Pre-Algebra Notes – Unit 15: Graphs and Functions II
Using Graphs to Find Rates of Change
CCSS 8.EE.5-1: Graph proportional relationships, interpreting the unit rate as the slope of
the graph.
A linear relationship expresses a constant rate of change. A proportional relationship expresses
a constant rate of change in which an input of 0 results in the output of 0. That is, a
proportional relationship is a special type of linear relationship.
Example: Darla keeps a record of the number of hours she babysits and the money she
earns.
Number of hours
Amount earned ($)
Discuss:
Week 1
1
10
Week 2
3
30
Week 3
6
60
Week 4
8
80
What is the input variable? (Number of hours)
What is the output variable? (Amount earned)
Find the rate of change:
Week 1 to Week 2
change in $
30 − ? ?
=
= = ?
change in hours 3 − 1 2
Week 2 to Week 3
change in $
? − 30 ?
=
= = ?
change in hours 6 − ? 3
Week 3 to Week 4
change in $
? −? ?
=
= = ?
change in hours ? − 6 2
Are the rates of change constant or variable?
(constant)
y
100
Make a graph that shows the money earned over the number of
hours.
Does the relationship between the number of hours worked and
the amount of money earned appear to be a proportional
relationship? (yes as the graph appears to be a straight line
that goes through the origin)
50
0
McDougal Littell, Chapter 1, Sections 8
Chapter 8, Sections 1-6
HS Pre-Algebra Unit 15: Graphs and Functions II
Revised 2013 - CCSS
5
10
Page 1 of 11
x
Find the unit rate:____________
($10 per hour—can look at graph and see the point (1, 10 ) )
Compare the rate of change to the unit rate. _______________________ (They are the same.)
We know that slope is the
rise
,
run
y
50
40
30
20
run of 1
10
rise of 10
1
0
2
3
4
5
x
Sample Question from CCSD semester exam:
Catherine charges $10 per hour to babysit.
Which graph shows how much she will earn
for a given time worked?
(A) Total earned ($)
60
(C) Total earned ($)
50
60
40
50
30
40
20
30
10
20
0
1
2
3
4
5
6 Hours worked
10
0
(B) Total earned ($)
60
2
3
4
5
6 Hours worked
3
4
5
6 Hours worked
(D) Total earned ($)
50
60
40
50
30
40
20
30
10
20
0
1
1
2
3
4
5
6 Hours worked
10
0
McDougal Littell, Chapter 1, Sections 8
Chapter 8, Sections 1-6
1
HS Pre-Algebra Unit 15: Graphs and Functions II
Revised 2013 - CCSS
2
Page 2 of 11
CCSS 8.EE.5-2: Compare two different proportional relationships represented in different
ways.
Example: Determine which train is traveling at a faster speed.
Train 1: Train 1’s distance is recorded in the table.
Time (hr)
1
3
6
10
Distance from station (mi) 75 225 450 750
Train 2:
Leaves the station at a time t, and its
distance from the station is
d = 65t .
Answer: Train 1’s rate of change (slope) is 75 miles per hour. Train 2’s rate of change is 65
miles per hour. Therefore, Train 1 is traveling faster.
Sample Question from CCSD semester exam:
The table below shows the number of shoes machine A has assembled.
Number of
Time (h)
Shoes Assembled (s)
8 a.m.
0
10 a.m.
40
noon
80
2 p.m.
120
4 p.m.
160
6 p.m.
200
Machine B’s performance can be modeled by s = 35h .
Which machine can produce more shoes per hour?
(A)
Machine A
(B)
Machine B
(C)
Both machines can produce at least 40 pairs of shoes per hour.
(D)
It cannot be determined from the information given.
Exploring Slope
McDougal Littell, Chapter 1, Sections 8
Chapter 8, Sections 1-6
HS Pre-Algebra Unit 15: Graphs and Functions II
Revised 2013 - CCSS
Page 3 of 11
CCSS 8.EE.6-1: Use similar triangles to explain why the slope m is the same between any two
distinct points on a non-vertical line in the coordinate plane.
Example: You can create many right triangles using two points on the graph of a line.
y
10
The lengths of the legs of each triangle
is given. The length of the
hypotenuse could be found using the
Pythagorean Theorem.
E
5
D
AFD=
AF 9=
FD 12=
AD 15
C
BJD=
BJ 6=
JD 8=
BD 10
-5
-10
J
5
10
x
B
CJD=
CJ 3=
JD 4 =
CD 5
-5
BHF=
BH 9=
HF 12=
BF 15
G
H
A
F
-10
Use the side lengths of the triangles to
determine whether the triangles
formed with the line are similar.
Are
AFD and
BJD similar?
(Yes; their sides are proportional)
Are
BJD and
CJD similar?
(Yes; their sides are proportional)
Are
CJD and
BHF similar?
(Yes; their sides are proportional)
Sample Question from CCSD semester exam:
Use the diagram below.
McDougal Littell, Chapter 1, Sections 8
Chapter 8, Sections 1-6
HS Pre-Algebra Unit 15: Graphs and Functions II
Revised 2013 - CCSS
Page 4 of 11
y
A
C
B
x
E
D
Choose the two statements must be true.
I. The slope of AC is the same as the slope of CE .
II. The length of AC is half the length of CE .
III. ∆ABC  ∆CDE
IV. The area of ∆ABC is half the area of ∆CDE .
(A) I and III
(B) I and IV
(C) II and III
(D) II and IV
Deriving the Slope-Intercept Formula
CCSS 8.EE.6-2: Derive the equation y = mx for a line through the origin and equation
=
y mx + b for a line intercepting the vertical axis at b.
Let l be a line with slope m and y-intercept b.
We know a point that must be on the line is ( 0, b ) (coordinates of the y-intercept).
McDougal Littell, Chapter 1, Sections 8
Chapter 8, Sections 1-6
HS Pre-Algebra Unit 15: Graphs and Functions II
Revised 2013 - CCSS
Page 5 of 11
Let ( x, y ) be a different point on l.
Slope is the ratio of change in y to change in x.
So m =
y −b
.
x−0
We want y by itself on one side of the equation:
y −b
x−0
y −b
y=
x
y −b
⋅x
m ⋅=
x
x
mx= y − b
mx + b = y − b + b
mx + b =
y
or
=
y mx + b
m=
slope
simplify the denominator
multiply both sides by x
simplify
add b to both sides of the equation
simplify
rewrite equation with y on the left
Sample Question from CCSD semester exam:
12 .
Find the y-intercept of the line with equation 2 x + 3 y =
(A) (0, 3)
(B) (0, 4)
(C) (2, 3)
(D) (6, 0)
McDougal Littell, Chapter 1, Sections 8
Chapter 8, Sections 1-6
HS Pre-Algebra Unit 15: Graphs and Functions II
Revised 2013 - CCSS
Page 6 of 11
Writing Equations to Describe Functions
CCSS 8.F.4-1: Construct a function to model a linear relationship between two quantities.
(This example is taken from Arizona Academic Content Standards (2010), Explanations and
Examples.)
The table below shows the cost of renting a car. The company charges $45 a day for the car as
well as charging a one-time $25 fee for the car’s navigation system (GPS). Write an expression
for the cost in dollars, c, as a function of the number of days, d.
Students might write the equation c = 45d + 25 using the verbal description or by first making a
table.
Days (d)
1
2
3
4
Cost (c) in dollars
70
115
160
205
Students should recognize that the rate of change is 45 (the cost of renting the car) and that initial
cost (the first day charge) also includes paying for the navigation system. Classroom discussion
about one time fees vs. recurrent fees will help students model contextual situations.
Sample Question from CCSD semester exam:
Coach is buying trophies and jackets for his baseball team. The jackets cost $45 each. The trophies cost $7
each with an initial set up fee of $12.
Let x represent the number of players on the team and y represent the total cost of jackets and trophies.
Choose the function that best describes the total cost the coach will spend based on the number of players he
has.
(E) y = 7x + 12
(F) y = 38x + 12
(G) y = 45x + 12
(H) y = 52x + 12
CCSS 8.F.4-2: Determine the rate of change and initial value of the function from a
description of a relationship or from two ( x, y ) values, including reading these from a
table or from a graph.
Example, writing an equation from a table.
Use table in example above.
Choose any two ordered pairs from the table to find the slope.
McDougal Littell, Chapter 1, Sections 8
Chapter 8, Sections 1-6
HS Pre-Algebra Unit 15: Graphs and Functions II
Revised 2013 - CCSS
Page 7 of 11
y2 − y1 160 − 115 45
=
=
x2 − x1
3− 2
1
=
m
Use =
y mx + b and a point from the table.
=
y mx + b
=
205 45(4) + b
205
= 180 + b
25 = b
Using the slope and y-intercept values to write and equation:
=
y 45 x + 25
Example, writing an equation from a graph.
y
Use the graph to the right.
5
First, determine the y-intercept.
The line intersects the y-axis at 1.
-5
Next, choose 2 points on the graph. Determine
the slope.
Count 2 units up and 1unit to the right….slope
2
is or 2.
1
5
x
-5
y mx + b , we have the equation for the line
Using =
to be =
y 2 x + 1.
Sample Question from CCSD semester exam:
The table below shows the charges for a taxi cab fare.
Miles driven (m)
0
5
10
15
20
Total cost (C)
$3.00
$10.50
$18.00
$25.50
$33.00
Choose the equation that models this situation.
(A) C = 5m + 7.50
(B) C = 5m + 10.50
(C) C = 1.50m + 3
(D) C = 1.50m + 33
McDougal Littell, Chapter 1, Sections 8
Chapter 8, Sections 1-6
HS Pre-Algebra Unit 15: Graphs and Functions II
Revised 2013 - CCSS
Page 8 of 11
CCSS 8.F.4-3: Interpret the rate of change and initial value of a linear function in terms of
the situation it models, and in terms of its graph or a table of values.
The constant rate of change of a line is called the slope of a line. If we determine the slope of the
line, then we should be able to explain what it shows. For instance, let’s create a table that
shows how much money a student earned by babysitting.
Time (h)
2
4
6
8
Money earned
$10
$20
$30
$40
We could make a graph and then determine the slope of
the line. Or we could look at the rate of change in the
chart. For every $10 earned, the student worked 2
hours. We could also conclude that the student was
earning $5 for each hour worked.
Sample Question from CCSD semester exam:
The equation h = 1.25t + 6 models the growth of a plant under the sunlight. The total height of the plant in
millimeters is represented by h and t is the number of hours the plant was in the sunlight. Interpret what the
slope 1.25 represents in this model.
(A) The height of the plant started at 1.25 mm.
(B) Every hour in the sunlight the plant grows 1.25 mm.
(C) The plant costs $1.25.
(D) Every 1.25 hours in the sunlight the plant will grow 6 mm.
CCSS 8.F.5: Describe qualitatively the functional relationship between two quantities by
analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear).
Sketch a graph that exhibits the qualitative features of a function that has been described
verbally.
Qualitative graphs are graphs used to represent situations that many not have numerical values or
graphs in which numerical values are not included.
One can start by supplying a table with data, asking some questions (that are answered by
looking at the table), then actually graph and connect the points. Look again at what was
happening at key places on the graph. For instance, let’s consider Mario slowly walking so that
students can record his time in minutes and distance in feet.
Time
(min)
Distance
Walked
(feet)
0
1
2
3
4
5
6
7
0
10
25
25
40
40
65
100
McDougal Littell, Chapter 1, Sections 8
Chapter 8, Sections 1-6
HS Pre-Algebra Unit 15: Graphs and Functions II
Revised 2013 - CCSS
Page 9 of 11
1.
During what period(s) of time did his distance not change? What
does that mean he was doing?
2. During which period of time did his distance change the most?
What does that tell you about his speed in walking?
Graph and connect the ordered pairs. Describe verbally what was
happening in each interval without looking at the table.
y
100
90
80
70
60
50
40
30
20
10
1
2
3
4
5
6
7
x
Sample Question from CCSD semester exam:
The outdoor temperature changes throughout the day. Early in the morning, temperature increases slowly. At 11
a.m. the temperature rises sharply. During the afternoon, the temperature stays the same for several hours. As
night falls, the temperature decreases slightly. Choose the graph that best represents this situation.
Temperature
Temperature
(A)
(B)
Hours
Hours
McDougal Littell, Chapter 1, Sections 8
Chapter 8, Sections 1-6
HS Pre-Algebra Unit 15: Graphs and Functions II
Revised 2013 - CCSS
Page 10 of 11
(C)
Temperature
Temperature
(D)
Hours
Hours
McDougal Littell, Chapter 1, Sections 8
Chapter 8, Sections 1-6
HS Pre-Algebra Unit 15: Graphs and Functions II
Revised 2013 - CCSS
Page 11 of 11