Math 60 ~ Chapter Zero Review Problems

Activity #1 (Intro to Algebra) Name:________KEY____________ Math 60

22 24

3 55

2 11

1 3

8 3

5 11

16 1

or 35 5

1 10

1 3

7 3

2 10

7 1

or 32 2

10 3

3 7

70 9 79 16 or 3

21 21

3

3

7

7

21

5 1

12 15

25 4 21 7

60 6

4

4

0

5

5

20

1. For each of the following fractions find the simplest form.

a. 5

25

5 1 1

5 5 5 b.

18

54

1 18 1

3 18 3 c.

62

93

2 31 2

3 31 3

2. Compute and write in simplest form (no decimals).

a. 10 21

3 20 b.

22 55

3 24

c. 10 3

3 7 d.

5 1

12 15

3. Simplify the following.

a.

24

8 3 b.

63

7 9 c.

22

0 undefined d.

0

3 0

4. Use exponential notation to rewrite the expression.

a. 3

10 10 1 100 b. 4

10 10 10 1 100

5. Write each expression in expanded form and then evaluate.

a. 4 2 2 2 22 16 b. 4 2 2 2 22 16

c. 4

2 2 2 22 16 d. 4

2 2 2 22 16

e. 3

2 2 2 2 8 f. 3

2 2 22 8

Activity #1 Page 2 of 3

33 4 42

3 64 42

192 42 150

2

1 1

4 16

22 7 7 5 2

9 49 5 2

40 5 2

8 2

6

6. Use the Order of Operations to simplify. Show all work. No decimals.

a. 9 17 8 b. 9 5

4 6

c. d. 1

43 11 2 14 72

e.

5 4 24

2 3 5 f.

4 18 1

27 15 3

7. Simplify.

a. 5

7

4

4 b.

2

2

( 5)

5

251

25

c. 3

3 2 2 42 d. 2

2 7 5 12 5 2

9 5

4 6

27 10 37 1 or -3

1

2

2 1

2

2

3

3

12

2

17 4 3 8

8

2

1 3 2

12

4

3 9

2 9

1

15 3

8 1

45 3

18 8 15 7

45 3 45

15

15 45

5 4 5

2 3 24

5 5

2 18

5 5

2 18

45 5 40 20 2 or 2

18 18 9

9

9

9

134 14 7

2

17 14 7

17 2 19


8. An Expression is a __meaningful collection of numbers, variables and operation_____ .

Identify whether the following is an expression or not.

a. 25 5 No b. 25 5 Yes c. 3 6 12x No

9. Below is a list of words.

Circle the words that relate to the mathematical operation “+”.

Box the words that relate to the mathematical operation “–”.

Cross out the words that relate to the mathematical operation “ ”.

Draw a line through the words that relate to the mathematical operation “ ”.

Less than double quotient sum subtract

Multiply per increased by times minus

Divide add of plus product

Difference more than less triple ratio

10. Rewrite the following word expressions into an algebraic expression.

Word Expression Algebraic Expression

The sum of the lengths of three sides of a triangle divided by 2.

2

a b c

Eighty more dollars than triple the cost of a chair. 80 3x

Fifteen dollars less than the original price. 15x

The sum of a number and 7. 7x

The quotient when 3 less than some number is divided by 3 more than some other number.

3

3

x

y

The sum of two consecutive integers. 1x x OR 2 1x

If x feet are cut off from a board that is 23 ft long, express how much of the board is left.

23 x

Activity #2 (Expressions) Name:________KEY____________ Math 60

2800 310n

Leasing the car for 60 months will cost $21,400.

12 5.3

2 2

24 10.6

3 4.6

P

2 2

12

6

7

14

6

2

P

1. The perimeter of a rectangle with length L and width W is given by the equation,

2 2P L W . Find the perimeter of the rectangles given the following lengths and widths. Show all work and include proper units.

a. 12 cm and 5.3 cmL W b. 6 inches and 7 inchesL W

2 2P L W 2 2P L W 2. A car lease costs $2,800 down and $310 per month for n months.

a. Write an expression that represents the total cost for n months of leasing the car.

C = ___________________

b. Use your expression from part a. to find how much to lease the car for 60 months. Interpret your result with a complete sentence.

2800 310 60 2800 18,600 $21,400C

3. Evaluate the following expressions if , , and .5 1 3 y zx Show all work.

a. x y 3 5 2

b. xy z 151 153 16

c. x yz 33 1 55 8

d. 2 3

2

x y

z

2 3 6 15 9 1OR

5

1

3 4

2 22 2

e. 22y x z 2

2 10 3 1 133 115 12

f. 2

4 3x y 2 2 2

4 3 12 15 353 9

The perimeter of the rectangle is 34.6 cm.

The perimeter of the rectangle is 26 inches.


5 2 7 10

3 7 10

21 30

x

x

x

5 2 7 10

5 14 20

14 15

x

x

x

2 2

2

2

3 7

4

4

9

5

4

9

x

x

x

x xx

x

x

2 2

2

2

3 7

4

4

9

5

4

9

x

x

x

x xx

x

x

2 2

2 2

2

6 10

6 10

7

4

5 7

2

5d dc c

c

d

d cd

c

2

2 2

2 2

2 2

6 10

6 1

5 7

5 7

5 7

2 16

0

6 10

c c

c c

c

d d

d

d

c

d

d d

c

4. Simplify the following expressions. Show all work.

a. 2 24 5 3 7x x x x b. 2 24 5 3 7x x x x

c. 2 25 6 7 10d c d c d. 2 25 6 7 10d c d c

e. 5 2 7 10x f. 5 2 7 10x

g. Add 3 2b and 5 10b h. Subtract 3 2b from 5 10b

3 2 5 10

3 2 5 10

8 12

b b

b b

b

5 10 3 2

5 10 3 2

2 8

b b

b b

b

5. A triangle has a side with length of x inches. The second side is 4 times as long as the first. The third side is 150 inches shorter than the first. Draw a picture of the situation and find a simplified expression for the perimeter.

Let x = the length of one side. Then second side is of length 4x. And the third side is of length x – 150. 4x x – 150 So Perimeter = x + 4x + x – 150 x = (6x – 150) inches

(note: figure is not drawn to scale)

Activity #3 (Solving Equations Part 1) Name:________KEY__________ Math 60 1. Solve each equation showing all your work. Circle your solution.

a. 4 9x

4 4

5x

b. 3 2 5x x

3 5

2

2 2

5 5

x

x x

x

c. 5 5 3 3 8 2 4x x

2

24 24

1

5 15 24 6 4

25 15 24 2

15 2

7

15 15

x x

x x

x

x

x x

d. 7 3x

7 7

4x

e. 4 3 4 11 2x x

11 1

12 16 11

1

1

1

2

16 2

6 6

8

1

x x

x

x x

x

f. 3 3 5 8 2x x

3 3 5 8 2

3 8 8

3 3

8

2

8

0

8

8

x x

x x

x

x x

x

2. Define the variable, set up an equation, solve the equation and answer the question with

a complete sentence.

You have saved $498 to purchase a new washing machine that costs $1349.

How much more must you save? Let x represent the unknown.

Let x = how much more you need to save

Equation: 498 1349x

Solve the Equation:

4

49

98 498

5

8 1 4

1

3 9

8x

x

You need to save an additional $851 to purchase the washing machine.


3. Solve each equation showing all your work. Circle your solution.

a. 4 8x

4 8

4 4

2x

x

b. 7

218

x

8 8

7

2

21

8 1 7

4

7

x

x

c. 115

x

5 5

1 1

11

1 5

55

x

x

d. 3 24x

3 241 1

3 1 3

8

1

x

x

e. 14 5 36x x

9 36

9 36

1

1 1

9

4

91

x

x

x

f. 4

105

x

5 5

4

25 1 OR

4 1

4

0

5 1

122 2

x

x

4. Translate each statement into an equation. Let x represent the number in each case.

a. A number divided by 5 is -4. b. 3

4 of a number is 18.

45

x

318

4x

5. Define the variable, set up an equation, solve it and answer the question with a complete

sentence.

At Grace Elementary School, we have found that on average 3

4of the students ride the bus

to school. If 570 students ride the bus, how many students attend the school?

Let x = the number of students who attend the school.

3570

4x now solve the equation:

4 3 570 4

3 4 1 3x

so 760x students.

There are 760 students who attend the school.

Activity #4 (Solving Equations Part 2) Name:________KEY__________ Math 60

Here are two solutions. Both are correct but one is much shorter than the other.

3

6 10 7 6 21

6 3 6 21

6 9

3

21 21

9

21

9

9

27

x

x

x

x

x x x

x

x

3 x

10 1

15 20 10 20

20 20

2

0

0

20 0

x x

x

x

x x

515 32

9

27 3

32

9 9

5 5

2

9

32

5

x

x

x

1

3 6 1 2 5 2

18 3 10 4

8 3 4

8 7

0 10

3 3

8

x x

x x

x x

x

x

8

x

7

8

7

8x

515 32

9

5 515 32

9 9

5 16015

9 9

160 515

9 9

135 16

160 1

9 9

5 5

0 5

9 9 9

295 5

9

60

9 9

9

59

x

x

x

x

x

x

x

1. Solve each equation and show all work. Circle your answers.

a. 3 1 17x b. 5 4 25x

1 1

3

3

x

18

3

6x

5 5

4

4

x

20

4

5x

c. 5 3 4 10 2x x d. 3 6 1 2 5 2x x

e. 2 3 5 7 6 21x x x f. 5

15 329

x


41 120 2 1680

39 12

120

0 1

0

680

39

12

x x

x

39

x

1560

40 (number of 41 cent stamps)

So there are 60 60 40 20 (number of 2 cent sta )

39

mps

x

x

3 5 2 4

3 9 2

9

2 2

x x

x x

x

x

x

2. Applications: Define the variable, write an equation, solve the equation and interpret your

results with a complete sentence.

a. The sum of twice a number and 4 is 20. Find the number.

Define the variable: Let x = a number

Write the equation: 2 4 20x

Solve the equation:

4 4

2

2

x

16

2

8x

Sentence: The number is 8.

b. Find two consecutive odd integers such that 3 times the first integer is 5 more than twice the second integer.

Define the variable: Let x = first odd integer____ then x + 2 = second odd integer

Write the equation: 3(first odd) 5 2(second odd)

3 2 5 2x x

Solve the equation:

Sentence: The two consecutive odd integers are 9 and 11.

c. Carla bought a total of 60 stamps in 41¢ and 2¢ denominations. If she paid $16.80, how

many of each type of stamp did she purchase?

Define the variable: Let x = number of 41¢ stamps, then _60 – x _ = number of 2¢ stamps.

Write the equation: 41 2 60 1680x x

Solve the equation:

Sentence: There are forty 41¢ stamps and twenty 2¢ stamps. Check it and see that it works!!!

Activity #5 (Solving Inequalities) Name:________KEY__________ Math 60

5 5

2 55 9

7 7

5 9

7 7

5

4

5

x x

x x

x

x

4 2 7 16

2 3

4 4

16 1

16

18

3

6

3

3

x x

x x

x

x

6 x

8 6 3 4 2

8 6 3 4 8

5 6 4 8

6

2

4

6 6

8

4

x x x

x x

x

x

x x

x

x

x

5

7 3 5 1

7 3 5 5

2 3 5

2

5

3

2

3

x x

x x

x

x

x

x

2

2

1x

3

5 2 3 4

5 2 3 12

2 2 12

2

3

2 2

2

x x

x x

x

x

x

x

10

2

5x

2 3 4 4 30

6 8 4 30

8 8

6 12 30

30

1

12

2

30

24

x x

x x

x x

x

12

x

2 x

1. Solve each inequality showing all work and graph the solution set on a number line.

a. 3 2x b. 4 12x c. 23

x

3

3

3

2

5

x

x

4 12

4 1

4

2

4

3

x

x

x

3 3

1 1

23

2

3 1

6

x

x

x

-5 -3 –6

d. 2 5

5 97 7

x x e. 8 6 3 4 2x x x f. 4 2 7 16x x

-4 -2 –6

g. 2 3 4 4 30x x h. 7 3 5 1x x i. 5 2 3 4x x

2 1 5

Multiplying by a negative, switch sense of inequality


2. Write the proper inequality for each statement.

a. x is positive x > 0

b. x is nonnegative x ≥ 0

c. x is no more than -4 x ≤ -4

d. x is at least 25 x ≥ 25

3. Applications: Define the variable, write an inequality, solve the inequality, and interpret

your results with a complete sentence

a. Cassie has $175 to purchase two identical skirts, a sweater and a pair of shoes. If the sweater costs $50 and the shoes cost $45 how much does she have to spend on each skirt? Define the variable: Let x = the cost of a skirt.

Equation: 2 skirts + 1 sweater + 1 pair of shoes ≤ Spending money

2x 50 45 1 75

2x 95 1 75

2x 80

2

-95 5

-9

2

x

80

40

2

x

Sentence: Cassie has $40 to spend on each skirt. Check it and see that it works!!!

a. Tim’s grade in Biology is based on three 100 point exams and one 200 point final.

He must earn 70% of the total points to receive a C in the course. Currently his test

scores are 63, 72 and 70. What must he earn on the 200 point final to receive a C in

the course?

Define the variable: Let x = Tim’s score on the final.

Equation: Test 1 points + Test 2 points + Test 3 points + Final points ≥ 70% of total points

63 72 70 0.70(500)

20

205 205

5 350

145

x

x

x

Sentence: Tim must score 145 or higher on the final exam to receive a C in the course. Note: 145/200 = 72.5% on the final.

Activity #6 (Formulas) Name:________KEY__________ Math 60

1. Solve each formula for the variable indicated.

a. d rt for t b. KT

VP

for T

r r

d rt

dt

r

KTV

P

P

K K

PV

K

P

T

c. y mx b for b d. y mx b for m

mx m

y mx b

y mx

x

b

y mx b

y b

b b

x

x

y

x

m

bm

x

e. 5

329

C F for F f. 1

2A h B b for b

532

32

9 9

32

9

932

5

3

5

2

5

9

5

C F

C F

C F

1

2

2

2

2 2 OR

2 2

1 1

h h

A h B b

h B bA

AB b

h

A A BhB b b

h h

g. 1B P rt , for t h. , for C s

D sn

P

Pr

Pr

Pr

r Pr

B P t

B P t

B P

P P

t

1 1

C sD

n

nD C s

n n

nD n

nD s C

s

s s

C

D

nD


2. Applications. Define your variable (by completing the chart), write the equation, solve the equation and interpret your results with a complete sentence. a. At noon a jogger leaves one point, running 8 mi/hr. One hour later a bicyclist leaves

the same point, traveling 20 mi/hr in the opposite direction. At what time will they be 36 miles apart? Let x = time of jogger

Equation: Distance Jogger + Distance of Bicyclist = Total Distance

8 x + 20(x – 1) = 36 8 x + 20x – 20 = 36

28x – 20 = 36 +20 + 20

28x = 56 x = 2

The jogger ran for 2 hours and the bicyclist rode for 1 hour.

b. With the wind behind it, a plane made a flight between two towns in 2 hr. Returning

against the wind, the plane flew 60 mph slower. The return flight took 3 hr. What was the plane’s speed in each direction? Define your variable, write the equation, and solve. Let x = speed of plane going Interpret your results with a complete sentence.

Equation: Distance Going = Distance Returning 2(x) = 3(x – 60) 2x = 3x – 180 -2x -2x 0 = x – 180

+180 +180 180 = x

Also x – 60 = 180 – 60 = 120 mph

Distance = (Rate) (Time)

Jogger 8 x 8 x

Bicyclist 20(x – 1) 20 x – 1

Distance = (Rate) (Time)

Going Trip 2(x) x 2

Returning Trip

3(x – 60) x – 60 3

The plane was traveling 180mph on the going trip and 120 mph on the return trip.

They will be 36 miles apart at 2:00pm.

Activity #7 (Sets and Set Notation) Name:________KEY__________ Math 60

1. List the elements of each set using the roster method.

List using the Roster Method

The set of factors of 60 {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}

The set of positive

multiples of 6 {6, 12, 18, 24…}

The set of integers

between 3 and 9, inclusive {3, 4, 5, 6, 7, 8, 9}

2. Use set-builder notation to describe each set. Graph each set on a number line.

Set Builder Notation Number Line

The set of real numbers less than 2

2x x

) 2

The set of real numbers

between 1 and 5, inclusive

1 5x x [ ]

1 5

The set of real numbers

greater than or equal to 27

27x x [ 27

3. Use set-builder notation, interval notation and a number line to describe the following sets.

Set Builder Notation Interval Notation Number Line

The set of real numbers greater

than or equal

to -3.7 and less than 14.8

3.7 14.8x x 3.7 ,14.8 [ )

-3.7 14.8

The set of real

numbers less than -205

205x x , 205

) -205


Basic Set Notation (where a and b represent any real numbers)

Set Set-Builder

Notation Interval Notation

Graph

All real numbers greater than b

x x b ,b b (

All real numbers less than or equal to b

x x b ,b b ]

All real numbers greater than or equal to a and less than b

x a x b ,a b a b [ )

All real numbers greater than a

and less than b x a x b ,a b

a b ( )

All real numbers less than or equal to 4

4x x ,4

] 4

All real numbers

greater than 15.25

15.25x x 15.25,

( 15.25

All real numbers greater than -15 and

less than -1.

15 1x x 15, 1 ( ) -15 -1

All real numbers less than or equal to -100.

100x x , 100

-100 ]

All real numbers greater than or equal to -9

and less than 20.

9 20x x 9,20 [ ) -9 20

All real numbers greater than 12.

12x x 12,

( 12

All real numbers

less than 205. 205x x ,205

) 205

Activity #8 (Two Variable Equations) Name:__________KEY_____________ Math 60

Let

2 2

7

22

7

7

7

,

0

0

0 is a solution

x

x

x

y

Let

1

2

, is a solut1

2io0 n

0

x

y

Let

1

2

, is a solut1

2io5 n

5

x

y

1. Complete each ordered pair so that it is a solution for the given equation.

a. 5 15x y b. 3 4y x c. 3 9y x

, since 5 15

, since 5 15

,

3 3

5 5

5 sin

0 0

2 2

4 45 15ce 5

4 4

4 4

3 3

, since 3 4

, since

0 0

0 0 3

5 5

3 3

4

, sin 31 1ce 4

, since 3 9

, since 3 9

0 0

0 0

2,

-3 -3

9 9

7 since 37 92

3 3

2. Find two solutions for each equation (answers are no unique). Find two solutions (ordered pairs) for each equation (answers are not unique). If you wish to use your graphing calculator, then list three solutions (ordered pairs) a. 4 2 8x y b. 3 12x y

Let

4 2 8

4

4, is

0

a

0

solut n0 io

y

x

y

2

Let

4 2 8

, is

0

0

0 a solu ion2 t

x

x

y

Let

3 12

4

, is 4 a soluti

0

0 on

0 y

y

x

Let

3 12

12

0

0

0

, is a solution12

y

x

x

c. 2

27

y x d. 1

2x

Let

2 2

7

2

, is a soluti2

0

0 o

0

n

y

y

x


3. The Tea-Totaler specializes in fancy Tea drinks. When x iced teas are sold, their Revenue is

3.25R x and their Cost is 0.75 30C x .

a. Algebraically find a simplified equation for the Profit. Profit = Revenue – Cost.

Profit Revenue Cost

3.25 0.75 30

3.25 0.75 30

2.50 30

P x x

x x

x

2.50 30P x

b. Use the profit equation you found in part a. to find the profit associated with selling 10 iced

teas (show the process of substitution). State your answer as an ordered pair. Using both x and y coordinates, interpret the ordered pair with a complete sentence.

2.50 30 25 30 510P

c. Use the profit equation you found in part a. to find the profit associated with selling 40 iced

teas (show the process of substitution). State your answer as an ordered pair. Using both x and y coordinates, interpret the ordered pair with a complete sentence.

2.50 30 100 30 0 04 7P

d. Algebraically determine how many iced teas need to be sold in order for the profit to be

zero. State your answer as an ordered pair. Using both x and y coordinates, interpret the ordered pair with a complete sentence.

2.5

2.5 30

30 2.

2.5

5

2

0

1

t

t

t

e. Algebraically determine how many iced teas need to be sold in order for the profit to be

$370. State your answer as an ordered pair. Using both x and y coordinates, interpret the ordered pair with a complete sentence.

2.

2.5 30

400 2

5 2.5

5

0

370

.

16

t

t

t

If there are only 10 iced teas sold, the company loses $5.

If 40 iced teas are sold, the company earns a profit of $70.

If 12 iced teas are sold, the profit will be $0.

Ordered pair: (10, -5)

Ordered pair: (40, 70)



If 160 iced teas are sold, the profit will be $370.

Activity #9 (Points, Relations, Functions) Name:________KEY__________ Math 60

-8 -6 -4 -2 2 4 6 8

-4

-2

2

4

x

y

(-6,2)

(0,0)

(-9,1) (-3,1) (7,1)

(1,-4) (6,-4)

1. Give the Quadrant in which each point is located or the axis on which the point lies.

a. 6, 2 Quad II d. 9, 1 Quad III

b. 7,1 Quad I e. 6, 4 Quad IV

c. 0,3 y-axis f. 3,1 Quad II

2. Plot each point on the grid below. Connect the line segments to make a famous

constellation.

a. 9,1

b. 6, 2

c. 3,1

d. 0,0

e. 1, 4

f. 6, 4

g. 7,1

3. State the domain and range of each relation and then tell whether the relation is a function of x. If the relation is NOT a function, explain why.

Relation Domain Range function of x?

The set of ordered pairs

0,1 , 1,1 , 2,1 , 3,1

0,1, 2, 3 1 YES

The set of ordered pairs

4,0 , 4,1 , 4,4 , 1,4

4, 4,1 0,1, 4 NO

(-4, 1) and (-4,4)

x y

8 9

4 5

5 4

2 3

3 2

8, 4, 5, 2, 3 9, 5, 4, 3, 2 YES


8.3 8

20.3

1

1.5( ) 8.3

12 1.5( )

.5 1

8

.3

.5

x

x

x

2( ) 3( ) 7

4 6

2

7

2

5

2( ) 3( )

4

7

6

2

7

2

17

2( ) 3( ) 7

100 30

0 10

7

1

77

21( ) 3(

100 30 7

0 ) 710

137

4. Given the function 2( ) 3 7f x x x , find:

a. (2)f b. ( 2)f

c. (10)f d. ( 10)f

e. ( )f a 2 3 7a a f. ( )f 2 3 7

5. The number of accidents in 1 month involving drivers x years of age can be approximated

by the function 2( ) 2 125 3000f x x x .

a. Use the function to find the number of accidents in 1 month that involved 17 years-

olds and interpret your results with a complete sentence.

2( ) 2( ) 125( ) 300017 17 17 1453f

There are approximately 1453 accidents in 1 month involving 17 year-olds.

b. Use the function to find the number of accidents in 1 month that involved 26 years-olds and interpret your results with a complete sentence.

2( ) 2( ) 125( ) 300026 26 26 1102f

There are approximately 1102 accidents in 1 month involving 26 year-olds.

6. Suppose that the weight, in pounds, of a baby boy x months old is predicted, for his first

10 months, by the function ( ) 1.5 8.3f x x .

a. Use the function to find the predicted weight at the age of 4 months and interpret your results with a complete sentence.

Let x = 4: ( ) 1.5( ) 8.34 4 14.3f

A 4 month old is predicted to weigh about 14.3 pounds.

b. Use the function to find the predicted age of a baby boy weighing 20.3 pounds and interpret your results with a complete sentence. Let ( ) 20.3f x :

A 20.3 pound baby boy is predicted to be about 8 months old.

Activity #10 (Tables and Graphs) Name:________KEY__________ Math 60

-4 -3 -2 -1 1 2 3 4

-3

-2

-1

1

2

3

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-8

-6

-4

-2

2

4

6

8

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

x

y

-4 -2 2 4

-4

-2

2

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-8

-6

-4

-2

2

4

6

8

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-8

-6

-4

-2

2

4

6

8

x

y

1. Determine whether each relation is a function or is not a function.

Then state its domain (D) and its range (R).

a. b.

Function? Yes or No Function? Yes or No

D: , R: 4 , D: 8, 6, 0, 4 R: 2,0, 2, 4

c. d.


D: 4, 2, 0, 2 R: 2 D: , 6 R: ,

e. f.


D: 4 , 4 R: 3 , 3 D: 8 , R: 4 , 6


-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

-8 -6 -4 -2 2 4 6 8

-4

-2

2

4

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-4

-2

2

4

6

8

x

y

i. j.


D: 4 , 8 R: 4 , 2 D: 2 , 6 R: 6 , 2

2. Use the given graph of y f x to answer the following, if possible.

If not possible, write “undefined”.

a. Find 10f = 6

b. Find 8f = 4

c. Find 6f = 2

d. Find 2f = 8

e. Find. 0f = 6

f. Find 8f = -2

g. Find 10f = undefined. .

h. Find all x such that 4f x when x = -2; x = 4; x = 8

i. Find all x such that 6f x when x = 0; x = 10

j. Find all x such that 2f x when x = -8


2 4 6 8 10

100

200

300

400

t

h(t)

(8, 240)

(4.94, 390)

(3, 330)

3. A ball is thrown upward with an initial velocity of 158 ft/s. Its height h in feet after t

seconds is given by 216 158h t t t . The ball is in the air for 9.9 seconds.

The graph of the height function is given.

a. Label the units on the horizontal axis.

b. Label the units on the vertical axis. c. Does the graph represent a function? Yes, the graph represents a function d. State the domain. [0, 9.9] seconds e. Maximum height occurs at 4.94 seconds. Find the maximum height to the nearest foot. Plot your results on the graph as an ordered pair.

216 14.94 4.94 458 .94

390ft

h

The maximum height is about 390 feet. f. State the range. [0, 390] feet

g. What is the height of the ball at 3 seconds? Plot the results on the graph as an ordered pair.

216 153 3 3

3

8

30 ft

h

At 3 seconds the ball is 330 feet off the ground (on its way up). h. Interpret h(8) = 240 in a complete sentence. Plot as an ordered pair.

At 8 seconds the ball is 240 feet off the ground (on its way down).

seconds

feet

Activity #11 (Graphing Linear Functions) Name:________KEY__________ Math 60

-6 -4 -2 2 4 6

-8

-6

-4

-2

2

4

6

8

x

y

(-6,3)

(-3,0)

(0,-3)

(3,-6)

-6 -4 -2 2 4 6

-8

-6

-4

-2

2

4

6

8

x

y

(5,0)

(0,-5)

(2,-3)

-6 -4 -2 2 4 6

-8

-6

-4

-2

2

4

6

8

x

y

(0,0)

(2,6)

(-2,-6)

-6 -4 -2 2 4 6

-8

-6

-4

-2

2

4

6

8

x

y

(0,0)

(-2,8)

(2,-8)

-6 -4 -2 2 4 6

-8

-6

-4

-2

2

4

6

8

x

y

(6,-5)(-4,-5) (2,-5)

-6 -4 -2 2 4 6

-8

-6

-4

-2

2

4

6

8

x

y

(-6,-6)

(0,-2)

(6,2)

-6 -4 -2 2 4 6

-8

-6

-4

-2

2

4

6

8

x

y

(2,3)

y=2x-1y+x=5

-6 -4 -2 2 4 6

-8

-6

-4

-2

2

4

6

8

x

y

(-5,4)

y=(-2/5)x+2

x = -5

1. Graph each equation in the space provided.

a. 3x y b. 5x y c. 3y x

d. 4y x e. 5y f. 2 3 6x y

2. Graph both lines on the same axes. Identify the point of intersection.

a. 2 1y x b. 2

25

y x

5y x 5x

Activity #11 Page 2 of 2 Math 60

0 30 60 90 120 150 1800

50

100

150

miles

dollars

(100,90)

(40,54)

2. A moving truck company rents their trucks for a flat fee of $30 plus $0.60 per mile.

a. We want to write the cost as a function of miles. First we should define our variables.

x represents _______miles_ __ y represents _______dollars__

b. State an equation representing the total cost, y, of renting a truck for x miles.

0.6( ) 30y x

c. Write the equation from part b. using function notation. Evaluate 100f .

0.6( ) 30f x x

100 1000.6 30 90f

d. Graph the equation from part b. on the axis below. Be sure to scale and label the axes.

e. Plot and label the point (40,54) on the graph in part d.

Using a complete sentence, what does this point represent? If you travel 40 miles, it will cost $54.

f. Plot and label the point associated with the 100 90f on the graph in part d.

Using a complete sentence, what does this point represent?

100 90f is associated with the ordered pair (100, 90).

If you travel 100 miles, it will cost $90.

Activity #12 (Slope) Name:________KEY__________ Math 60

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

(0,-3)

(6,1)

(-6,-7)

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

(0,4)

(2,-4)

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

(0,-1)

(2,5)

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

(0,3)

(4,0)

(-4,6)

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

(0,5)

(5,0)

1. Find the slope of the line that passes through each pair of points.

a. 2,6 and 3,9 b. 2,6 and 3,6 c. 2,6 and 2,3

9 6

3 2

3

1

3

m

6 6

3 2

0

1

0

m

6 3

2 2

3

0

undefined

m

Slope is 3. Slope is 0. Slope is undefined. 2. State the slope and vertical intercept of the line given by each equation. 3. Graph the line described in each exercise and give the equation of each line, in slope-

intercept form.

a. Slope –4; y-intercept: 0,4 b. Slope 3; y-intercept: 0, 1

Equation: 4 4y x Equation: 3 1y x

4. Graph each equation.

a. 2

33

y x b. 3

34

y x

c. 5y x

Equation of line Slope Vertical Intercept

23

5y x

2

5m 0 , 3

4 4y xx 1m 0 , 4

0 44y x 0m 0 , 4

0y xx 1m 0 , 0

10

1

2 2xy x

1

2m 0 , 0


5. In the year 2000, the average cost of an oil change was $28. By 2010, the average of an

oil change was $35. Let x represent the number of years since the year 2000.

a. Use this information to determine two points: (number years since 2000, cost of oil change).

The year 2000 means x = 0 when the cost is $28 gives the point (0, 28). The year 2010 means x = 10 when the cost is $35 gives the point (10, 35).

b. Using function notation, write a linear equation that approximates the cost of an oil

change in a given year.

Step 1. Find slope: 35 28 7 $0.70

10 0 10 1 yearm

Step 2. Find equation: y mx b

0.7 28f x x

c. Identify the vertical intercept in the equation found in (a). Interpret the meaning of the vertical intercept in the context of this story problem. The vertical intercept is (0, 28).

d. Identify the slope of the equation found in (a). Interpret the meaning of the slope in the context of this story problem.

The slope is 0.70.

e. Use the function in part (a) to predict the average cost of an oil change in 2012.

Interpret your result with a complete sentence.

The year 2012 means x = 12.

So 12 0.7 12 28 36.4f

f. Use the function in part (a) to predict when the average cost of an oil change will be

$42. Interpret your result with a complete sentence.

0.7 28

42 0.7 2

0.7

8

0.714

20

0.7

f x x

x

x

x

x = 20 means the year 2020. We predict an oil change to cost $42 in the year 2020.

In the year 2012, we estimate the cost of an oil change to be $36.40.

This means that the cost of an oil change will increase 70 cents each year.

This means in 2000, the cost of an oil change is $28.

Activity #13 (Forms of Linear Equations) Name:________KEY_______________ Math 60

3 2 4

2 3 4

3 4

2 2

32

3

2

2

3

2

x y

y x

x

y

x

x

x

y

5 4 8

4 5 8

5 8

4 4

52

4

5

4

5

4

x y

y

x x

x

xy

xy

( 8,4)

4 ( 8)

3

4and

4 6

3

4

4 6

10

310

4

y mx

b

b

b

x

mb

b

y

1 1 ( 8,4and

34 8

4

34 6

4

)

4

31

3

4

3

4

04

8

y y m x x

y x

y x

y

xy

m

x

Use the point and to find b:

2 11

21

3

2

3

,3

3

so 33 3

1

m

y mx b

b

b

1. Write each equation in slope-intercept form: y mx b .

a. 3 2 4x y b. 5 4 8x y

2. Find the slope-intercept equation of the line with slope 3

4and passes through the point (-8,4)

a. using slope-intercept form: y mx b b. using point-slope form: 1 1y y m x x

3. Consider the line that passes through the points 2,5 and 1,3 .

Write the equation of the line in slope-intercept form.

To find slope:

3 5 2

1 2 3m

The equation of the line

is 2 11

3 3y x

.


65 75

65 75

6

247.25

172.25 6

5 65

5

2.65

f x x

x

x

x

1

1 3 2slope of first line:

3 2 5m

2

5 1 4 2slope of second line:

7 3 10 5m

4. Mike bills a customer at the rate of $65 per hour plus a fixed service call charge of $75.

a. Write a linear equation that models the total cost f(x) where x represents the number of

hours it takes to complete a job.

65 75f x x

b. Use your function from part a. to determine the cost of a job that takes 3.5 hours to complete. Interpret your result with a complete sentence.

3.5 3.565 75 302.50f

c. Use your function from part a. to determine the number hours to complete the job if the total bill were $247.25. Interpret your result with a complete sentence.

5. Tell whether the pairs of lines are parallel, perpendicular, or neither.

a. L1 through (-2,-3) and (3,-1); L2 through (-3,1) and (7,5).

b. L1 with equation 2x + 4y = 8; L2 with equation 4x + 8y = 10

1

1equation L : 2

2y x 2

1 5equation L :

2 4y x

The slope of first line is -1/2. The slope of second line is -1/2.

A $247.25 bill means that the job took 2.65 hours to complete.

These two lines have the same slope so they are parallel.

These two lines have the same slope so they are parallel.

A job that takes 3.5 hours to complete will cost $302.50.


6. Find the slope-intercept form of the equation of the line with the given properties:

a. passes through the point 0,2 and parallel to the line 2

43

y x .

Our slope is -2/3 and b = 2 m = -2/3

So equation of line is 2

23

y x

b. passes through the point 0,2 and perpendicular to the line 2

43

y x

Our slope is 3/2 and b = 2 m = -2/3

So equation of line is 3

22

y x

c. passes through the points 1, 1 and 4,8 .

8 1 9

4

1 5m

Use 1, 1 :

9 4so 1

5 5

9 4equati

91 1

n: 5 5

5

o

y mx b

b

b

y x

OR

1 1

9 9

1 1

15 5

9 91

5 5

9 4equation:

5

9

5

5

y y m x x

y x

y x

y x

y x

d. passes through the points 4, 1 and 4,8 .

8 1 9undefined

4 4 0 m

Equation of line is 4x .

e. passes through (-3,4) and has an undefined slope

Equation of line is 3x .

f. has y-intercept (0,4) and a 0 slope

0 4

y mx b

y x

Equation of line is 4y .

Activity #14 (Rate of Change) Name:__________KEY_____________ Math 60 1. A driver used 10.3 gallons of gas driving 337.84 miles. The same driver used 5.4 gallons

of gas driving 177.12 miles.

a. Use this information to determine two points (gallons, miles) and calculate the rate of change. Interpret the rate of change with a complete sentence. Use the points: (10.3, 337.84) and (5.4, 177.12)

337.84 177.12 160.72

Rate of Change10.3

32.8 miles

gallon5.4 s4.9

b. Model the miles driven, y, as a linear function of the number of gallons used, x.

Write your equation using function notation.

Let x = # gallons and y = # of miles: y = 32.8x + b To find b: Use the point (5.4, 177.12)

177.12 = 32.8(5.4) + b

177.12 = 177.12 + b so 32.8f x x

0 = b (Doesn’t it make sense that (0, 0) should be a solution?)

c. Use the model to determine how much gas would be required to drive 225 miles

(round to one decimal place). Interpret your results in the context of this application.

Implies: 225f x

32.8f x x

32.8 so 6.9 g2 a25 llonsx x

It will take about 6.9 gallons of gas to drive 225 miles.

d. Use the model to determine how far can the driver go on 12 gallons of gas (to the

nearest mile)? Interpret your results in the context of this application.

Implies: 12x

32.8 393.612 1 m s2 ilef

With 12 gallons, the driver can travel about 393.6 miles.


The function is

1650 30900f x x

1650 30900

1650 30900

19100 1650

11

5

6

0000

.

f x x

x

x

x

2. A small company awards it employees annual raises based years of employment. The annual salary of an employee is shown in the table.

a. Use this information to determine two points and calculate the rate of change.

Interpret the rate of change with a complete sentence.

The ordered pair (x,y) means (Years employed, Annual Salary $) Use the points: (2, 34200) and (6, 40800)

40800 34200 6600

Rate of Change6

$1650

1year2 4

b. Model the annual salary as a linear function of the number of years employed.

y = 1650x + b To find b: Use the point (2, 34200)

1650

34200 3300 so 1650 30900

3090

2

0

34200 b

b y x

b

c. State the vertical intercept.

Interpret the vertical intercept in the context of this application.

The vertical intercept is (0, 30900) and it means that an employee’s initial salary will be $30,900.

d. Use your model to when the employee will be earning $50,000. Round to the nearest

tenth of a year. Interpret your results in the context of this application.

Implies 50,000f x so

e. Use your model to determine how much the employee will earn in 15 years. Interpret your results in the context of this application.

Implies x = 15 so 1650 3090015 1 5 , 505 5 6f

The employee can expect to make $55,650 in 15 years.

Years employed Annual salary in dollars

2 34,200

6 40,800

The employee will earn $50,000 in about 11.6 years.

The employee can expect to make $55,650 in 15 years.

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Math 60 ~ Chapter Zero Review Problems