

by V

a=cap2aCEci.tance

of

C

farads

is

given

a. Rewrite and simplify the equation for the case of a 0.0002 farad capacitor.

V = 100 E

b. Graph the equation you found in part a.

Period (sec)

2

1

350

300

0

4 8 12 16 20

250 Pendulum Length (ft)

200

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 = 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?

approximately 284 ft/s

3. ERROR ANALYSIS Gregory is drawing the graph of y = -5x + 1. He describes the range and domain as {x ? x -1}, { y ? y 0}. Explain and correct the mistake that Gregory made.

The domain is actually {y ? y 0} because the graph has been reflected across the x-axis.

150 100 50

0

2 4 6 8 10

Energy (joules)

c. How would the graph differ if you wished to store E + 1 joules of energy in the capacitor instead?

translation of V = 100 E one unit to the left

d. How would the graph differ if you applied a voltage of V + 1 volts instead?

translation of V = 100E one unit down

Chapter 10

9

Glencoe Algebra 1

Voltage (volts)

Lesson 10-1

Answers (Lesson 10-1)

Glencoe Algebra 1

A4

Chapter 10

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NAME

10-1 Enrichment

DATE

PERIOD

Cubic Root Functions

A cubic root function contains the cubic root of a variable. The cubic root of a number x are the numbers y that satisfy the equation y ? y ? y = x (or, alternatively, y = 3 x ). Unlike square root functions, cubic root functions return real numbers when the radicand is negative.

Example Graph y = 3 x.

Step 1 Make a table.

x

y

-5

-1.71

-3

-1.44

-1

-1

0

0

1

1

3

1.44

5

1.71

Step 2 Plot points and draw a smooth curve.

y

0

x

Exercises

Graph each function, and compare to the parent graph.

1. y = 2 3 x

2. y = 3 x + 1

3. y = 3 x + 1

y

y

y

0

x

0

x

0

x

dilation of y = 3 x

4. y = 3 x - 1 + 2

y

translation of y = 3 x up 1 unit

5. y = 3 3 x - 2

y

0

x

0

x

translation of y = 3 x left 1 unit

6. y = - 3 x + 3

y

0

x

translation of y = 3 x up 2 units and right 1 unit

Chapter 10

dilation of y = 3 x translated right 2 units

10

reflection of y = 3 x across the x-axis translated up 3 units

Glencoe Algebra 1

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Lesson 10-2

NAME

DATE

10-2 Study Guide and Intervention

PERIOD

Simplifying Radical Expressions

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 For any numbers a and b, where a 0 and b 0, ab = a b.

Example 1 Simplify 180.

180 = 2 2 3 = 22 32 = 2 3 5 = 65

3 5 5

Prime factorization of 180 Product Property of Square Roots Simplify. Simplify.

Example 2 Simplify 120a2 ? b5 ? c4.

120a2 b5 c4 = 23 3 5 a2 b5 c4 = 22 2 3 5 a2 b4 b c4 = 2 2 3 5 a b2 b c2 = 2ab2c230b

Exercises

Simplify each expression.

1. 28

2 7

5. 162

9 2

9. 4a2

2 a

13. 410 36

24 15

17. 24a4b2

2a2 b 6

2. 68

2 17

6. 3 ? 6

3 2

10. 9x4

3x2

14. 3x2 33x4

9x3

18. 81x4y2 9x2 y

3. 60

2 15

7. 2 ? 5

10

11. 300a4

10a2 3

15. 20a2b4

2 a b25

19. 150a2b2c

5ab 6c

20. 72a6b3c2

6a3bc 2b

21. 45x2y5z8

3 x y2z45y

22. 98x4y6z2 7x2y3z 2

4. 75

5 3

8. 5 ? 10

5 2

12. 128c6

8c3 2

16. 100x3y 10 x xy

Chapter 10

11

Glencoe Algebra 1

Answers (Lesson 10-1 and Lesson 10-2)

Glencoe Algebra 1

A5

Chapter 10

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NAME

DATE

PERIOD

10-2 Study Guide and Intervention (continued)

Simplifying Radical Expressions

Quotient Property of Square Roots 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

For any numbers a and b, where a 0 and b > 0,

a b

=

a . b

Example

Simplify

56 45

.

56 = 4 14

45

9 5

= 2 14 3 15

= 214 5 35 5

= 270 15

Exercises

Simplify each expression.

Simplify the numerator and denominator. Multiply by 5 to rationalize the denominator.

5 Product Property of Square Roots

1.

9 18

2 2

3. 100 121

10 11

5. 82 2

28

7.

3 4

5 30 2 4

9. 3a2 10b6

|a| 30 10| b 3 |

11. 100a4 5a2 144b8 6b4

13. 4 3 + 5

3 - 5

2

15. 5 5 - 25

5 + 5

Chapter 10

2. 8 24

3 3

4. 75 5

3

6.

2 5

6 23 5 5

8.

5 7

2 14 5 7

10. x6 x3 y4 y2

12. 75b3c6 5bc33b

a2

a

14. 8 42 - 26

2 + 3

16.

8

45 - 14

27 + 410

33

12

Glencoe Algebra 1

Answers

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Lesson 10-2

NAME

DATE

10-2 Skills Practice

Simplifying Radical Expressions

Simplify each expression.

1. 28 27

2. 40 210

PERIOD

3. 72 62

4. 99 311

5. 2 ? 10 25

6. 5 ? 60 103

7. 35 ? 5 15

8. 6 ? 424 48

9. 23 ? 315 185

11. 81a2d4 9| a |d2

13. 75m5P2 5m2| P |3m

15.

1 6

6 6

17.

q 12

3q

6

19. 12 23 b2 | b |

21. 2 4 + 5

8 - 25 11

23. 5 7 + 7

35 - 57 42

10. 16b4 4b2

12. 40x4y6 2x2y310

14.

5 3

15 3

16.

6 ? 7

1 3

14 7

18.

4h 5

2 5h 5

20. 45 35 4m4 2m2

22. 3 6 + 33

2 - 3

24. 4 3 - 2

12 + 42 7

Chapter 10

13

Glencoe Algebra 1

Answers (Lesson 10-2)

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