2.2 Find Slope and Rate of Change

Slope

The slope m of a nonvertical line is the ratio of vertical change (the rise) to horizontal change (the run).

Parallel

Two lines in a plane that do not intersect

Perpendicular

Two lines in a plane that that intersect to form a right angle

Rate of change

How much one quality changes, on average, relative to the change in another quantity

SLOPE OF A LINE

Words

The slope m of a nonvertical line is the ratio of __vertical__ change (the rise) to __horizontal__ change (the run).

Algebra	Graph

Find slope

What is the slope of the line passing through the points (1, 3) and (6, 7)?

Let (x₁, y₁) = (1, 3) and (x₂, y₂) = (6, 7).

6-1

7-3

5

4

CLASSIFICATION OF LINES BY SLOPE

The slope of a line indicates whether the line _rises_ from left to right, _falls_ from left to right, is _horizontal_, or is _vertical_.

Positive slope	Negative slope	Zero slope	Undefined slope
Rises from left to right	Falls from left to right	Horizontal	Vertical

Example 2

Classify lines using slope

Without graphing, tell whether the line through the given points rises, falls, is horizontal, or is vertical.

a. (-6, -2), (1, 3)

b. (2, -1), (2, 2)

Solution

_a.

3-(-2)

1-(-6)

_{_____}

Because m __>__ 0, the line __rises__.

=

=

3

m

=

2-(-1)

b.

0

2 - 2

Because m is _undefined_, the line is _vertical_.

Complete the following exercises.

1. Find the slope of the line passing through the points (4, 2) and (7, 9).

2. Without graphing tell whether the line through the points (-3, 2) and (1, 4) rises, falls, is horizontal, or is vertical.

rises

SLOPES OF PARALLEL AND PERPENDICULAR LINES

Consider two different nonvertical lines l₁ and l₂ with slopes m₁ and m₂.

Parallel lines The lines are parallel if and only if they have the_same_ slope. m1 __=__ m2
Perpendicular lines The lines are perpendicular if and only if their slopes are__negative reciprocals__ m1 = ______ or m1 m2 = __-1__

Example 3

Tell whether the lines are parallel or perpendicular.

Line 1: through (-3, -1) and (2, 5)

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

Find the slopes of the two lines.

=

m₁

=

2- (-3)

5- (-1)

=

m₂

=

-3 -3

1- (-4)

Because m₁m₂ =__-1__, m1 and m2 are negative __reciprocals__ are of each other. So, the lines are __perpendicular__