1
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
TASK 2.2.1: AVERAGE RATES OF CHANGE
Solutions
One of the ways in which we describe functions is by whether they are increasing,
decreasing, or constant on an interval in their domain. If the graph of f consistently rises
on the interval then we conclude that the function f is increasing on the interval. If the
graph of f consistently falls on the interval then we conclude that the function f is
decreasing on the interval. If the graph of f remains constant (horizontal) then we
conclude that the function f is constant on the interval.
This can be expressed symbolically as:
•
•
•
A function f is increasing on an interval if for any x1 and x2 in the interval, where
x2 > x1, then f (x2) > f (x1).
A function f is decreasing on an interval if for any x1 and x2 in the interval, where
x2 > x1, then f (x2) < f (x1).
A function f is constant on an interval if for any x1 and x2 in the interval, where x2
> x1, then f (x2) = f (x1).
A single function can be increasing on some intervals, decreasing on others, and constant
on yet others. Consider the function shown below:
f is increasing on the interval (-1,3) because for any x1 and x2 in the interval (-1,3), where
x2 > x1, then f (x2) > f (x1). f is constant on an interval (3,6) because for any x1 and x2 in the
interval (3,6), where x2 > x1, then f (x2) = f (x1). f is decreasing on the interval (6,10)
because for any x1 and x2 in the interval (6,10), where x2 > x1, then f (x2) < f (x1).
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
2
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
It is useful to not only determine whether or not a function is increasing or decreasing
over an interval but also how fast the function is either increasing or decreasing. To
investigate how fast a function is increasing or decreasing over an interval, we calculate
the change in f(x) relative to the change in x over the interval. This yields the average rate
of change over the interval.
To make this notion more mathematically precise we define the average rate of change
of f from c to x as follows:
If c is in the domain of a function y=f(x), the average rate of change of f from c to
x is defined as:
!y f (x) " f (c)
Average rate of change =
=
, x#c
!x
x"c
This expression is also called the difference quotient of f at c.
Graphically, we can see that the average rate of change of f from c to x corresponds to the
slope of the secant line passing through (c, f(c)) and (x, f(x)) as in the figure below:
For a given value of c, the average rate of change of f from c to x is a function of x. Thus,
we can say that for a given value of c, the function rc given by:
!y f (x) " f (c)
=
is a function of f and represents the average rate of change of f
!x
x"c
from c to x for any x not equal to c.
rc (x) =
•
If a function is increasing on the interval (c,x) then the sign of rc is positive.
Why? When f is increasing x>c implies f(x)>f(c) so both (x-c) and (f(x)-f(c)) are
positive.
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
3
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
•
•
If a function is decreasing on the interval (c,x) then the sign of rc is negative.
Why? When f is decreasing x>c implies f(x)<f(c) so (x-c) is positive and (f(x)f(c)) is negative and then their ratio is negative.
If a function is constant on the interval (c,x) then rc (x)=0. Why? When f is
constant x>c implies f(x)=f(c) so (x-c) is positive and (f(x)-f(c)) is zero and then
their ratio is zero.
Note that rc (x)>0 does not imply that f is increasing on (c,x). However, if we find that
rc (x)>0 for any x in the interval (c,d) then we can conclude that f is increasing on the
interval (c,d).
The following illustrates this point:
Consider the function f with graph
f (8) ! f (2) 5 ! 4 1
=
= is positive, but f is not
8! 2
6
6
increasing on the interval (2,8). However, for any x in the interval (2,3), r2 (x) is
positive—which would imply that f is increasing on the interval (2,3).
If we let c=2 and x = 8, then r2 (8) =
Let’s Try This!
( )
Complete the function tables for each of the following functions by finding f xi and
( )
( )
f xi+1 ! f xi
for the given values of xi . You can use the graphing
xi+1 ! xi
calculator to make your calculations by following these steps:
rx (xi+1 ) =
i
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
4
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
•
Enter the x values for the given function in L1.
Press STAT, then ENTER to access the list editor. Enter the xi values in L1.
•
Cursor to list L2 and move the cursor up to highlight the L2 at the top of the
column.
•
Type the function, replacing the x in the function with L1 (2nd 1). For example,
for x + 2, enter L1 + 2.
The screen shots are for Problem 1, f (x) = 2x – 1:
•
Press ENTER. This will calculate the function values and store them in L2.
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
5
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
•
Move the cursor to list L3 and cursor up to highlight the L3 at the top of the
column.
•
Press 2nd STAT (LIST), OPS, ΔList (7)
Type L2 (2nd 2), and close the parenthesis.
•
Press ENTER. This will calculate Δf (x) for each pair of successive function
values in list L2. This is also r(x) since the x values given are such that Δx = 1.
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
6
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
Graph each function on the graphing calculator and sketch the graph on the grid
provided.
Press Y= and enter Y1= 2x -1. Press ZOOM 6.
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
7
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
f (x) = 2x ! 1
1.
i
xi
( )
f xi
rx (xi+1 ) =
( )
( )
f xi+1 ! f xi
xi+1 ! xi
i
0
0
-1
2
1
1
1
2
2
2
3
2
3
3
5
2
4
4
7
2
5
5
9
2
6
6
11
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
8
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
f (x) = !3x + 2
2.
i
xi
( )
f xi
rx (xi+1 ) =
( )
( )
f xi+1 ! f xi
xi+1 ! xi
i
0
0
2
-3
1
1
-1
-3
2
2
-4
-3
3
3
-7
-3
4
4
-10
-3
5
5
-13
-3
6
6
-16
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
9
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
3.
i
f (x) =
1
x!2
2
xi
f xi
( )
rx (xi+1 ) =
( )
( )
f xi+1 ! f xi
xi+1 ! xi
i
0
0
-2
.5
1
1
-1.5
.5
2
2
-1
.5
3
3
-.5
.5
4
4
0
.5
5
5
.5
.5
6
6
1
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
10
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
f (x) = x 2
4.
i
xi
( )
f xi
rx (xi+1 ) =
( )
( )
f xi+1 ! f xi
xi+1 ! xi
i
0
0
0
1
1
1
1
3
2
2
4
5
3
3
9
7
4
4
16
9
5
5
25
11
6
6
36
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
11
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
f (x) = x 3
5.
i
xi
( )
f xi
rx (xi+1 ) =
( )
( )
f xi+1 ! f xi
xi+1 ! xi
i
0
0
0
1
1
1
1
7
2
2
8
19
3
3
27
37
4
4
64
61
5
5
125
91
6
6
216
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
12
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
f (x) = 2 x
6.
i
xi
( )
f xi
rx (xi+1 ) =
( )
( )
f xi+1 ! f xi
xi+1 ! xi
i
0
0
1
1
1
1
2
2
2
2
4
4
3
3
8
8
4
4
16
16
5
5
32
32
6
6
64
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
13
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
Reflect and Summarize
1.
What observations can you make about the average rate of change of linear
functions compared to the average rate of change of non-linear functions?
Linear functions have a constant average rate of change. Non-linear functions do
not have a constant average rate of change.
2.
Using the average rate of change of a function of f from c to x, rc (x) =
f (x) ! f (c)
,
x!c
and the slope-intercept form of the equation of a line, y = mx + b, prove that the
slope, m, is the average rate of change of the function on any interval x1 , x2 .
(
)
The rate of change of f from x1 to x2 is given by
( )
rx x2 =
1
( )
( ) for x
f x2 ! f x1
x2 ! x1
1
" x2
()
Since f x = mx + b, we can express
( )
f (x1 ) = mx1 + b and f (x2 ) = mx2 + b, and substitute into rx x2 , giving
1
( )
rx x2 =
1
( mx
2
) (
+ b ! mx1 + b
)
x2 ! x1
Simplifying the numerator, we have
( )
rx x2 =
1
mx2 + b ! mx1 ! b mx2 ! mx1 + b ! b mx2 ! mx1
=
=
x2 ! x1
x2 ! x1
x2 ! x1
factoring out the m, we have
( )
rx x2 =
1
(
m x2 ! x1
x2 ! x1
) = m for x
" x2
1
Since x1 and x2 had no other restrictions other than x1 " x2 ,
we have shown that the average rate of change of f from
x1 to x2 , for any x1 and x2 is m.
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
14
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
Math notes
This task builds toward the understanding that the average rate of change of a linear
function over any interval is constant and is equal to the slope of the linear function. We
learn later in calculus that the slope of a linear function also corresponds to the
instantaneous rate of change at any given point.
Teaching notes
It is helpful to introduce this task by sketching several graphs of functions and asking
participants to note that parts of the graphs are rising, parts are falling, and parts are
horizontal and that we have a mathematical way of describing this. Remind participants
that we are examining the change in y over the change in x when we calculate values
f (x) ! f (c)
for rc (x) =
. Have participants work in individually initially and then move
x!c
them into groups to compare findings. Assign each group one of the questions 1-6 to
present on chart paper. After group presentations of questions 1-6, have the participants
work in groups on Reflect and Summarize task. After hearing participant responses to
question 2 of the Reflect and Summarize task, the facilitator should have participants
think about how they might describe the instantaneous rate of change of a linear function
by building from their work that for an arbitrary interval (c,d) ( c ! d ) the average rate of
change (slope of the secant line through (c,f(c)) and (d, f(d)) is always constant and equal
to the slope of the linear function. Have the participants graphically determine the
instantaneous rate of change as dc.
Extensions
For each function in exercises 1-5, have participants derive a general formula for the
f (x) ! f (c)
function rc (x) =
. The facilitator may foreshadow calculus concepts of
x!c
instantaneous rates of change (derivatives) and limits by asking participants what the
“limiting” function rc (x) would be as we consider smaller and smaller intervals (c,x).The
facilitator may encourage participants to rewrite the function rc (x) in terms of the change
f (c + !x) " f (c)
in x: !x = x " c . They would then examine rc ( c + !x ) =
.
!x
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
15
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
TASK 2.2.1: AVERAGE RATES OF CHANGE
TASK 2.2.1: AVERAGE RATES OF CHANGE
One of the ways in which we describe functions is by whether they are increasing,
decreasing, or constant on an interval in their domain. If the graph of f consistently rises
on the interval then we conclude that the function f is increasing on the interval. If the
graph of f consistently falls on the interval then we conclude that the function f is
decreasing on the interval. If the graph of f remains constant (horizontal) then we
conclude that the function f is constant on the interval.
This can be expressed symbolically as:
•
•
•
A function f is increasing on an interval if for any x1 and x2 in the interval, where
x2 > x1, then f (x2) > f (x1).
A function f is decreasing on an interval if for any x1 and x2 in the interval, where
x2 > x1, then f (x2) < f (x1).
A function f is constant on an interval if for any x1 and x2 in the interval, where x2
> x1, then f (x2) = f (x1).
A single function can be increasing on some intervals, decreasing on others, and constant
on yet others. Consider the function shown below:
f is increasing on the interval (-1,3) because for any x1 and x2 in the interval (-1,3), where
x2 > x1, then f (x2) > f (x1). f is constant on an interval (3,6) because for any x1 and x2 in the
interval (3,6), where x2 > x1, then f (x2) = f (x1). f is decreasing on the interval (6,10)
because for any x1 and x2 in the interval (6,10), where x2 > x1, then f (x2) < f (x1).
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
16
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
It is useful to not only determine whether or not a function is increasing or decreasing
over an interval but also how fast the function is either increasing or decreasing. To
investigate how fast a function is increasing or decreasing over an interval, we calculate
the change in f(x) relative to the change in x over the interval. This yields the average rate
of change over the interval.
To make this notion more mathematically precise we define the average rate of change
of f from c to x as follows:
If c is in the domain of a function y=f(x), the average rate of change of f from c to
x is defined as:
!y f (x) " f (c)
Average rate of change =
=
, x#c
!x
x"c
This expression is also called the difference quotient of f at c.
Graphically, we can see that the average rate of change of f from c to x corresponds to the
slope of the secant line passing through (c, f(c)) and (x, f(x)) as in the figure below:
For a given value of c, the average rate of change of f from c to x is a function of x. Thus,
we can say that for a given value of c, the function rc given by:
!y f (x) " f (c)
=
is a function of f and represents the average rate of change of f
!x
x"c
from c to x for any x not equal to c.
rc (x) =
•
•
If a function is increasing on the interval (c,x) then the sign of rc is positive.
Why?
If a function is decreasing on the interval (c,x) then the sign of rc is negative.
Why?
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
17
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
•
If a function is constant on the interval (c,x) then rc (x)=0. Why?
Note that rc (x)>0 does not imply that f is increasing on (c,x). However, if we find that
rc (x)>0 for any x in the interval (c,d) then we can conclude that f is increasing on the
interval (c,d).
The following illustrates this point:
Consider the function f with graph
f (8) ! f (2) 5 ! 4 1
=
= is positive, but f is not increasing on
8! 2
6
6
the interval (2,8). However, for any x in the interval (2,3), r2 (x) is positive—which would
imply that f is increasing on the interval (2,3).
If we let c=2, then r2 (8) =
Let’s Try This!
( )
Complete the function tables for each of the following functions by finding f xi and
( )
( )
f xi+1 ! f xi
for the given values of xi . You can use the graphing
xi+1 ! xi
calculator to make your calculations by following these steps:
rx (xi+1 ) =
i
•
•
•
•
Enter the x values for the given function in L1.
Cursor to list L2 and move the cursor up to highlight the L2 at the top of the
column.
Type the function, replacing the x in the function with L1 (2nd 1). For example,
for x + 2, enter L1 + 2.
Press ENTER. This will calculate the function values and store them in L2.
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
18
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
•
•
•
•
Move the cursor to list L3 and cursor up to highlight the L3 at the top of the
column.
Press 2nd STAT (LIST), OPS, ΔList (7)
Type L2 (2nd 2), and close the parenthesis.
Press ENTER. This will calculate Δy for each pair of successive function values
in list L2. This is also rx x2 since the x values given are such that Δx = 1.
i
( )
Graph each function on the graphing calculator and sketch the graph on the grid
provided.
f (x) = 2x ! 1
1.
i
xi
( )
f xi
rx (xi+1 ) =
i
0
0
1
1
2
2
3
3
4
4
5
5
6
6
( )
( )
f xi+1 ! f xi
xi+1 ! xi
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
19
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
f (x) = !3x + 2
2.
i
xi
( )
f xi
rx (xi+1 ) =
i
0
0
1
1
2
2
3
3
4
4
5
5
6
6
( )
( )
f xi+1 ! f xi
xi+1 ! xi
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
20
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
f (x) =
3.
i
xi
1
x!2
2
( )
f xi
rx (xi+1 ) =
i
0
0
1
1
2
2
3
3
4
4
5
5
6
6
( )
( )
f xi+1 ! f xi
xi+1 ! xi
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
21
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
f (x) = x 2
4.
i
xi
( )
f xi
rx (xi+1 ) =
i
0
0
1
1
2
2
3
3
4
4
5
5
6
6
( )
( )
f xi+1 ! f xi
xi+1 ! xi
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
22
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
f (x) = x 3
5.
i
xi
( )
f xi
rx (xi+1 ) =
i
0
0
1
1
2
2
3
3
4
4
5
5
6
6
( )
( )
f xi+1 ! f xi
xi+1 ! xi
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
23
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
f (x) = 2 x
6.
i
xi
( )
f xi
rx (xi+1 ) =
i
0
0
1
1
2
2
3
3
4
4
5
5
6
6
( )
( )
f xi+1 ! f xi
xi+1 ! xi
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
24
Algebra II: Strand 2. Linear Functions; Topic 2. Slope and Rate of Change; Task 2.2.1
Reflect and Summarize
1.
What observations can you make about the average rate of change of linear
functions compared to the average rate of change of non-linear functions?
2.
Using the average rate of change of a function of f from c to x, rc (x) =
f (x) ! f (c)
,
x!c
and the slope-intercept form of the equation of a line, y = mx + b, prove that the
slope, m, is the average rate of change of the function on any interval x1 , x2 .
(
)
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.