Slopes of Equations and Lines

1
Slopes of Equations and Lines

• Honors Geometry
• Chapter 2
• Nancy Powell, 2007

2
Objectives

• Calculate and interpret the slope of a line
• Graph lines given a point and the slope
• Use the point-slope form of a line
• Find the equation of a line given two points
• Write the equation of a line in slope-intercept
  form

3
SLOPE...
Let and be two
distinct points with . The slope m of
the non-vertical line is defined by the
formula
NOTE
If , then is a vertical line
and the slope m is undefined (since this results
in division by 0).
4
SLOPE...
Slope can be thought of as the ratio of the
vertical change ( ) to the
horizontal change ( ), often termed
RISE
over RUN
y
RUN
RISE
x
5
SLOPE...
y
RUN
RISE
x
Because this line rises from left to right with a
RISE and a RUN , well call this a RISING line.
x
6
SLOPE...
y
-RISE
RUN
x
Because this line falls from left to right or has
a RISE and a RUN, well call this a FALLING
line.
7
Plotting the two points results in the graph of a
VERTICAL line with the equation
.
y
x
x1
If , then is zero and
the slope is UNDEFINED. ( or - RISE and Zero
RUN)
8
Plotting the two points results in the graph of a
HoRIZ0ntal line with the equation .
y
If , then is zero and
the slope is ZERO. ( Zero RISE and or - RUN)
x
9
Example Find the slope of the line joining the
points (3,8) and (-1,2).
10
Example Draw the graph of the line passing
through (1,4) with a slope of -3/2.
Step 1 Plot the given point. Step 2 Use the
slope to find another point on the line (vertical
change , horizontal change ).
2
- 3
y
Find another point on this line. ( ____ , ____ )
(1,4)
-3
(3,1)
2
x
11
Example Draw the graph of the equation x 2.
What do you know about the slope of this line?
The slope is undefined.
y
x 2
x
12
Theorem Point-Slope Form of an Equation of a Line
An equation of a non-vertical line of slope m
that passes through the point (x1, y1) is
13
Example Find an equation of a line with slope -2
passing through (-1,5).
14
A HoRIZ0ntal line is given by an equation of the
form y b, where (0,b) is the y-intercept.
Slope
0
Example Graph the line y4.
y 4
15
(No Transcript)
16
Example Find the slope m and y-intercept (0,b)
of the graph of the line 3x - 2y 6 0.
Solve for y in terms of x and find the slope and
the y-intercept!
So, (0,3) is the y-intercept
17
Objectives (part 2)

• Define parallel, perpendicular, and oblique lines
• Find equations of parallel Lines
• Find equations of perpendicular Line
• Determine whether lines are parallel,
  perpendicular or oblique

18
Definition Parallel Lines
l

• Two distinct non-vertical lines are parallel if
  and only if they
• are in the same plane,
• have the same slope and
• have different y-intercepts.

m
n
19
Find the equation of the line parallel to
y -3x 5 passing through (1,7).
Since parallel lines have the same slope, the
slope of the parallel line must also be equal to
-3.
x1 1 and y1 7
Isnt this Slope-intercept form?
20
Definition Perpendicular Lines
Two lines are said to be perpendicular if they
intersect at a right angle.

• Two non-vertical lines are perpendicular if
• Their slopes are opposite reciprocals of each
  other like
• The product of their slopes is -1.

21
Example Find the equation of the line
perpendicular to y -3x 10 passing through
(1,5).
l
Slope of perpendicular line
Isnt this Slope-intercept form?
m
n
22
Definition Oblique Lines

• Oblique lines are lines that intersect but are
  not perpendicular to each other.
• Oblique lines do not form right angles.
• This means that the slopes of oblique lines are
  not the same and they are not opposite
  reciprocals.

m
n
l
23
Slopes and Oblique, Parallel, and Perpendicular
Lines

• Which of the following pairs of slopes are slopes
  of
• Oblique lines?
• Parallel lines?
• Perpendicular lines?
• How do you know?

b. -2/3 and -3/2
a. 2/3 and 3/2 c. 2 and - 0.5 e. 1.25
and -1.25 g. ¾ and 0.75

• b. -2/3 and -3/2
• 5/4 and 2/3
• 7/4 and 3/7
• h. 0 and undefined

a. 2/3 and 3/2

• 5/4 and 2/3

c. 2 and - 0.5
e. 1.25 and -1.25

• 7/4 and 3/7

g. ¾ and 0.75
h. 0 and undefined