2.1 Lines and Slopes The of the line through the

2.1 Lines and Slopes
The slope of the line through the
distinct points ( x1 , y1 ) and ( x2 , y2 )
Change in y Rise y2 − y1
=
=
is
Change in x Run x2 − x1
where x2 − x1 ≠ 0.
Example 1: Find the Slope
Find the slope of the line passing through
the pair of points (2,1) and (3, 4).
Solution
Let ( x1 , y1 ) = (2,1) and ( x2 , y2 ) = (3, 4).
y2 − y1 4 − 1 3
Slope = m =
= = 3.
=
x2 − x1 3 − 2 1
Possibilities for a Line’s Slope
Positive Slope Negative Slope
m>0
Line rises from left to right.
m<0
Line falls from left to right.
Possibilities for a Line’s Slope
Zero Slope
m=0
Line is horizontal.
Undefined Slope
m is
undefined.
Line is vertical.
Example 2: Find the Slope
Find the slope of the line passing through
the pair of points (−1,3) and (2, 4) or state
that the slope is undefined. Then indicate
whether the line through the points rises,
falls, is horizontal, or is vertical.
Solution
Let ( x1 , y1 ) = (−1,3) and ( x2 , y2 ) = (2, 4).
4−3
1
y2 − y1
=
=
Slope = m =
x2 − x1 2 − (−1) 3
The slope is positive, and
the line rises from left to right.
Practice Exercise
Find the slope of the line passing through
the points (4, −1) and (3, −1) or state
that the slope is undefined. Then indicate
whether the line through the points rises,
falls, is horizontal, or is vertical.
Answer
The slope m is zero.
Thus, the line is a horizontal line.
Point-slope Form of the
Equation of a Line
The point-slope equation of a nonvertical
line of slope m that passes through the
point ( x1 , y1 ) is
y − y1 = m( x − x1 ).
Example 3: Writing the PointSlope Equation of a Line
Write the point-slope form of the equation
of the line passing through (1,3) with a slope
of 4. Then slove the equation for y.
Solution
We use the point-slope equation of a line
with m = 4, x1 = 1, and y1 = 3.
m = 4, x1 = 1, and y1 = 3
y − y1 = m( x − x1 )
y − 3 = 4( x − 1)
y − 3 = 4x − 4
y = 4x −1
Practice Exercise
Write the point-slope of the equation
of the line passing throuhg the points
(3,5) and (8,15). Then solve the
equation for y.
Answer
Point-slope form of the equation:
y − 5 = 2( x − 3).
Then solve for y gives:
y = 2x −1
Example 4: Writing the PointSlope Equation of a Line
Write the point-slope form of the equation
of the line passing through the points (3,5)
and (8,15). Then slove the equation for y.
Solution
First find the slope to use the
point-slope form.
Given (3,5) and (8,15).
15 − 5 10
m=
=
=2
5
8−3
We can take either point on the line to
be ( x1 , y1 ). Let's use ( x1 , y1 ) = (3,5).
y − 15 = 2( x − 3)
y − 15 = 2 x − 6
y = 2x + 9
Practice Exercises
1. Write the point-slope form of the equation
of the line passing through (4, −1) with a slope
of 8. Then slove the equation for y.
2. Write the point-slope form of the equation
of the line passing through the points ( −2, 0)
and (0, 2). Then slove the equation for y.
Answers to Practice Exercises
1. y = 8 x − 33
2. y = x + 2
The Slope-Intercept Form of
the Equation of a Line
y
The slope-intercept
equation of a
nonvertical line
with slope m and
y -intercept b is
y = mx + b
Y-intercept is b
(0, b)
Slope is m
x
A line with slope m
and y -intercept b.
Graphing y=mx+b Using the
Slope and y-Intercept.
Plot the y-intercept on the y-axis.
This is the point (0,b).
„ Obtain a second point using the
slope, m. Write m as a fraction,
and use rise over run starting at
the y-intercept to plot this point
„
Graphing y=mx+b Using the
Slope and y-Intercept.
„
Use a straightedge to draw a line
through the two points. Draw
arrowheads at the ends of line to
show that the line to show that the
line continues indefinitely in both
directions.
Example 5: Graphing by Using
the Slope and y-Intercept
Give the slope and the y -intercept of the
line y = 3 x + 2. Then graph the line.
Solution
y = 3x + 2
The slope
is 3
The y -intercept
is 2.
2 Rise
Slope = m = 2 = =
1 Run
First use the y -intercept 2, to
plot the point (0, 2). Starting
•
•
at (0, 2), move 2 units up and
1 unit to the right. This gives
us the second point of the line.
Use a straightedge to draw a
line through the two points.
The graph of y = 3 x + 2.
Practice Exercises
Give the slope and y -intercept
of each line whose equation is
given. Then graph the line.
1. y = −3 x + 2
3
2. y = x − 3
4
Answers to Practice Exercises
1. m = −3, b = 2
3
2. m = , b = −3
4
Equation of a Horizontal Line
A horizontal line
is given by an
m=0
Y-intercept
is 4
•
equation of the
form
y=b
where b is the
y -intercept.
The graph of y = 4
Equation of a Vertical Line
A vertical line is
given by an
X-intercept
is -5
equation of the
form x = a
where a is the
x-intercept.
•
Slope is
undefined
The graph of x = -5
Example 6: Graphing a
Horizontal Line
Graph y = 5 in the
rectangular coordinate system.
Y-intercept
is 5.
Solution
All points on the graph
of y = 5 have a value of
y that is always 5. Thus
it is a horizontal line
with y -intercept 5.
•
Example 7: Graphing a
Vertical Line
Graph x = −5 in the
rectangular coordinate system.
Solution
No matter what the
y -coordinate is, the
corresponding
x-coordinate for every
point on the line is 5.
X-intercept
is –5.
•
Practice Exercises
Graph each equation in the rectangular
coordinate system.
1. y = 4
2. x = 0
Answers to Practice Exercises
1.
2.
General Form of the Equation
of a Line
Every line has an equation that can
be written in the general form
Ax + By + C = 0
where, A, B, and C are three
real numbers, and A and B
are not both zero.
Equations of Lines
1. Point-slope form:
2. Slope-intercept form:
y − y1 = m( x − x1 )
y = mx + b
3. Horizontal line:
y=b
4. Vertical line:
x=a
5. General form: Ax + By + C = 0
Example 8: Finding the Slope
and the y-Intercept
Find the slope and the y -intercept of the
line whose equation is 4 x + 6 y + 12 = 0.
Solution
First rewrite the equation in slope-intercept
form y = mx + b. We need to solve for y.
4x + 6 y +12 = 0
6 y = −4 x − 12
The coefficient of x,
4
12
y = − x−
6
6
the constant term, 2,
2
y = − x−2
3
− 23 , is the slope and
is the y -intercept.
m = − , b = −2.
2
3
Practice Exercises
a. Rewrite the given equation in
slope-intercept form.
b. Give the slope and y-intercept.
c. Graph the equation.
1. 6 x − 5 y − 20 = 0
2. 4 y + 28 = 0
Answers to Practice Exercises
1. y = x − 4
6
5
6
slope = m =
5
y -intercept = b = −4.
Answers to Practice Exercises
2. y = −7
slope = m = 0
y -intercept = b = −7.