Transcript y - Sun Valley Charter School

Unit 3 ~ Lesson 4B
Proportional Relationships
Warm-up
Solve
s
1. - 7= - 7
3
2. 74=14-5k
3. 8x +3=- 35
r
4. 9+ =15
-5
Proportional Relationships are represented in the following ways:

Verbal Descriptions

d = 50t
A truck travels at a constant
rate of speed and is traveling
50 miles every hour. How far
would it travel in 5 hours?

Tables
Equations
d = total distance
r = 50 mph
t = hours driven
Time t (hours)
0
1
2
….
5
x - axis
Distance d (in miles)
0
50
100
….
?
y - axis
y
Graphs
Distance is on the y - axis
Miles

300
250
200
150
100
50
0
0
Use the data
from the table
to plot the
coordinates.
(x, y)
x
2
4
6
Hours
8
Time is on the x - axis
Now solve the problem for distance traveled in 5 hours.
Verbal Descriptions

A truck travels at a constant
rate of speed and is traveling
50 miles every hour. How far
would it travel in 5 hours?

Solution:
50 miles • 5 hours = 250 miles
Equations
d = 50t
d = total distance
r = 50 mph
t = hours driven
d = 50 miles • 5 hours
d = 250 miles
Now solve the problem for distance traveled in 5 hours.

Tables
Time t (hours)
0
1
2
….
5
x - axis
Distance d (in miles)
0
50
100
….
250
?
y - axis
Reading the table: what did you multiply “ x ” by to get “ y ”?
x • 50
So multiply 5 by 50
y
Graphs
Miles

300
250
200
150
100
50
0
0
Use the data
from the table
to plot the
coordinates.
(x, y)
x
2
4
6
8
Hours
Reading the graph: locate 5 hours on the x-axis and
estimate the number of miles based on the graph.
Verbal Descriptions & Equations
You can use an equation to represent a verbal description of a
proportional relationship.
Equation Format:
y = kx
k is the constant of proportionality or unit rate
 Amy charges $10 an hour to babysit.
 Write an equation in this format:
Assign values
to the variables
y = kx
y = total
k = constant of proportionality (unit rate)
x = quantity
y = total $ earned
k = hourly rate
x = time (hours)
 Equation: y = x  $10 or y = $10x or c = $10h
So what is the constant of
proportionality again?
It is the constant rate of change
or UNIT RATE: $10 per hour
Recall the first example: d = 50t
It was also written in the same
format: k is the UNIT RATE
because it is 50 miles per hour
Equations & Constant of Proportionality
Write an equation for each situation below & identify the constant
of proportionality.
Equation Format: y = kx
Assign values
to the variables
y = total
k = constant of proportionality (unit rate)
x = quantity
1. The football ticket cost is $7.
y = total ticket sales
k = $7
x = # of tickets sold
y = $7x
2. The state playoff football ticket cost $12.
y = total ticket sales
k = $12
x = # of tickets sold
y = $12x
3. Andrew charges $9.50 an hour to cut grass.
y = total income
k = $9.50
x = # of hours worked
y = $9.5x
For these equations to be
directly proportional,
k & x must be multiplied!
Equations & Constant of Proportionality
Write an equation for each situation below & identify the constant
of proportionality.
Equation Format: y = kx
Assign values
to the variables
y = total
k = constant of proportionality (unit rate)
x = quantity
4. My car gets 26 miles per gallon.
y = total distance
k = 26
x = # of gallons of gas
y = 26x
For these equations to be
directly proportional,
k & x must be multiplied!
5. The height of the building is equal to the number of floors times
12 feet.
y = total height of the building
k = 12
x = # of floors
y = 12x
6. Write your own situation that could be represented by the
equation y = 35x. State what x, y, and 35 represent in your
problem and explain how you know that the problem represents a
proportional relationship.
Name the Constant of Proportionality
7.
y = -9x
8.
0.4x = y
9.
y = -x
Tables
10. This table displays the rate at which water is flowing from a faucet into a
bathtub.
Write an equation. _________________
How many gallons are in the bathtub in 4 minutes? ______
How many gallons of water would be in the bathtub in10 minutes? _______
Does it have a proportional relationship? ________ Why or Why Not?
Time m (minutes)
0
1
2
3
4
x - axis
Gallons g (in gallons)
0
2
4
6
?
y - axis
*Assign x and y values on the table
*What did you multiply “ x ” by to get “ y ”?
Tables
11. A cheetah runs 87 meters in 3 seconds.
Write an equation. ______________
How far can it run in 10 seconds? _____
How far can it run in 1second? _______
Does it have a proportional relationship? ________ Why or Why Not?
Time s (seconds)
3
4
5
….
10
Distance m (in meters)
87
116
145
….
?
Tables & Graphs…..complete the table & graph the results
Time m (minutes)
0
1
2
3
4
x - axis
Gallons g (in gallons)
0
2
4
6
8?
y - axis
10
Which point on the graph shows
the unit rate?

8
Gallons
12.
HINT: When x is 1, y is the unit rate.

6
(1, 2)

4
What is the unit rate?
2

2

0
2
4
Minutes
6
8
Tables & Graphs…..complete the table & graph the results
Time s (seconds)
3
4
5
….
10
x - axis
Distance m (in meters)
87
116
145
….
290
?
y - axis
Meters
13.

290
261
232
202
174
145
116
87
58
29
0


2



Which point on the graph
shows the unit rate?
(1, 29)

What is the constant
of proportionality?
29
4
6
Seconds
8
10
Tables & Graphs…..complete the table & graph the results
14. The equation y = 7.50x relates to the number of hours worked at Ingles and
the total amount of money earned. Use this equation to complete the
table below.
Time h (hours)
0
1
2
3
4
5
6
Income d (in dollars)
0
7.5
15
22.5
30
37.5
45
50

Income
40
10
0
In 4 hours, you will
earn $30

30
20
What does the point
(4, 30) represent?

2
4
Hours
6
8
Proportional Relationships
A graph shows a directly proportional relationship if it is linear (straight line)
that passes through or touches the origin (0,0).
Do the graphs below show a proportional relationship?
15.
16.
17.
18.