Lesson 1-2 Ordered Pairs - Weebly

Ordered Pairs as Points 13

Lesson

1-2Ordered Pairs

as Points

Lesson 1-2

BIG IDEA The ordered pairs of real numbers and the lines that you graphed in algebra are points and lines in Euclidean plane coordinate geometry.

Lines are one-dimensional, while the pipeline problem of Lesson 1-1 takes place in two dimensions. Two-dimensional objects lie in planes. We think of a plane as a fl at surface, like a table top or a tennis court without any boundaries or edges. Floors, roads, and even the surface of some regions on Earth are fl at like a plane.

What Is Coordinate Geometry?

Number lines show us that a point on a 1-dimensional line can be located by a single number. Around the year 1630, Pierre de Fermat and René Descartes realized that it is useful to identify a location in a plane by an ordered pair of real numbers. At the right, the three points (0, 0), (4, 1), and (–2.1, –1.5) are graphed. The study of geometric fi gures using points as ordered pairs of real numbers is called plane coordinate geometry.

See Quiz Yourself 1 at the right.

In algebra you graphed equations of lines. These graphs consisted of all the ordered pairs that solved a linear equation. Since we can view points as ordered pairs, we can view lines as sets of ordered pairs. This type of geometry is called Euclidean plane coordinate geometry.

The Standard Form of an Equation of a Line

There are many ways to write an equation of a line. One form that occurs frequently is the standard form, Ax + By = C. In the standard form, the values of A, B, and C determine the slope and location of the line. When A = 0, the equation of the line is of the form By = C, and the line is horizontal. When B = 0, the equation of the line is of the form Ax = C, and the line is vertical. When neither A nor B is zero, the line is oblique (neither horizontal nor vertical).

Vocabulary

plane coordinate geometry

horizontal line

vertical line

oblique line

standard form of an equation

of a line

slope-intercept form of an

equation of a line

y-intercept

slope

QUIZ YOURSELF 1

The point B above has integer coordinates. What are they?

y

x

(�2.1, �1.5)

(0, 0)

(4, 1)

2 4

2

4

�2

�4

�2�4

B

Give the coordinates of 6 points on the line with equation x + 4y = –10.

Mental Math

SMP_SEGEO_C01L02_013-018.indd 13SMP_SEGEO_C01L02_013-018.indd 13 11/8/07 3:02:10 PM11/8/07 3:02:10 PM

14 Points and Lines

Chapter 1

The Slope-Intercept Form of an Equation of a Line

Another way to write the equation of a line is the slope-intercept form. Recall from algebra that the slope-intercept form of an equation of a line is the form y = mx + b. In this form, m represents the slope, and b represents the y -intercept. The y-intercept is the value of y when x = 0. The slope is the ratio of the difference between y -coordinates and the difference between x-coordinates. It can be calculated as shown at the right.

Examine the slope formula. When y2 = y1, the line is horizontal and its slope is 0. When x2 = x1, the line is vertical and its slope is undefi ned.

Example 1

Graph the line with equation y = 2x - 2 and check your answer.

Solution 1 When x = 0, y = –2. When x = 1, y = 2 · 1 - 2 = 0.

The two points on the graph are (0, –2) and (1, 0). Plot these points and connect them.

Check It looks like the line passes through the point with coordinates (2, 2). 2 · 2 - 2 = 2, so we probably have a good graph of the equation.

Solution 2 Start at the y-intercept, –2, and advance according to the slope, 2 (up 2, right 1). The graphs are the same, so the slope checks.

Euclidean Plane Coordinate Geometry

Description of a point

A point is an ordered pair of real numbers.

Description of a line

A line is the set of ordered pairs of real numbers (x, y) satisfying an equation of the form Ax + By = C, where A and B are not both zero.

Slope = m =

y2 – y1x2 – x1

y2 – y1

x2 – x1

(x1, y1)

(x2, y2)

�

y

x

(0, �2)

(1, 0)2

2

�2

�2y

x2

2

2

�2

�2 1

Horizontal Line

y = 1.5

y

x

(3, 1.5)(0, 1.5)(�4, 1.5)

Vertical Line

x = –3

y

x

(�3, 4)

(�3, 3)

(�3, 0)

Oblique Line

x + y = 3

y

x

(0, 3)

(3, 0)



Lesson 1-2

Example 2

Write an equation of the line that contains the points (3, 1) and (–2, 11).

Solution To write the equation of the line, we need to fi nd the values of m and b in the equation y = mx + b.

m = y

2 - y

1

_____

x2 - x

1

= 11 - 1

______

–2 - 3 =

10

__

–5 = –2

You now know that this particular line is of the form y = –2x + b. To fi nd the value of b, substitute the coordinates of one of the points for x and y. We choose the point (3, 1).

y = –2x + b

1 = –2(3) + b

1 = –6 + b

7 = b

Substitute the value of b into the equation y = –2x + b. The equation is

y = –2x + 7.

See Quiz Yourself 2 at the right.

Example 3

Write an equation for the line containing the points (4, 9) and (–2, 6).

Solution First fi nd the slope using the formula

m = y

2 - y

1

_____

x2 - x

1

= 6 - ?

_____

–2 - ?

= ? .

Now you know that this line has an equation of the form y = 1 __ 2 x + b.

Substitute the coordinates of one of the given points to fi nd b.

y = 1

_

2 x + b

? = 1

_

2 ( ? ) + b

b = ?

So an equation of the line that contains the points (4, 9) and

(–2, 6) is y = ? x + ? .

A line that is in standard form can be “converted” into slope-intercept form. If B does not equal zero in the standard from, it is possible to solve for y. The resulting equation is y = Ax

__

B +

C

__

B . If we let m = –

A __

B and b = C

__ B , then we have

the form y = mx + b. Converting from standard form to slope-intercept form is useful if you are trying to use a graphing utililty to graph the equation of a line.

GUIDED

QUIZ YOURSELF 2

How can you tell that the answer to Example 2 is correct?


16 Points and Lines

Chapter 1

Example 4

Convert the equation 3x – 2y = 6 into slope-intercept form and graph it.

Solution 3x – 2y = 6 Given

–2y = –3x + 6 Add –3x to both sides.

y = 3

_

2 x – 3 Divide both sides by –2.

Start at the y-intercept, –3, and advance according to the slope: up 3, right 2. Plot the point (2, 0) and draw the line from it to (0, –3).

Ramps, braces, slides, and many other structures are often modeled by lines. With an equation for a line, you can determine locations of points on the structure.

Example 5

The maintenance crew at the Splash ’n’ Slide water park have found that

their main attraction, the Big Kahuna Water Slide, needs some additional

vertical bracing. There already exists a 10-foot vertical brace near the

bottom and a 46-foot vertical brace near the top that is 72 feet from that

bottom brace. Two more vertical braces are needed that are equally spaced

between these two braces. Model the slide with a line, letting the bottom of

the slide be the point (0, 10).

a. What are the possible coordinates for the top of the slide?

b. Find an equation for the line modeling the slide.

c. How long should each of the additional braces be?

Solution

a. Draw a picture.Since the top of the bottom brace is at (0, 10) and the 46-foot tall brace is ? feet away, the coordinates for the top of the slide are ( ? , ? ).

b. From Part a, the line modeling the slide contains (0, 10) and ( ? , ? ). Its slope is ? - 10

_____ ? - 0 = ? . The y-intercept of this line is ? . Therefore, the equation of the line is y = ? x + ? .

c. The horizontal distance between the original two braces is 72 feet, so two equally spaced braces between them would have to be located at ? feet and ? feet away from the shorter brace.

Substituting these values for x in the equation results in a vertical length of ? for the shorter of the two new braces, and a vertical length of

? for the longer of the two.

GUIDED

10’?

?

72’

46’

y

x

(0, �3)

(2, 0)2 4

2

4

�2

�4

�2�4



Lesson 1-2

Questions

COVERING THE IDEAS

1. a. What is the idea behind a plane in geometry?b. Give several examples of real-world objects that resemble planes.

2. In a coordinate plane, how can the location of points be described?

3. What is the slope of the graph of y = 3x - 1?

4. Write an equation for the line through (6, 8) and (2, –10).

In 5–9, classify the line with the given equation as horizontal, vertical, or oblique.

5. x = 89 6. x + y = 89 7. y = x + 89 8. y = 89 9. x = 0

In 10−13, convert the equation to slope-intercept form and

sketch a graph.

10. y = 2x - 5 11. 2x + 3y = 6 12. x = 3 13. 3x + y = –1

APPLYING THE MATHEMATICS

14. a. Write an equation for the vertical line that passes through (0, 5). b. Write an equation for the horizontal line that passes through (0, 5). c. Write an equation in slope-intercept form for an oblique line that

passes through (0, 5).

15. a. Graph the line with equation y = 3x + 2. b. Graph the line with equation y = 8. c. Find the point of intersection of the two lines.

16. Fire escapes zigzag down the sides of many tall buildings. Each set of stairs of the fi re escape in the building pictured here goes up about 12 feet and over about 6 feet. Model the bottom fi re escape stairs by a line beginning at (0, 30). (The bottom of the fi re escape is 30 feet off the ground.)

a. What point stands for the top of the bottom fi re escape stairs? b. Give an equation for the line modeling the bottom fi re

escape stairs.

17. A wheelchair ramp usually can be no steeper than the ramp pictured at the right, by the Americans with Disabilities Act (ADA) specifi cations.

a. If the left point of this ramp is at (–12, 0), where is the right point?

b. What is the slope of this ramp? c. What is an equation for this ramp? d. How high is the ramp at its middle?

24 ft

2 ft


18 Points and Lines

Chapter 1

18. a. Graph the line with equation 4x + 3y = 6. b. Graph the line with equation x = 3. c. Find the point of intersection of the two lines.

REVIEW

19. Find BC, where B and C are the points marked below. (Lesson 1-1)

�4 �3 �2 �1 0 1 2 3

C B

20. What are the coordinates of all the points at distance 3 from the point 2.12 on the number line? (Lesson 1-1)

21. Evaluate the expression √ _______

a2 + b2 when a = –8 and b = –15. (Previous Course)

EXPLORATION

22. When driving in a city, taxicabs are restricted to city streets, and so distances traveled are measured in the number of blocks the taxi has to travel to get from one place to another. In the map of part of Chicago at the right, each north-south block is one taxicab block, and each east-west block is half a taxicab block. Thus, the distance from point A, at W. Byron St. and N. Sacramento Ave., to point B, at W. Irving Park Rd. and N. Francisco Ave., is 2 taxicab blocks. Find all other street intersections that are 2 taxicab blocks from point A. To make your task easier, put a grid on the map, with A being at the origin, so B is at (1, 1). Then you can name the other intersections by their coordinates. On what shape do all the points seem to lie?

QUIZ YOURSELF ANSWERS

1. (–2, 3)

2. Answers vary. Sample: Substitute 3 for x and 1 for yin y = –2x + 7 to see if the point (3, 1) is on the line.

N California Ave

N M

ozart St

N Francisco Ave

N Richm

ond St

N Sacram

ento Ave

N W

hipple St

N A

lbany Ave

W Montrose Ave

W Cullom Ave

W Berteau Ave

W Irving Park Rd

A

B

W Byron St

W Grace St

W Belle Plaine Ave

W Waveland Ave

W Addison St

N A

lbany Ave

N Troy St

N Kedzie Ave

19

N

S

W E


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Lesson 1-2 Ordered Pairs - Weebly