Section 3.1:

Slope of a Line and Applications of Slope

Objective 1: Determine the slope of a line.

3.1 Lecture Guide: Slope of a Line and Applications of Slope

Algebraically Verbally Numerical Example

The slope of a line is the ratio of the change in y to the change in x. 

2 1

2 1

y ym

x x

for 1 2.x x

ym

x

m

run

1 3

2 1

3 1

4 3

x y

A 1-unit change in x produces a 2-unit change in y.

For the points(2, – 1) and (3, 1),

and reduces to: m

m

Slope of a Line Through 1 1,x y 2 2,x yand

Graphical Example

x

y

2 unit change in y

1 unit change in x

2, 1

3,1

Slope of a Line Through 1 1,x y 2 2,x yand

Calculate the slope of the line through each pair of points then graph a line that passes through the points.

1. (– 2, 7) and (3, 5)

Calculate the slope of the line through each pair of points then graph a line that passes through the points.

2. (1, – 8) and (7, – 3 )

3. (– 5, 3) and (2, 3)

Calculate the slope of the line through each pair of points then graph a line that passes through the points.

4. (5, 3) and (5, – 2 )

Calculate the slope of the line through each pair of points then graph a line that passes through the points.

Classifying Lines by Their Slopes

Numerically Verbally

m is positive The line slopes ______________ to the right.

m is negative The line slopes ______________ to the right.

m is zero The line is ________________________.

m is undefined The line is ________________________.

5. Calculate the slope of the line in the graph.

6. Calculate the slope of the line in the graph.

-5

5

-5 5

x

y

-5

5

-5 5

x

y

7. Determine the slope of the line in the graph.

8. Determine the slope of the line in the graph.

-5

5

-5 5

x

y

-5

5

-5 5

x

y

9. Calculate the slope of the line containing the points in the table.

10. Calculate the slope of the line containing the points in the table.

0 2

5 5

10 8

15 11

20 14

25 17

30 20

x y

3 4

0 2

3 0

6 2

9 4

12 6

15 8

x y

11. Complete the table so that the points all lie on a line having a slope .

12. Complete the table so that the points all lie on a line having a slope .

0 3

4

8

12

16

x y0 3

4

8

12

16

x y

5

4m

3

4m

(a) (b)

13. For the equation 4 6 24x y Find the x-intercept. Find the y-intercept.

(c) Use the points to determine the slope of the line.

Objective 2: Use slopes to determine whether two lines are parallel, perpendicular, or neither.

If l1 and l2 are distinct nonvertical* lines with slopes m1 and m2 respectively, then:

Algebraically Verbally Graphically

l1 and l2 are parallel because they have the ___________ slope.

or

l1 and l2 are perpendicular because their slopes are negative ______________.

* Also note all vertical lines are parallel to each other, and all vertical lines are perpendicular to all horizontal lines.

x

x

2l

Parallel and Perpendicular Lines

1l

1l

2l

y

y

1 2 1mm

12

1m

m

1 2m m

14. (a) If l1 and l2 are parallel and then _______.

1

3

5m

2m

(b) If l1 and l2 are perpendicular and then _______.

1

3

5m

2m

15. If l1 and l2 are perpendicular and m1= – 4 then m2 = _______.

16. If l1 and l2 are perpendicular and m1= 0, then m2 is _______________.

2,1 and 3,5

6, 2 and 2,3

17.

Determine whether the line that passes through the first pair of points is parallel to, perpendicular to, or neither parallel nor perpendicular to the line that passes through the second pair of points.

Determine whether the line that passes through the first pair of points is parallel to, perpendicular to, or neither parallel nor perpendicular to the line that passes through the second pair of points.

and

and

18. 1,2 3,5

4,0 2, 3

19.

Determine whether the line that passes through the first pair of points is parallel to, perpendicular to, or neither parallel nor perpendicular to the line that passes through the second pair of points.

(– 2, 5) and (0,1)

(7, 3) and (6, 5)

20.

Determine whether the line that passes through the first pair of points is parallel to, perpendicular to, or neither parallel nor perpendicular to the line that passes through the second pair of points.

(– 3, 4) and (1, 7)

(0, – 6 ) and (3, – 2)

21.

Determine whether the line that passes through the first pair of points is parallel to, perpendicular to, or neither parallel nor perpendicular to the line that passes through the second pair of points.

(– 3, 4) and (6, 4)

(– 2 , 5) and (– 2, 0)

22.

Determine whether the line that passes through the first pair of points is parallel to, perpendicular to, or neither parallel nor perpendicular to the line that passes through the second pair of points.

(– 3, 4) and (6, 4)

(– 2, 1 ) and (3, 1)

23. Compute the missing values in the table.

Change in x Change in y Slope

−5 3

7

4 2

3 0

3 Undefined

17

Using the given point and slope, determine another point on the line and graph the line.

24. Through (0, – 3) with

Point: ___________.

12

m

Using the given point and slope, determine another point on the line and graph the line.

25. Through (0, 2) with

Point: ___________.

23

m

26. Through (4, – 3) with m = 0.

Point: ___________.

Using the given point and slope, determine another point on the line and graph the line.

27. Through (4, – 3) with an undefined slope.

Point: ___________.

Using the given point and slope, determine another point on the line and graph the line.

28. A local high school purchases a copy machine for $1200. Due to depreciation, the value of the machine decreases with time. The table below lists the value y of the copy machine after x months.

Months Value

0 $1200

6 $1050

12 $900

18 $750

24 $600

30 $450

36 $300

(a) Determine the rate of change of the value with respect to time.

Objective 3: Calculate and interpret rates of change.

(b) Interpret the meaning of this value.

28. A local high school purchases a copy machine for $1200. Due to depreciation, the value of the machine decreases with time. The table below lists the value y of the copy machine after x months.

(c) At this rate, how long after the copy machine was purchased will the machine have no value?