Comparing Steepness: Slope of Lines, Study notes of Algebra

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Slope of a Line

Section 1.

Introduction

Two ladders leaning

against a building.

Which is steeper?

We compare the vertical

distance from the base of

the building to the

ladder’s top with the

horizontal distance from

the ladder’s foot to the

building.

C o m p a r i n g t h e S t e e p n e s s o f T w o O b j e c t s

To compare the steepness of two objects such as two

ramps, two roofs, or two ski slopes, compute the

ratio

for each object. The object with the larger ratio is

the steeper object.

Property of Comparing the Steepness of Two Objects

vertical distance

horizontal distance

Property

C o m p a r i n g t h e S t e e p n e s s o f T w o O b j e c t s

Road A climbs steadily for 135 feet over a horizontal

distance of 3900 feet. Road B climbs steadily for

120 feet over a horizontal distance of 3175 feet.

Which road is steeper? Explain.

• These figures are of the two roads, however they

are not to scale

Comparing the Steepness of Two Roads

Example

Solution

C o m p a r i n g t h e S t e e p n e s s o f T w o O b j e c t s

The grade of a road is the ratio of the vertical to the

horizontal distance written as a percent.

What is the grade of roads A?

Ratio of vertical distance to horizontal distance is for

road A is 0. 038 = 0. 038 ( 100 %) = 3. 8 %.

Comparing the Steepness of Two Roads

Definition

Solution

Example

F i n d i n g a L i n e ’ s S l o p e

Let’s use subscript 1 to label x 1 and y 1 as the

coordinates of the first point, ( x 1 , y 1 ). And x 2 and y 2

for the second point, ( x 2 , y 2 ).

Run : Horizontal Change = x 2 – x 1

Rise : Vertical Change = y 2 – y 1

The slope is the ratio of the rise to the run.

Slope of a Non-vertical Line

We will now calculate the

steepness of a non-vertical line

given two points on the line.

Pronounced x sub 1 and y sub 1

Pronounced x sub 1 and y sub 1

F i n d i n g a L i n e ’ s S l o p e

A formula is an equation that contains two or more

variables. We will refer to the equation a

Slope of a Non-vertical Line

2 1

2 1

y y m x x

as the slope formula.

Sign of rise or run

run is positive run is negative rise is positive rise is negative

Direction (verbal)

goes to the right goes to the left goes up goes down

(graphical)

Definition

F i n d i n g a L i n e ’ s S l o p e

Find the slope of the line that contains the points

(1, 2) and (5, 4).

( x 1 , y 1 ) = ( 1 , 2 )

( x 2 , y 2 ) = ( 5 , 4 ).

Finding the Slope of a Line

m

Example

Solution

F i n d i n g a L i n e ’ s S l o p e

Find the slope of the line that contains the points

(2, 3) and (5, 1).

Finding the Slope of a Line

rise 2 2

run 3 3

m

By plotting points, the run

is 3 and the rise is – 2.

Example

Solution

F i n d i n g a L i n e ’ s S l o p e

Increasing: Positive Slope Decreasing: Negative Slope

Definition

Positive rise

Positive run

m =

= Positive slope

negative rise

positive run

m =

= negative slope

I n c r e a s i n g a n d D e c r e a s i n g L i n e s

Find the slope of the two

lines sketched on the right.

Comparing the Slopes of Two Lines

For line l 1 the run is 1 and the

rise is 2.

rise 1 2 run 2

m = = =

Example

Solution

I n c r e a s i n g a n d D e c r e a s i n g L i n e s

Note that the slope of l 2 is

greater than the slope of l 1 ,

which is what we expected

because line l 2 looks steeper

than line l 1.

Comparing the Slopes of Two Lines

rise 4 4 run 1

m = = =

For line l 2 the run is 1 and the

rise is 4.

Solution Continued

I n c r e a s i n g a n d D e c r e a s i n g L i n e s

Find the slope of the line that

contains the points (4, 2) and

(4, 5).

Investigating the slope of a Vertical Line

Plotting the points (above) and calculating the slope

we get 5 2 3 , division by zero is undefined. 4 4 0

m

The slope of the vertical line is undefined.

Example

Solution

H o r i z o n t a l a n d Ve r t i c a l L i n e s

• A horizontal line has slope of zero (left figure).
• A vertical line has undefined slope (right figure).

Property

Property

H o r i z o n t a l a n d Ve r t i c a l L i n e s