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This document from Lehmann's Intermediate Algebra, 4ed, explains how to compare the steepness of two objects or lines by calculating their slope as the ratio of vertical distance to horizontal distance. examples of comparing ladders, roads, and lines.
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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.
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.
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.
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
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 ).
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).
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
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.
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.
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
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