Download Ch.2, Section 3

Chapter 2, Section 3
Slope of a line
Often known as “m” from the formula y = mx + b.
The slope is calculated using a very precise formula.
Given two points, (a, b) and (c, d), the slope is
db
.
ca
If you label your points ( x 1 , y1 ) and ( x 2 , y 2 ), the formula is
( y 2  y1 )
.
(x 2  x1 )
This becomes an exercise in remembering
 which subtraction is on top and
 handling negative numbers and
 keeping track of the order of the points.
A handy mnemonic device is to remember the words “rise over run” – the phrase has the
r-words in alphabetical order. Thus “rise” is the y subtract and “run” is the x subtract.
Let’s get some practice.
If the line includes the points (2,1) and (5, 6). You may chose either point to be the first
point and the other will be the second point.
Chose a point one:
Write the formula:
Fill in the formula and calculate:
Now if you have negative numbers, be careful not to confuse the negative sign with the
subtract notation.
(2,1) (4,3)
Chose a point one:
Write the formula:
Fill in the formula and calculate:
Let’s look at a more difficult situation:
Suppose we have y = 5?
Sketch the graph for the line “y is always 5”.
Pick two points:
Chose a point one:
Write the formula:
Fill in the formula and calculate:
Suppose we have x = 2?
Sketch the graph for x is always 2.
Pick two points:
Chose a point one:
Write the formula:
Fill in the formula and calculate:
Summarizing:
Horizontal lines have m = 0.
Vertical lines have an undefined “m”.
All other lines have a slope that is a real number.
Can a slope be an irrational number?
Sure, just consider the line through () and ( 0,0)!
Comparing two lines:
How do lines relate to one another?
Well they’re parallel or they intersect. And, if the do intersect, then the angle of
intersection is 90 or it’s not.
When lines are parallel they have the same slope and different y intercepts.
When lines intersect they have different slopes.
If intersecting lines meet in an angle of 90 then the slope of one is the
negative reciprocal of the other. The lines that intersect this way are called
perpendicular.
Let’s look at two lines and decide how they relate to one another:
Given:
Line one:
(0, 3) and (1, 5)
Line two:
(0, 5) and (1, 7)
Calculate the slopes:
Sketch the graphs:
Let’s look at
Line one:
Line two:
(0, 1) and (1, 4)
(0, 3) and (3, 20
Calculate the slopes:
Sketch the graphs
Practice
1.
1 1
3
What is the slope for the line through  ,  and  ,
2 6
4
2
?
3
What is the slope of a line that is parallel to this?
What is the slope of a line that is perpendicular to this?
2.
What is the slope of the line through ( 1, 8) and ( 2, 13)?
What is the slope of a line that is parallel to this?
What is the slope of a line that is perpendicular to this?
3.
 1
What is the slope of the line through 1,  and 2, 0 ?
 2
What is the slope of a line that is parallel to this?
What is the slope of a line that is perpendicular to this?
4.
What is the slope of the line through (0, 3) and (5, 4)?
What is the slope of a line that is parallel to this?
What is the slope of a line that is perpendicular to this?
5.
What is the slope of the line through (0, 7) and (1, 5)?
What is the slope of a line that is parallel to this?
What is the slope of a line that is perpendicular to this?
6.
 14 
What is the slope of the line through 1,  and 3,  4  ?
 3
What is the slope of a line that is parallel to this?
What is the slope of a line that is perpendicular to this?