### Find the Slope of a Line from its Graph Find the

```Math 102
2.3 "Determining the Equation of a Line"
Objectives:
*
Find the slope of a line determined by two points.
*
Write the equation of a line, given the slope and a point contained in the line.
*
Write the equation of a line, given two points contained in the line.
*
Write the equation of a line parallel or perpendicular to another line and containing a speci…ed point.
*
Solve application problems involving the slope of a line.
Find the Slope of a Line from its Graph
De…nition:
"Slope of a Line"
The slope of a line (m) is a ratio that compares the vertical change to the corresponding horizontal change as we
move along the line from one point to another:
Example 1: (Finding the slope of a line from its graph)
Find the slope of the line graphed below.
y
-2
2
4
6
x
-5
Find the Slope of a Line given Two Points
De…nition:
"Slope of a Line"
If P1 and P2 are any two di¤erent points on a line, P1 with coordinates (x1 ; y1 ) and P2 with coordinates (x2 ; y2 ) ;
then the slope of the line (denoted by m) is
where x1 6= x2 :
Example 2: (Finding the slope of a line given two points)
Find the slope of the line that passes through the given points.
a) (3; 1) ; ( 6; 2)
b) ( 3; 6) ; (4; 8)
Page: 1
Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
2.3
Find the Slope of Horizontal and Vertical Lines
Example 3: (Finding the slope of a horizontal line)
Find the slope of the following lines.
a) x =
3
b) y = 4
y
-4
y
4
4
2
2
-2
2
4
-2
-4
x
-2
2
4
x
-2
-4
-4
Slopes of Horizontal and Vertical Lines:
Horizontal lines have a slope of
Vertical lines have
Types of Slopes:
y
y
y
x
Positive Slope
m>0
y
x
Negative Slope
x
Zero Slope
m<0
mH = 0
x
Unde…ned Slope
mV = undef ined
Application of Slope
The concept of slope has many real-world applications even though the word "slope" is often not used. For example,
hospital beds are constructed so that both the head end and the foot end can be raised or lowered; that is, the slope of either
end of the bed can be change. A roofer, when making an estimate to replace a roof, is concerned not only about the total
area to be covered but also about the pinch of the roof. The concept of slope is also used in the construction of stairways.
In highway construction, the word grade is used to describe the slope.
Page: 2
Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
2.3
Example 4: (Applied problem)
A certain highway has a 3% grade. How many feet does it rise in a horizontal distance of 1 mile?
Use Slope-Intercept Form to Write the Equation of a Line
Slope-Intercept Form:
The equation of the line with slope m and y
intercept (0; b) is
Example 5: (Using the slope-intercept form)
Use the slope-intercept form to write an equation of the line with slope
5
and y intercept
4
0;
1
4
: Write the answer in
standard form.
Example 6: (Using the slope-intercept form)
Find the slope and y-intercept of the line graphed below. Then use the slope-intercept form to write the equation of the line.
Write the answer in standard form.
y
5
4
3
2
1
-4
-2
2
-1
4
x
Page: 3
Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
2.3
Example 7: (Using the slope-intercept form)
Use the slope-intercept form to write an equation of the line that has the given slope and passes through the given point.
a) slope 3; and passes through ( 2; 5)
b) slope
2; and passes through ( 2; 8)
Example 8: (Using the slope-intercept form)
Find the slope and the y intercept of each line .
a) 2x + y = 9
b) x
5y = 0
Use the Point-Slope Form to Write the Equation of a Line
Point-Slope Form:
The equation of the line passing through (x1 ; y1 ) and with slope m is
Example 9: (Using the point-slope form)
Find an equation of the line that has slope
5
and passes trough (2; 6). Write the equation in standard form.
4
Page: 4
Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
2.3
Example 10: (Using the point-slope form)
Find an equation of the line passing through ( 2; 5) and (4; 3). Write the equation in slope-intercept form.
Example 11: (Application problems)
After purchasing a new drill press, a machine shop owner had his accountant prepare a depreciation worksheet for tax purposes.
Depreciation
Drill press
...............
Worksheet
\$1,970
a) Assuming straight-line depreciation, write an equation that gives
(new)
Salvage value
the value (v) of the drill press after x years of use.
........
\$270
b) Find the value of the drill press after 2 21 years of use.
(in 10 years)
Use Slope as an Aid When Graphing
If we know the slope and the y intercept of a line, we can graph the line without constructing a table of solutions.
Example 12: (Use slope as an aid when graphing)
Find the slope and y-intercept of the line with the equation 3x
2y =
Page: 5
4 and graph the line.
Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
2.3
Parallel and Perpendicular Lines
Properties of Parallel and Perpendicular Lines:
If two nonvertical lines have slopes of m1 and m2; then
1: The two lines are parallel if and only if
2: The two lines are perpendicular if and only if
Example 13: (Determining whether lines are parallel or perpendicular)
Determine whether the following lines are parallel, perpendicular or neither.
a)
2x
4x
y=4
b)
2y = 17
5x
3y = 12
3x + 5y = 27
Example 14: (Finding equations of parallel and perpendicular lines)
Write the equation of each line that satis…es the given conditions. Write the answer in standard form.
a) Contains the point (1; 2) and is parallel to the line 3x
y=5
b) Contains the point ( 3; 5) and is perpendicular 3x + 7y = 4
Page: 6
Notes by Bibiana Lopez
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