2 – 2: Slope and 2 – 2: Slope and Rate of ChangeRate of Change

Objective:

CA Standard 7.0: Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial

denominators and simplify complicated rational expressions, including those with

negative exponents.

The Slope of a line The Slope of a line

The slope of a non-vertical line passing through (x1, y1) and (x2, y2) is

given by

2 1

2 1

y ym

x x

Example Example 11

Find the slope of the line passing through (-3, 5) and (2, 1)

2 1

2 1

y ym

x x

1 5

2 3m

4

5m

Classification of Lines Classification of Lines by Slopeby Slope

A line with a positive slope rises from the left to the right m > 0

2

-2

-5 5

A line with a negative slope rises from the falls from left to right.

m < 0

2

-2

-5 5

A line with a slope of zero is horizontal (m = 0)

2

-2

-5 5

A line with an undefined slope is a vertical line. (m is undefined)

2

-5 5

Classify Lines by their Slopes Tell whether the line through the given points

rises, falls, is horizontal or vertical.

(3, -4), (1, -6)

6 4 6 4

1 3 2m

2

12

The slope of the line is positive. The line rises.

(2, -1), (2, 5)

5 1 6

2 2 0m

The slope of the line is undefined, the line is vertical.

Tell whether the line through the given points rises, falls, is horizontal or vertical.

Comparing Comparing Steepness of LinesSteepness of Lines

If two lines have positive slopes the line with the greater slope is

steepest

If two lines have negative slopes the line with the slope of greater

absolute value is steeper.

Tell which line is steeper. Line 1: through (2, 3) and (4, 7)

Line 2: through (-1, 2) and (4, 5)

1

2

7 3 42

4 2 25 2 3

4 1 5

m

m

32

5

Line 1 is steeper.

Two lines are parallel if they do not intersect.

Two lines are perpendicular if they intersect to form right

angles.

Slope can be used to determine whether two distinct lines (non-

vertical) lines are parallel or perpendicular.

Slopes of Parallel and Slopes of Parallel and Perpendicular Lines.Perpendicular Lines.

Consider two different lines l1, and l2 with

slopes m1, m2

2

-2

-4

-5 5

Parallel Lines: have the same slope.

m1 = m2

Perpendicular Lines: slopes are negative reciprocals of each other.

1 1 22

1 or 1m mm

m

2

-2

-4

-5 5

l2

l1

Classifying Parallel and Classifying Parallel and Perpendicular Lines Perpendicular Lines Tell whether the lines are parallel,

perpendicular or neither.

Line 1 passes through (-3, 3) and (3, -1)

Line 2 passes through (-2, -3) and (2, 3)

1

1 3 4 2

3 3 6 3m

2

3 3 6 3

2 2 4 2m

1 2

2 3Because 1

3 2

the lines are perpendicular.

mm

Line1 passes through (-3, 1) and (3, 4)

Line 2 passes through (-4, -3) and (4, 1)

1

2

4 1 3 1

3 3 6 2

1 3 4 1

4 4 8 2

m

m

1 2Because

The two lines are parallel.

m m

HOMEWORKHOMEWORK

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Geometrical Use of Geometrical Use of SlopeSlope

The slope of a road or grade is usually expressed as a percent. For example if a road has a grade of 3%, it rises 3 feet for every 100

feet of horizontal distance.

Find the grade of a road that rises 75 feet over a horizontal distance of 2000 feet

75

2000 100

x

7500

2000x

3.75%x

Slope as Rate of Slope as Rate of ChangeChange

The number of US cell phones subscribers increased from 16 million in 1993 to 44 million in 1996. Find the average rate of change and use it to estimate the number of subscribers

in 1997.

Average rate of change = Change in subscribers

Change in time

44 16

1996 1993

28 19

3 3

The number of subscribers will be approximately 44 + 9 1/3 = 53 1/3 million

subscribers.