Section 3.4 & 3.5
SLOPE AND RATE OF CHANGE
EQUATIONS OF LINES
Slope and Rate of Change
 Find the slope of a line given two points on the line.
 Find the slope of a line given its equation.
 Find the slopes of vertical and horizontal lines.
 Compare the slopes of parallel and perpendicular
lines.
 Use the slope-intercept form to write an equation of
a line.
 Use the slope-intercept form to graph a linear
equation.
 Find equations of vertical and horizontal lines.
Section 3.4 & 3.5
Finding the Slope of a Line Given Two Points of the Line
 A key feature of a line is its slant or steepness. This
measure is called the slope of the line.
 Slope is the ratio of the vertical change to the
horizontal change between two points as we move
along the line.
 Can be found on a graph by counting the vertical rise
and horizontal run.
Section 3.4 & 3.5
Finding the Slope of a Line Given Two Points of the Line
 Find the slope of the line
between
A to B
1.
Down 3  -3
3

 1
Right 3  +3
3


B to A
Up 3  +3
Left 3  -3



Section 3.4 & 3.5
3

 1
3
The direction traveled does
not matter, the slope will be
the same either way.
2.
A and C  72
3.
B and C 4
Summary of Slope
 The slope of a line is always interpreted by reading
the line as it moves from left to right.
Upward Line
Downward Line
Positive Slope
Negative Slope
m>0
m<0
Horizontal Line
Vertical Line
Zero Slope
Undefined Slope
m=0
m does not exist
Section 3.4 & 3.5
Finding the Slope of a Line Given Two Points of the Line
 Consider the points (1, 0) and (4, 3) in the coordinate
plane.
 Count the “rise” and the “run” to determine the slope
of the line between the points.
y
5
4–
31
4
(4, 3)
3
2
3–0
13
(1, 0)
–5
–4
–3
–2
–1
–1
–2
–3
–4
–5
Section 3.4 & 3.5
1
2
3
4
5
x
Finding the Slope of a Line Given Two Points of the Line
 The Slope Formula
 Given two points  x1 , y1 and  x2 , y2, where x1  x2 , the slope of the
line connecting the two points is given by the formula
y2  y1
m
x2  x1
 Find the slope of the line connecting the given
points.
1.
2.
(6, -2) and (5, 5) 7
x
y
-5
7
-2
5
1
3
4
1
Section 3.4 & 3.5
 23
Using the Slope-Intercept Form to Graph an Equation
 Graph the line passing
through the point (-2, 5)
with a slope of -3.
1.
To graph, plot the known
point.

2.
3.
Slope is the “rise” and
“run” needed to get from
one point to another along
a line.
3
Follow the slope to reach
3 
a second point.
1
Connect the points with
a straight line.
Section 3.4 & 3.5
Using the Slope-Intercept Form to Graph an Equation
 Through the given point, draw a line with the
given slope.
1. (3, 2) m = 4
2. (-2, -1) m = 2/7
3. (2, -7) m = 0
4. x-int: -3, m = -1/2
Section 3.4 & 3.5
Finding the Slope of a Line Given Its Equation
 Slope-Intercept Form
 When a linear equation in two variables is written in slope-intercept
form, y  mx  b , m is the slope of the line and (0, b) is the yintercept of the line.
 Find the slope and y-intercept of the line.
1.
2.
y  x3
m  95 ; (0,3)
7x  2 y  8
m  72 ; (0, 4)
5
9

3.
y 1

4.
Equation must be written in y = mx + b
All horizontal lines have a slope of zero.
x  2

m  0; (0,1)
m  undefined ; does not exist
All vertical lines have undefined slope.
Section 3.4 & 3.5
Using the Slope-Intercept Form to Write an Equation
 Slope-Intercept Form
 When a linear equation in two variables is written in slopeintercept form, y  mx  b , m is the slope of the line and
(0, b) is the y-intercept of the line.
 Find the equation of the line with y-intercept (0, 7)
and slope of ½.
y  12 x  7
 Find the equation of the line through the points (1, 3)
and (0, 4)
Section 3.4 & 3.5
y  x  4
Finding Equations of Vertical and Horizontal Lines
 Find an equation of the horizontal line through the
point (1, 4).
y4
 Find an equation of the vertical line through the
point (1, 4).
Section 3.4 & 3.5
x 1
Using the Slope-Intercept Form to Graph an Equation
 If an equation is in slope-intercept form, both the
slope and a point on the line are known.
 Slope-Intercept Form
y  mx  b
slope
Section 3.4 & 3.5
y-intercept
(0, b)
Using the Slope-Intercept Form to Graph an Equation
 To graph an equation in
slope-intercept form
y = mx + b:
1.
2.
3.
Plot the y-intercept (0, b).
Plot a second point by
rising the number of units
indicated by the
numerator of the slope
then running the number
of units indicated by the
denominator of the slope,
m.
Draw a straight line
through the points.
Section 3.4 & 3.5
 Find the slope and y-
intercept of each line,
then graph.
5.
6.
y  3x  4
y   13 x  2
Using the Slope-Intercept Form to Graph an Equation
 Determine the slope and y-intercept, then graph.
8. 4y = -8x
7. 2x + y = 8
y  2 x  8
9. 4x – 3y = 9
y  43 x  3
Section 3.4 & 3.5
y  2 x
10. y = 8
Using the Slope-Intercept Form to Graph an Equation
 Determine the slope and y-intercept, then graph.
12. x = 4
11. 2x – y = 4
y  2x  4
13. 3x – 4y = 4
y  x 1
3
4
Section 3.4 & 3.5
14. x = 3/2 y
y  23 x
Slopes of Parallel and Perpendicular Lines
 Parallel

Parallel lines have the
same slope.
Section 3.4 & 3.5
 Perpendicular

Perpendicular lines have
slopes that are negative
reciprocals, or a
product of -1.
Slopes of Parallel and Perpendicular Lines
 Determine if the pair of lines is parallel,
perpendicular, or neither.
1.
2.
3.
y  5 x  1
x  5 y  10
m1  5; m2  15
x  y  11
2 x  y  11
m1  1; m2  2
2 x  3 y  21
6 y  4 x  2
m1   23 ; m2   23
Section 3.4
perpendicular
neither
parallel