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OBJECTIVE:

1. Interpret a line graph. 2. Plot ordered pairs. 3. Find ordered pairs that satisfy a given equation. 4. Graph lines. 5. Find x- and y-intercepts. 6. Recognize equations of horizontal and vertical lines and lines passing through the origin. 7. Use the midpoint formula.

1. Interpret a line graph. 2. Plot ordered pairs. 3. Find ordered pairs that satisfy a given equation. 4. Graph lines. 5. Find x- and y-intercepts. 6. Recognize equations of horizontal and vertical lines and lines passing through the origin. 7. Use the midpoint formula.

1.
Copyright © 2010 Pearson Education, Inc. All rights reserved

Sec 4.1 - 1

Sec 4.1 - 1

2.
Chapter 4

Graphs, Linear Equations,

and Functions

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Sec 4.1 - 2

Graphs, Linear Equations,

and Functions

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Sec 4.1 - 2

3.
4.1

The Rectangular Coordinate

System

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Sec 4.1 - 3

The Rectangular Coordinate

System

Copyright © 2010 Pearson Education, Inc. All rights reserved

Sec 4.1 - 3

4.
4.1 The Rectangular Coordinate System

Objectives

1. Interpret a line graph.

2. Plot ordered pairs.

3. Find ordered pairs that satisfy a given equation.

4. Graph lines.

5. Find x- and y-intercepts.

6. Recognize equations of horizontal and vertical lines

and lines passing through the origin.

7. Use the midpoint formula.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 4

Objectives

1. Interpret a line graph.

2. Plot ordered pairs.

3. Find ordered pairs that satisfy a given equation.

4. Graph lines.

5. Find x- and y-intercepts.

6. Recognize equations of horizontal and vertical lines

and lines passing through the origin.

7. Use the midpoint formula.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 4

5.
4.1 The Rectangular Coordinate System

Rectangular (or Cartesian, for Descartes) Coordinate System

y

8

y-axis

Origin 6

x-axis

Quadrant II 4Quadrant I

2

x

0 0

-8 -6 -4 -2 0 -2 2 4 6 8

Quadrant III -4

Quadrant IV

-6

-8

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 5

Rectangular (or Cartesian, for Descartes) Coordinate System

y

8

y-axis

Origin 6

x-axis

Quadrant II 4Quadrant I

2

x

0 0

-8 -6 -4 -2 0 -2 2 4 6 8

Quadrant III -4

Quadrant IV

-6

-8

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 5

6.
4.1 The Rectangular Coordinate System

Rectangular (or Cartesian, for Descartes) Coordinate System

y

Ordered Pair

Quadrant

(x, y)

A

D A (5, 3) Quadrant I

x

B (2, –1) Quadrant IV

B

C C (–2, –3) Quadrant III

D (–4, 2) Quadrant II

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 6

Rectangular (or Cartesian, for Descartes) Coordinate System

y

Ordered Pair

Quadrant

(x, y)

A

D A (5, 3) Quadrant I

x

B (2, –1) Quadrant IV

B

C C (–2, –3) Quadrant III

D (–4, 2) Quadrant II

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 6

7.
4.1 The Rectangular Coordinate System

Caution

The parentheses used to represent an ordered pair are also used to

represent an open interval (introduced in Section 3.1). The context of

the discussion tells whether ordered pairs or open intervals are being

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 7

Caution

The parentheses used to represent an ordered pair are also used to

represent an open interval (introduced in Section 3.1). The context of

the discussion tells whether ordered pairs or open intervals are being

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 7

8.
4.1 The Rectangular Coordinate System

EXAMPLE 1 Completing Ordered Pairs

Complete each ordered pair for 3x + 4y = 7.

(a) (5, ? )

We are given x = 5. We substitute into the equation to find y.

3x + 4y = 7

3(5) + 4y = 7 Let x = 5.

15 + 4y = 7

4y = –8

y = –2

The ordered pair is (5, –2).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 8

EXAMPLE 1 Completing Ordered Pairs

Complete each ordered pair for 3x + 4y = 7.

(a) (5, ? )

We are given x = 5. We substitute into the equation to find y.

3x + 4y = 7

3(5) + 4y = 7 Let x = 5.

15 + 4y = 7

4y = –8

y = –2

The ordered pair is (5, –2).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 8

9.
4.1 The Rectangular Coordinate System

EXAMPLE 1 Completing Ordered Pairs

Complete each ordered pair for 3x + 4y = 7.

(b) ( ? , –5)

Replace y with –5 in the equation to find x.

3x + 4y = 7

3x + 4(–5) = 7 Let y = –5.

3x – 20 = 7

3x = 27

x=9

The ordered pair is (9, –5).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 9

EXAMPLE 1 Completing Ordered Pairs

Complete each ordered pair for 3x + 4y = 7.

(b) ( ? , –5)

Replace y with –5 in the equation to find x.

3x + 4y = 7

3x + 4(–5) = 7 Let y = –5.

3x – 20 = 7

3x = 27

x=9

The ordered pair is (9, –5).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 9

10.
4.1 The Rectangular Coordinate System

A Linear Equation in Two Variables

A linear equation in two variables can be written in the form

Ax + By = C,

where A, B, and C are real numbers (A and B not both 0). This form is

called standard form.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 10

A Linear Equation in Two Variables

A linear equation in two variables can be written in the form

Ax + By = C,

where A, B, and C are real numbers (A and B not both 0). This form is

called standard form.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 10

11.
4.1 The Rectangular Coordinate System

Intercepts

y

y-intercept (where the line intersects

the y-axis)

x-intercept (where the

line intersects

the x-axis)

x

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 11

Intercepts

y

y-intercept (where the line intersects

the y-axis)

x-intercept (where the

line intersects

the x-axis)

x

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 11

12.
4.1 The Rectangular Coordinate System

Finding Intercepts

When graphing the equation of a line,

let y = 0 to find the x-intercept;

let x = 0 to find the y-intercept.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 12

Finding Intercepts

When graphing the equation of a line,

let y = 0 to find the x-intercept;

let x = 0 to find the y-intercept.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 12

13.
4.1 The Rectangular Coordinate System

EXAMPLE 2 Finding Intercepts

Find the x- and y-intercepts of 2x – y = 6, and graph the equation.

We find the x-intercept We find the y-intercept

by letting y = 0. by letting x = 0.

2x – y = 6 2x – y = 6

2x – 0 = 6 Let y = 0. 2(0) – y = 6 Let x = 0.

2x = 6 –y = 6

x=3 x-intercept is (3, 0). y = –6 y-intercept is (0, –6).

The intercepts are the two points (3,0) and (0, –6).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 13

EXAMPLE 2 Finding Intercepts

Find the x- and y-intercepts of 2x – y = 6, and graph the equation.

We find the x-intercept We find the y-intercept

by letting y = 0. by letting x = 0.

2x – y = 6 2x – y = 6

2x – 0 = 6 Let y = 0. 2(0) – y = 6 Let x = 0.

2x = 6 –y = 6

x=3 x-intercept is (3, 0). y = –6 y-intercept is (0, –6).

The intercepts are the two points (3,0) and (0, –6).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 13

14.
4.1 The Rectangular Coordinate System

EXAMPLE 2 Finding Intercepts

Find the x- and y-intercepts of 2x – y = 6, and graph the equation.

The intercepts are the two points (3,0) and (0, –6). We show these ordered

pairs in the table next to the figure below and use these points to draw the

graph. y

x y

x

3 0

0 –6

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 14

EXAMPLE 2 Finding Intercepts

Find the x- and y-intercepts of 2x – y = 6, and graph the equation.

The intercepts are the two points (3,0) and (0, –6). We show these ordered

pairs in the table next to the figure below and use these points to draw the

graph. y

x y

x

3 0

0 –6

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 14

15.
4.1 The Rectangular Coordinate System

EXAMPLE 3 Graphing a Horizontal Line

Graph y = –3.

Since y is always –3, there is no value of x corresponding to y = 0, so the

graph has no x-intercept. The y-intercept is (0, –3). The graph in the figure

below, shown with a table of ordered pairs, is a horizontal line.

y

x y

x

2 –3

0 –3

–2 –3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 15

EXAMPLE 3 Graphing a Horizontal Line

Graph y = –3.

Since y is always –3, there is no value of x corresponding to y = 0, so the

graph has no x-intercept. The y-intercept is (0, –3). The graph in the figure

below, shown with a table of ordered pairs, is a horizontal line.

y

x y

x

2 –3

0 –3

–2 –3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 15

16.
4.1 The Rectangular Coordinate System

EXAMPLE 3 con’t

Graphing a Vertical Line

Graph x + 2 = 5.

The x-intercept is (3, 0). The standard form 1x + 0y = 3 shows that every

value of y leads to x = 3, so no value of y makes x = 0. The only way a straight

line can have no y-intercept is if it is vertical, as in the figure below.

y

x y

x

3 2

3 0

3 –2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 16

EXAMPLE 3 con’t

Graphing a Vertical Line

Graph x + 2 = 5.

The x-intercept is (3, 0). The standard form 1x + 0y = 3 shows that every

value of y leads to x = 3, so no value of y makes x = 0. The only way a straight

line can have no y-intercept is if it is vertical, as in the figure below.

y

x y

x

3 2

3 0

3 –2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 16

17.
4.1 The Rectangular Coordinate System

Horizontal and Vertical Lines

To avoid confusing equations of horizontal and vertical lines remember

1. An equation with only the variable x will always intersect the

x-axis and thus will be vertical.

2. An equation with only the variable y will always intersect the

y-axis and thus will be horizontal.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 17

Horizontal and Vertical Lines

To avoid confusing equations of horizontal and vertical lines remember

1. An equation with only the variable x will always intersect the

x-axis and thus will be vertical.

2. An equation with only the variable y will always intersect the

y-axis and thus will be horizontal.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 17

18.
4.1 The Rectangular Coordinate System

EXAMPLE 4 Graphing a Line That Passes

through the Origin

Graph 3x + y = 0.

We find the x-intercept We find the y-intercept

by letting y = 0. by letting x = 0.

3x + y = 0 3x + y = 0

3x + 0 = 0 Let y = 0. 3(0) + y = 0 Let x = 0.

3x = 0 0+y=0

x=0 x-intercept is (0, 0). y=0 y-intercept is (0, 0).

Both intercepts are the same ordered pair, (0, 0). (This means

the graph goes through the origin.)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 18

EXAMPLE 4 Graphing a Line That Passes

through the Origin

Graph 3x + y = 0.

We find the x-intercept We find the y-intercept

by letting y = 0. by letting x = 0.

3x + y = 0 3x + y = 0

3x + 0 = 0 Let y = 0. 3(0) + y = 0 Let x = 0.

3x = 0 0+y=0

x=0 x-intercept is (0, 0). y=0 y-intercept is (0, 0).

Both intercepts are the same ordered pair, (0, 0). (This means

the graph goes through the origin.)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 18

19.
4.1 The Rectangular Coordinate System

EXAMPLE 4 Graphing a Line That Passes

through the Origin

Graph 3x + y = 0.

To find another point to graph the line, choose any nonzero

number for x, say x = 2, and solve for y.

Let x = 2.

3x + y = 0

3(2) + y = 0 Let x = 2.

6+y=0

y = –6

This gives the ordered pair (2, –6).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 19

EXAMPLE 4 Graphing a Line That Passes

through the Origin

Graph 3x + y = 0.

To find another point to graph the line, choose any nonzero

number for x, say x = 2, and solve for y.

Let x = 2.

3x + y = 0

3(2) + y = 0 Let x = 2.

6+y=0

y = –6

This gives the ordered pair (2, –6).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 19

20.
4.1 The Rectangular Coordinate System

EXAMPLE 4 Graphing a Line That Passes

through the Origin

Graph 3x + y = 0.

These points, (0, 0) and (2, –6), lead to the graph shown below.

As a check, verify that (1, –3) also lies on the line.

y x-intercept

and

y-intercept

x y

x

0 0

2 –6

1 –3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 20

EXAMPLE 4 Graphing a Line That Passes

through the Origin

Graph 3x + y = 0.

These points, (0, 0) and (2, –6), lead to the graph shown below.

As a check, verify that (1, –3) also lies on the line.

y x-intercept

and

y-intercept

x y

x

0 0

2 –6

1 –3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 20

21.
4.1 The Rectangular Coordinate System

Use the midpoint formula

If the endpoints of a line segment PQ are (x1, y1) and

(x2, y2), its midpoint M is

x1 x2 y1 y2

, .

2 2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 21

Use the midpoint formula

If the endpoints of a line segment PQ are (x1, y1) and

(x2, y2), its midpoint M is

x1 x2 y1 y2

, .

2 2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 21

22.
4.1 The Rectangular Coordinate System

EXAMPLE 5 Finding the Coordinates of a Midpoint

Find the coordinates of the midpoint of line segment PQ with

endpoints P(6, −1) and Q(4, −2).

Use the midpoint formula with x1 = 6, x2 = 4, y1 = −1, y2 = −2:

6 4 1 ( 2) 10 3 3

, , 5,

2 2 2 2 2

Midpoint

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 22

EXAMPLE 5 Finding the Coordinates of a Midpoint

Find the coordinates of the midpoint of line segment PQ with

endpoints P(6, −1) and Q(4, −2).

Use the midpoint formula with x1 = 6, x2 = 4, y1 = −1, y2 = −2:

6 4 1 ( 2) 10 3 3

, , 5,

2 2 2 2 2

Midpoint

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 4.1 - 22