Algebra I Part 1
Unit 3 Day 5 – Rate of Change and Slope
p1
Jose is travelling to his grandmother’s house for a visit. His grandmother lives 450 miles away, and it took him 10
hours to drive there.
a.
What was Jose’s average speed?
b.
If x represents the time and y represents the miles driven, write an equation to represent the
situation described above.
c.
Make a table of x and y values based on your equation.
x
(time)
d.
y
(miles)
Use the points from your table to make a graph.
y
miles



















x

e.











time
How many miles had Jose driven after 3 hours? after 8 hours?
Algebra I Part 1
Unit 3 Day 5 – Rate of Change and Slope
p2
Kendra’s grandmother also lives 450 miles from her house. It took her 9 hours to drive to visit her grandmother.
a.
What was Kendra’s average speed?
b.
If x represents the time and y represents the miles driven, write an equation to represent the
situation described above.
c.
Make a table of x and y values based on your equation.
x
(time)
d.
y
(miles)
Use the points from your table to make a graph.
y
miles



















x

e.











time
Compare the line in Jose’s graph to the line in Kendra’s graph. What differences do you notice?
What similarities do you notice?
Algebra I Part 1
Unit 3 Day 5 – Rate of Change and Slope
p3
The speed in the two problems above can also be called a rate of change because it describes the change in the
distance over time. In a graph, the rate of change is also called the slope and it represents the steepness of the
line. Kendra’s rate of change was faster than Jose’s, and therefore her slope is larger and her line is steeper.
Rate of Change
If x is the independent variable, and y is the dependent variable, then the rate of change is
Ex.
change in y
.
change in x
The table shows how the cost changes with the number of minutes used. Use the table to find the
rate of change. Explain the meaning of the rate of change.
Minutes
Used
X
20
40
60
Cost
($)
Y
1
2
3
In real-world situations, rate of change is often not a constant.
The graph below shows the number of U.S. passports issued in 2002, 2004, and 2006.
a.
b.
Find the rates of change for 2002-2004 and 2004-2006.
Explain the meaning of the rate of change in each case.
12
Passports (millions)
Ex.
10
y
8.9
7.0
8
12.1
6
4
2
2002
c.
How are the different rates of change shown on the graph?
d.
Is this graph linear? Why or why not?
2004
Year
2006
x
Algebra I Part 1
Unit 3 Day 5 – Rate of Change and Slope
Ex.
p4
A. The graph shows the number of airplane departures in the United States in recent years. Find
the rates of change for 1995-2000 and 2000-2005.
A.
B.
C.
D.
B.
1,200,000 per year; 900,000 per year
8,100,000 per year; 9,000,000 per year
900,000 per year; 900,000 per year
180,000 per year; 180,000 per year
Explain the meaning of the slope in each case.
A.
B.
C.
D.
C.
For 1995-2000, the number of airplane
departures increased by about 900,000
flights each year. For 2000-2005, the
number of airplane departures increased by
about 180,000 flights each year.
The rate of change was the same for 1995-2000 and 2000-2005.
The number of airplane departures decreased by about 180,000 for 1995-2000 and
180,000 for 2000-2005.
For 1995-2000 and 2000-2005, the number of airplane departures was the same.
How are the different rates of change shown on the graph?
A.
B.
C.
D.
There is a greater vertical change for 1995-2000 than for 2000-2005. Therefore,
the section of the graph for 1995-2000 has a steeper slope.
They have different y-values.
The vertical change for 1995-2000 is negative, and for 2000-2005 it is positive.
The vertical change is the same for both periods, so the slopes are the same.
A graph is linear only if the rate of change is constant (the same) for all points on the graph.
A positive rate of change indicates that the output values are increasing.
A negative rate of change indicates that the output values are decreasing.
Determine whether each function is linear. Explain.
a.
X
1
2
3
4
Y
6
12
18
24
Use (1, 6) and (2, 12)
change in y 6
 6
change in x 1
Use (2, 12) and (3, 18)
change in y 6
 6
change in x 1
b.
X
-10
-2
6
14
Y
5
1
-4
-10
Use (-10, 5) and (-2, 1)
change in y 4
1


change in x
8
2
Use (-2, 1) and (6, -4)
change in y 5

change in x
8
Use (3, 18) and (4, 24)
change in y 6
 6
change in x 1
All rates of change are 6, so this is linear
The 1st 2 rates of change are
different, so this is not linear
Algebra I Part 1
Unit 3 Day 5 – Rate of Change and Slope
Ex.
Determine if each function is linear. Why or why not?
p5
Determine whether the function is linear. Explain.
X
5
10
15
20
Y
2
4
6
8
X
3
6
12
15
Y
12
16
20
24
Slope:
 The slope of a nonvertical line is the ratio of the change in the y-coordinates (y2 – y1 or y) to the change
in the x-coordinates (x2 – x1 or x) as you move from one point to another.
 It can be used to describe a rate of change.
 Slope describes how steep a line is.
 The greater the absolute value of the slope, the steeper the line.
change in y  coordinates y2  y1 y
rise


slope =
=
( is the math symbol meaning change)
change in x  coordinates x2  x1 x
run

Because a linear function has a constant rate of change, any two points on a nonvertical line can be used to
determine its slope.


The slope of a nonvertical line is the ratio of the rise to the run.
The slope (symbolized by the letter m) of a nonvertical line through any two points (x 1, y1) and (x2, y2) can
be found as follows.
y  y1
m= 2
x2  x1


The slope of a line can be positive, negative, zero, or undefined.
If the line is not horizontal or vertical, then the slope is either positive or negative.
Ex.
Find the slope of the line that passes through each pair of points.
a.
(-3, 2) and (5, 5)
b.
(-3, -4) and (-2, -8)
c.
(-3, 4) and (4, 4)
d.
(-2, 4) and (-2, 3)
Algebra I Part 1
Unit 3 Day 5 – Rate of Change and Slope
Graphs of Slope:
 The graphs of lines with different slopes are summarized below.
Positive Slope
Line slopes up from
left to right.

Negative Slope
Line slopes down from
left to right.
Slope of 0
Horizontal line.
p6
Undefined Slope
Vertical line.
Sometimes you are given the slope and must find a missing coordinate.
Find the value of r so that the line through (6, 3) and (r, 2) has a slope of ½.
y2  y1
x2  x1
23 1

r6 2
1
1

r6 2
cross multiply
-1(2) = 2(r – 6)
-2 = 2r – 12
+12
+ 12
10 = 2r
2
2
5=r
Ex.
Find the value of p so that the line through (p, 4) and (3, -1) has a slope of 
5
.
8