The Cartesian Coordinate Plane - Kyrene School …...722 • Chapter 11 The Cartesian Coordinate Plane © 2011 Carnegie Learning 12. Compare the ordered pair for the point you plotted

© 2

011

Car

negi

e Le

arni

ng

717

Air traffic

controllers use radar to track tens of

thousands of commercial airline flights. Controllers use quadrants to identify the locations, altitudes,

and speeds of many different flights.

11.1 FourQuadrantsExtending the Coordinate Plane ...................................719

11.2 GeometryandGraphsGraphing Geometric Figures ........................................725

11.3 Water,WaterEverywhereSolving Problems with Multiple Representations ...........731

11.4 EveryGraphTellsaStory!Interpreting Graphs ....................................................739

TheCartesianCoordinatePlane

© 2

011

Car

negi

e Le

arni

ng

718 • Chapter 11 The Cartesian Coordinate Plane

© 2

011

Car

negi

e Le

arni

ng

11.1 Extending the Coordinate Plane • 719

Key Terms x-axis

y-axis

origin

quadrant

ordered pair

Cartesian Coordinate Plane

Learning GoalsIn this lesson, you will:

Extend the coordinate system to four quadrants.

Name points on the plane.

Graph ordered pairs on the Cartesian Coordinate Plane.

Calculate the distance between points on the coordinate

plane which are on the same vertical or horizontal line.

You’ve worked with coordinate planes before, but you may not know how they

were invented. As the story goes, René Descartes (pronounced day-KART), a

French mathematician and philosopher, was having trouble falling asleep one

night. While trying to fall asleep, he looked up at the tiled ceiling and spotted a fly.

His mind began to wander and a question popped in his head: Could he describe

the path of the fly without tracing the actual path?

From that question came the revolutionary invention of the coordinate system—an

invention which made it possible to link algebra and geometry. Where have you

seen examples of coordinate planes? How do coordinate grids help you identify

the locations of objects?

FourQuadrantsExtending the Coordinate Plane

© 2

011

Car

negi

e Le

arni

ng


Problem 1 Expanding the Coordinate System

In earlier lessons, you graphed points on a coordinate plane where both the x- and

y-coordinates were zero or positive numbers.

x8

6

8

94

4

62

2

73 510

y

9

5

7

3

1

0

Now let’s include negative numbers in the coordinate plane.

1. To begin this process, first draw a horizontal line segment across the width of the grid

that splits the grid in half. Draw arrowheads at the ends of your line segment. Label

the line x. This horizontal line on the coordinate plane is called the x-axis.

2. Next, draw a line segment perpendicular to your first line segment from the top of the

grid that splits the grid in half so that the line segments intersect. Label this line y. This

vertical line on the coordinate plane is called the y-axis.

3. Label the point of intersection with 0. This point where the x-axis and y-axis intersect

on the coordinate plane is known as the origin. Then, using an interval of 1, label the

grid lines to the right and above 0 with positive numbers in numerical order. Finally,

label the grid lines to the left and below 0 with negative numbers in numerical order.

Perpendicular means that two

lines intersect each other at a right

angle.

© 2

011

Car

negi

e Le

arni

ng


4. How many regions are created when the coordinate plane is divided by the

perpendicular lines?

These regions on the coordinate plane are called quadrants. They are numbered with

Roman numerals from one to four (I, II, III, IV) starting in the upper right-hand quadrant and

moving counterclockwise.

5. Label each of the quadrants on your coordinate plane.

You can plot points on the coordinate plane using an ordered pair. An orderedpair is a

pair of numbers which can be represented as (x, y) that indicate the position of a point on

the coordinate plane. For example, the ordered pair

for the origin is (0,0).

6. Plot a point on the coordinate plane

anywhere in the first quadrant, and label

the point with its ordered pair.

7. Plot a point on the coordinate plane

anywhere in the second quadrant, and

label the point with its ordered pair.

8. Plot a point on the coordinate plane anywhere in the third

quadrant, and label the point with its ordered pair.

9. Plot a point on the coordinate plane anywhere in the fourth

quadrant, and label the point with its ordered pair.

10. Compare the ordered pair for the point you plotted in the first quadrant with the

ordered pairs your classmates plotted. What is similar about the points you graphed?

11. Compare the ordered pair for the point you plotted in the second quadrant with the


Remember, the x -coordinate always comes first and the y -coordinate

always comes second whether the numbers are positive or negative.


© 2

011

Car

negi

e Le

arni

ng

12. Compare the ordered pair for the point you plotted in the third quadrant with the ordered

pairs your classmates plotted. What is similar about the points you graphed?

13. Compare the ordered pair for the point you plotted in the fourth quadrant with the


This coordinate plane, called the CartesianCoordinatePlane, is named after

René Descartes.

14. Draw and label x- and y-axes on the coordinate plane shown. Then, plot and label

each point.

a. A (22, 3)

b. B (2, 23)

c. C (22, 23)

d. D (0, 24)

e. E (24, 0)

f. F ( 2 __ 3 , 22 )

© 2

011

Car

negi

e Le

arni

ng


Problem 2 Identifying Points and the Distances between Points

On the coordinate plane are points labeled from A to H.

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

A

CH

E

F

D

B

G

1. Identify the ordered pairs associated with each point.

A B C

D E F

G H

Use the coordinate plane shown to answer Questions 2 through 11.

86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

D

A

F

B

C

E

x

y

2. Identify the ordered pairs associated with each point.

A B C

D E F

3. Use the coordinate plane and scale from Question 2 to determine the distance from

point A to point B.


© 2

011

Car

negi

e Le

arni

ng

4. Describe how the ordered pairs for A and B are similar.

5. Use the x-coordinates of points A and B to calculate |4| 1 |29|.

6. How can an absolute value equation help you calculate the distance from one point to

another on the coordinate plane when the points are on the same horizontal line?

7. Use the grid and scale to calculate the distance from point A to point F.

8. Describe how the ordered pairs for A and F are similar.

9. Use the y-coordinates of points A and F to calculate |8| 1 |25|.

10. How can an absolute value equation help you calculate the distance from one point to

another on the coordinate plane when the points are on the same vertical line?

11. Write an absolute value equation and calculate each distance from:

a. point B to point C b. point D to point E

c. point E to point F

Be prepared to share your solutions and methods.

© 2

011

Car

negi

e Le

arni

ng

11.2 Graphing Geometric Figures • 725

GeometryandGraphsGraphing Geometric Figures


Plot points to form geometric figures.

Identify points on the coordinate plane to form geometric figures.

Identify geometric figures plotted on the coordinate plane.

“B5.”

“Hit! You sank it!”

These words are classic phrases you hear with the very popular game Battleship.

To begin the game, each player places their fleet of ships on a grid that has letters

and numbers. Then each player takes turn guessing points on their opponent’s

grid, and that opponent announces if that guessed coordinate “hit” one of the

ships, or if it was a “miss.” Both players record the point guessed on a grid.

Originally, this game was invented in the 1900s, but did not get popular until the

1930s. Have you ever played this game before? How is Battleship similar to the

coordinate planes you have been using? How is Battleship and the coordinate

planes you have been using different from each other?


© 2

011

Car

negi

e Le

arni

ng

Problem 1 Coordinate Geometry

One advantage of the Cartesian Coordinate Plane is that it enables mathematicians to use

coordinates to analyze geometric figures.

1. The points A (22, 4) and B (22, 22) are plotted on the coordinate plane shown. Plot

and label points C, D, E, and F so that squares ABCD and ABEF are formed.

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

A (–2, 4)

B (–2, –2)

2. Compare your square with your classmates.

a. Are all the squares the same?

b. How do you know that the other squares are drawn correctly?

3. On the coordinate plane, the line segment AB is graphed. Plot and label points C and

D to form parallelogram ABCD with a height of 4 units.

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

A (–4, –2) B (6, –2)

Remember a parallelogram is a quadrilateral in

which both pairs of opposite sides are

parallel.

© 2

011

Car

negi

e Le

arni

ng


4. Compare your parallelogram with your classmates.

a. Are all the parallelograms the same?

b. How do you know that the other parallelograms are drawn correctly?

5. Draw a trapezoid with a height of 5 units, and a base AB on the coordinate

plane shown.

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

A (–5, 4) B (–2, 4)

6. Compare your trapezoid with your classmates.

a. Are all the trapezoids the same?

b. How do you know that other trapezoids are drawn correctly?

Remember, a trapezoid is a

quadrilateral with exactly one pair of

parallel sides.


© 2

011

Car

negi

e Le

arni

ng

Problem 2 What Am I?

1. Graph the points on the coordinate plane. Connect the points with line segments, and

then identify the geometric figure.

x y

22 3

3 22

22 23

3 2

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

© 2

011

Car

negi

e Le

arni

ng


2. Graph the points on the coordinate plane. Connect the points with line segments and

identify the resulting figure.

x y

22 5

3 22

22 23

5 2

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y


© 2

011

Car

negi

e Le

arni

ng

Problem 3 Draw Me!

1. Points A and B are endpoints of one diagonal of a rectangle. Determine the locations

of vertices C and D of the rectangle, plot them, and then draw the rectangle.

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

B

A

2. Plot and identify five points that will create the vertices of a pentagon. Name the

pentagon PQRST.

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y


© 2

011

Car

negi

e Le

arni

ngWater,WaterEverywhereSolving Problems with Multiple Representations

Learning GoalIn this lesson, you will:

Analyze and solve problems with multiple representations.

Have you ever heard of the saying “sink or swim”? Well, for some competitive

swimmers, “sinking” is the name of the game. Actually, the technical term is “free

diving.” Free diving involves a person swimming into the ocean without the use of

breathing devices. So, how deep do you think the most successful free diver has

swum? Currently, William Trubridge of New Zealand holds the record for free

diving. He has free dove almost 288 feet into the ocean! And don’t forget, that

dive was without any breathing devices! So, how long do you think it took him to

reach that depth? How long do you think he can hold his breath? How long can

you hold your breath?

11.3 Solving Problems with Multiple Representations • 731


© 2

011

Car

negi

e Le

arni

ng

Problem 1 The Diver

A free diver is a person who dives into the ocean without the use of any breathing device

like scuba equipment. William Trubridge holds the record for free diving. Suppose

Trubridge dives at the rate of 1.37 feet per second.

1. How deep will he be after:

a. 10 seconds?

b. 25 seconds?

c. 1 minute?

d. 1 1 __ 2 minutes?

2. What quantity or quantities are changing in this situation?

3. Define variables for the independent and the dependent quantities.

4. Write an equation for this problem situation. Since he is diving below

the surface of the ocean, the distances he dives will be defined as

negative numbers to represent “below the surface.”

When a diver dives, he or she is going below sea

level.

© 2

011

Car

negi

e Le

arni

ng


5. Complete the table using your answers from Question 1.

Time (seconds)

Depth (feet)

6. Graph the equation using the values from the table. The convention for graphing

relationships between variables is to graph the independent variable on the horizontal

axis, or x-axis, and the dependent variable on the vertical axis, or y-axis.

25 50 75 100 125 150 175 200

–240

–210

–180

–150

–120

–90

–60

–30

–270

Time (seconds)

Dep

th (

feet

)

x

yRemember

what conventions are? Conventions are rules that are developed

to maintain an order.


© 2

011

Car

negi

e Le

arni

ng

7. At what depth was William Trubridge at 3 minutes and 30 seconds into his world-record

breaking dive?

a. Use your graph to determine his depth.

b. Use your equation to determine his depth.

8. Compare your answers from your graph and from your equation. Which answer is

more accurate? Why?

9. Explain why the points should be connected in this graph. For what value of time

should the graph begin and end?

© 2

011

Car

negi

e Le

arni

ng


Problem 2 In the Pool

1. Create a table of values for the points on this graph.

x y

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

Suppose the y-axis of this graph represents the water level of a pool (in inches) and the x-axis

represents time (in hours). The origin represents 3:00 pm, and the preferred water level.

2. Describe the meaning of each point:

a. the first point on the left

b. the second point on the left

c. the last point on the right


© 2

011

Car

negi

e Le

arni

ng

3. At what rate did the water go into the pool? Explain your reasoning.

4. Describe a situation that could match the graph.

5. Define variables for the quantities that are changing, and write an equation for

this situation.

6. Would it make sense to connect the points on the graph? If so, connect the points.

7. What would happen to the graph if the water did not go into the pool at the same

rate throughout?

© 2

011

Car

negi

e Le

arni

ng


Problem 3 No Place Like Home

Let’s consider another graph. The x-axis of this graph represents time in minutes from

12:00 pm, and the y-axis represents your distance from home in blocks. A point at the

origin represents you being home at 12:00 pm.

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

1. Describe your distance from home over the time from 9 minutes before 12:00 pm to

12:00 pm.

2. Describe your distance from home over the time from 12:00 pm to 6 minutes after

12:00 pm.


© 2

011

Car

negi

e Le

arni

ng

Let’s consider another graph comparing distance from home with time in minutes. Again,

a point at the origin represents you begin at home at 12:00 pm.

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

3. Describe your distance from home over the time from 9 minutes before 12:00 pm

to 12:00 pm.


© 2

011

Car

negi

e Le

arni

ng

11.4 Interpreting Graphs • 739

EveryGraphTellsaStory!Interpreting Graphs


Interpret information about a situation from a graphical representation.

Identify the graphs of situations.

You can read about business, but you also read graphs about financial markets.

Graphs are a part of almost every business section of newspapers, or financial

tabs on news web sites. Graphs will almost always display the Dow Jones, Nasdaq,

and Standard and Poors 500 (also known as the S & P 500). These three names

represent the financial markets that are publically traded every weekday in the

United States. And as a part of every weekday trading session, graphs are used to

show the trends of trading during the day. News of the U.S economy, press

releases from companies, and unemployment reports can quickly change the

trend of trading on a daily basis—and graphs can quickly capture these changes in

a visual way.

What other things use graphs to depict trends? Can graphs be used to show

trends in data for months and years too?


© 2

011

Car

negi

e Le

arni

ng

Problem 1 It’s Not a Tall Tale!

1. Write a few short sentences to describe each graph.

a. The Water Level in the Bathtub b. Money in Your Bank Account

Time (minutes)

Am

ount

of W

ater

(ga

llons

)

x

y

Time (weeks)

Am

ount

of M

oney

(do

llars

)

x

y

© 2

011

Car

negi

e Le

arni

ng


c. Running a Race d. Jogging for Exercise

x

y

Time

Spe

ed

x

y

Time

Dis

tanc

e fr

om H

ome


© 2

011

Car

negi

e Le

arni

ng

e. Create a graph. Be sure to label the axes, the intervals, and name your graph.

Then, ask your partner to tell the relationship it shows. Did your partner tell the

story you had in mind?

Problem 2 Hot Air

Nadja is coordinating a student council banquet. She asks Matthew to blow up balloons

for the event. The graphs shown represent his efforts.

Graph1 Graph2

21

1

2

3

4

5

6

7

8

3 4 5 6 7 8 9

9

Time (seconds)

Vol

ume

of A

ir in

the

Bal

loon

(cu

bic

feet

)

x

y

21

1

2

3

4

5

6

7

8

3 4 5 6 7 8 9

9

Time (seconds)

Vol

ume

of A

ir in

the

Bal

loon

(cu

bic

feet

)

x

y

© 2

011

Car

negi

e Le

arni

ng


Graph3 Graph4

21

1

2

3

4

5

6

7

8

3 4 5 6 7 8 9

9

Time (seconds)

Vol

ume

of A

ir in

the

Bal

loon

(cu

bic

feet

)

x

y

21

1

2

3

4

5

6

7

8

3 4 5 6 7 8 9

9

Time (seconds)

Vol

ume

of A

ir in

the

Bal

loon

(cu

bic

feet

)

x

y

1. Analyze each graph shown, and then answer each question.

a. What quantity is represented on the x-axis in each graph?

b. What quantity is represented on the y-axis in each graph?

c. What quantity or quantities change in each graph?

d. What quantity or quantities depends on the other quantity?


© 2

011

Car

negi

e Le

arni

ng

2. Match the descriptions with the appropriate graph.

a. Matthew blows air into a balloon at a steady rate, then ties it off when it is full.

b. Matthew blows air into a balloon, and then the balloon pops!

c. Matthew blows air into a balloon and then lets the air out.

d. Matthew blows air into a balloon slowly. As the balloon stretches out, it becomes

easier for him to blow into the balloon, and he can blow more air into the balloon.

He then ties off the balloon when it is full.

Problem 3 Match and Sort

In this activity, you will match a specific graph to a scenario. Follow the steps given.

1. Cut out each graph.

2. Tape each graph in the box with the appropriate scenario.

3. Label the axes with the appropriate quantities.

4. Cut out the scenarios, and sort them into similar groups.

© 2

011

Car

negi

e Le

arni

ng


A.

y

x

B.

y

x

C.

y

x

D.

y

x

E.

y

x

F.

y

x

✂

✂

✂


© 2

011

Car

negi

e Le

arni

ng

© 2

011

Car

negi

e Le

arni

ng


G. y

x

H.

y

x

I.

y

x

J.

y

x

✂

✂


© 2

011

Car

negi

e Le

arni

ng

© 2

011

Car

negi

e Le

arni

ng


1. You buy T-shirts to sell for your school. There is a $25 design charge for each T-shirt.

What is the total cost for different numbers of T-shirts?

2. A bus leaves school at the end of the day and stops to drops off its first passenger.

✂

✂


© 2

011

Car

negi

e Le

arni

ng

© 2

011

Car

negi

e Le

arni

ng


3. You have Fig Newtons for your class party. How many Fig Newtons will each

classmate receive (you don’t know how many classmates will show up)?

4. You are drinking your milk through a straw, and then the carton spills over.

✂

✂


© 2

011

Car

negi

e Le

arni

ng

© 2

011

Car

negi

e Le

arni

ng


5. Your telephone calling card charges $0.40 for the first minute of calls and $0.40 for

each additional minute of calls.

6. The video stores charges $3.00 for DVD rentals. How many DVDs can you rent for

different amounts of money?

✂

✂


© 2

011

Car

negi

e Le

arni

ng

© 2

011

Car

negi

e Le

arni

ng


7. You record the temperature for each hour on February 2, 2010.

8. There is a record of your growth since you were born. ✂

✂


© 2

011

Car

negi

e Le

arni

ng

© 2

011

Car

negi

e Le

arni

ng


9. On Monday, the rain fell at a steady rate. Then, it let up for a few hours before a

sudden downpour. Finally it let up.

10. You toss a basketball in the air. ✂

✂


© 2

011

Car

negi

e Le

arni

ng

© 2

011

Car

negi

e Le

arni

ng


5. How did you sort your graphs? Did your partner sort his or her graphs in the

same way?



© 2

011

Car

negi

e Le

arni

ng

© 2

011

Car

negi

e Le

arni

ng

Chapter 11 Summary • 761

Chapter 11 Summary

Key Terms x- axis (11.1)

y- axis (11.1)

origin (11.1)

quadrant (11.1)

ordered pair (11.1)

Cartesian Coordinate

Plane (11.1)

Cartesian Coordinate Plane

The Cartesian Coordinate Plane is formed by

two perpendicular number lines that intersect

at the zeros, or the origin. The intersecting

number lines divide the plane into four

regions, called quadrants.

Example

Point A at (24, 3) has been plotted. It is in Quadrant II.

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

A (–4, 3)

So, taxi drivers in London have a larger than normal

part of the brain that controls memory and spatial navigation, probably

because they are always learning new routes. I wonder if taking a different route to my next class will make my

brain grow. . .


© 2

011

Car

negi

e Le

arni

ng

Calculating the Distance Between Points

The distance between two points on a coordinate plane can be calculated by using the

coordinates of the two points.

Example

Points A and B are labeled on the coordinate plane.

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

A B

Identify the ordered pair associated with each point.

Point A (24, 5) Point B (3, 5)

Write an absolute value equation to calculate the distance between Point A and Point B.

| 24 | 1 | 3 |

4 1 3 5 7

© 2

011

Car

negi

e Le

arni

ng


Graphing Geometric Figures

One advantage of the Cartesian Coordinate Plane is that it enables mathematicians to use

coordinates to analyze geometric figures.

Example

On the coordinate plane, the line segment AB is graphed. Plot and label points C and D to

form parallelogram ABCD with a height of 6 units.

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

A B

D C


© 2

011

Car

negi

e Le

arni

ng

Analyzing and Solving Problems with Multiple Representations

Problem situations can be represented in multiple ways in order to analyze the problem.

Example

An airplane is taking off and climbing at a constant rate of 1000 feet per minute.

Complete the table to show the plane’s altitude, a, after t minutes.

Time (minutes) Altitude (feet)

1 1000

2 2000

4 4000

6 6000

8 8000

Write an equation for this situation.

a 5 1000t

Graph the equation using the values from the table.

y

x

Alti

tude

(fe

et)

Time (minutes)

Plane Altitude During Take-Off

8

6,000

8,000

9 104

4,000

62

2,000

73 510

9,000

10,000

5,000

7,000

3,000

1,000

0

© 2

011

Car

negi

e Le

arni

ng


Describing a Graph in Your Own Words

You can observe how a line or set of points in a graph move (up, down, horizontal,

steep, gradual) to describe the relationships between the quantities represented on the

x- and y- axes. You can describe the graph as increasing or decreasing quickly or

slowly, or as remaining constant.

Examples

y

x

Hou

rs o

f Stu

dy

Days

This graph represented the number of hours Clair studied in the days surrounding her

mid-semester exam. Prior to the exam, there was a peak in the number of hours she

studied. Then, after she took the exam, the amount of studying leveled off.

y

x

Dis

tanc

e (m

iles)

Time (hours)

A man drove quickly to work because he was running late. He stayed at work all day and

then drove home at a steady pace.


© 2

011

Car

negi

e Le

arni

ng

Documents

The Cartesian Coordinate Plane - Kyrene School …...722 • Chapter 11 The Cartesian Coordinate Plane © 2011 Carnegie Learning 12. Compare the ordered pair for the point you plotted