geometry unit 3.7

UNIT 3.7 EQUATIONS OF LINES IN THE COORDINATE PLANE

Slope (m)

The ratio of its vertical rise to its horizontal run.

Steepness

Slope = m =Vertical rise

Horizontal run

Find the slopes.

28m

28m

-8

2

42

4m 2

2

4

Slope (continued)

12

12

xxyy

m

of a line containing two points with coordinates (x1, y1) and (x2, y2) is given by the formula

Slopes

12

12

xxyy

m

3155

All horizontal lines have a 0

slope

All vertical lineshave an

undefined slope

40 0

12

12

xxyy

m

66

34

07 undefined

Positive Slopes

Rise (upward) as you move left to right

Line slopes up from left

to right

y

x

Negative Slope

Fall (downward) as you move left to right

Line slopes down from left to right

y

x

Find the slope using the slope formula.

12

12

xxyy

m

87

62

25

87

12

12

xx

yym

04

10

41

41

Rate of Change

Describes how a quantity is changing over time.

xy

The slope of a line can be used to determine the Rate of Change

Change in quantity (y) Change in time (x)

Recreation: For one manufacturer of camping equipment,

between 1990 and 2000 annual sales increased by $7.4 million

per year. In 2000, the total sales were $85.9 million. If the

sales increase at the same rate, what will be the total sales in

2010?12

12

xxyy

m

14.7

20002010

9.852

y

10

9.85

1

4.72y

+85.9 +85.9

159.9 mill. = y2

74.0 = y2 – 85.9

7.4(10) = y2 – 85.9

Forms of Linear Equations

Slope-Intercept Form -

y = mx + b

slope y-intercept

Point-Slope Form -

y – y1 = m(x – x1)

slope x-coordinatey-coordinate

Graph

13

2 xy

13

2 xy

1.) The equation is in slope-intercept form y = mx + b

32

The slope is

y-intercept (0, 1)

2.) Plot the point (0, 1)

32

3.) Use the slope , from

the point (0, 1) go up 2,right 3

Graph

13

2 xy

13 xy

1.) The equation is in slope-intercept form y = mx + b

The slope is 3

y-intercept (0, 1)

2.) Plot the point (0, 1)

3.) Use the slope 3, from the point (0, 1) go up 3,right 1

Graph y - 3 = -2(x + 3)

1.) The equation is in point-slope form y – y1 = m(x – x1)

The slope is -2

Point on line (-3, 3)

2.) Plot the point (-3, 3)

3.) Use the slope -2, from the point (-3, 3) go down 2, right 1

Graph )4(3

12 xy

Point on line (4, 2)

2.) Plot the point (4, 2)

31The slope is

3.) Use the slope , from

the point (4, 2) go down 1, right 3

31

1.) The equation is in point-slope form y – y1 = m(x – x1)

Writing Equations of Linear Lines

If we know the slope and at least one point

If we have the slope and y-intercept, use the slope-intercept form; y = mx + b

If we have the slope and a point, use the point-slope form; y – y1 = m(x – x1)

Write the equation of the line

What is an equation of the line with slope 3 and y-intercept -5?

Start with the slope-intercept form of the equation

y = mx + by = 3x + (-5) Substitute 3 for m, and -5

for b

Simplifyy = 3x - 5

Write the equation of the line

What is an equation of the line through point (-1, 5) with slope 2?

Start with the point-slope form of the equation

y – y1 = m(x – x1)

y – 5 = 2(x - (-1)) Substitute 2 for m, and -1 in for x1 and 5 in for y1

Simplifyy – 5 = 2(x + 1)

Write the equation of the line

21

What is an equation of the line with slope and y-intercept 2?

Start with the slope-intercept form of the equation

y = mx + b

Substitute for m, and 2 for b 2

1y = x + 221

Write the equation of the line

What is an equation of the line through point (-1, 4) with slope -3?

Start with the point-slope form of the equation

y – y1 = m(x – x1)

y – 4 = -3(x - (-1)) Substitute -3 for m, and -1 in for x1 and 4 in for y1

Simplifyy – 4 = -3(x + 1)

Writing Equations of Linear Lines

If we know two points on the line Find the slope using the formula Using the point-slope formula Plug in one of the two points Plug in the slope for m

Write the equation of the line

What is an equation of the line through point (-2, -1) and point (3, 5)?

12

12

xxyy

m Find the slope

y + 1 = (x + 2) or56

y - 3 = (x - 5)56

32

51

56

56

Start with the point-slope form of the equation

y – y1 = m(x – x1) Plug in the slope and one of the two points

Writing Equations Horizontal and Vertical Lines

We don’t need a slope All points on a horizontal line have the

same y-coordinate; so the equation is y = y1.

All points on a vertical line have the same x-coordinate; so the equation is x = x1.

Where (x1, y1)

Write the equation of the line

What are the equations for the horizontal and vertical lines through (2, 4)?

The horizontal is y = y1

y = 4 Substitute 4 for y1

The vertical is x = x1

x = 2 Substitute 2 for x1

Write the equation of the line

What are the equations for the horizontal and vertical lines through (4, -3)?

The horizontal is y = y1

y = -3 Substitute -3 for y1

The vertical is x = x1

x = 4 Substitute 4 for x1

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