Proportionality
Connecting proportional relationships to slope
Analyze linear equation y=mx
Analyze the y-intercept of a line
Analyze linear equation y=mx+b
100
A student purchases notebooks for school. Each notebook costs $3. What is the constant of proportionality?
Notebooks(x) Cost (y)
1 $3
2 $6
3 $9
k=3
or
y=3x
100
A student walks 3 miles every hour.
What is the constant of proportionality?
What is the slope?
Time (x) in hours Distance (y) in miles
1 3
2 6
3 9
4 12
k=3
m=3
y=3x
100
A line is represented by the equation:
y=3x
What is the slope of the line?
m=3
100
A line is represented by the equation: y=2x+4
What is the y-intercept?
b=4
100
A line is represented by the equation: y=3x+5.
What is the slope (m)?
What is the y-intercept (b)?
m=3
b=5
200
A cyclist travels 12 miles in 2 hours. Assume the cyclist travels at a constant speed.
What is the unit rate?
6 miles in 1 hour
200
A proportional relationship is represented by the equation: y=5x
What is the slope?
m=5
200
A student earns $6 for every hour of babysitting. The relationship is represented by:
y=6x
How much money will the student earn in 4 hours?
y=6(4)
y=$24
200
A line passes through the points (0, 6) and (3, 0).
What is the y-intercept?
b=6
200
A line is represented by: y=4x+2.
What is the rate of change?
What is the starting value?
rate of change= 4
starting value=2
300
Determine whether the relationship in the table is proportional. Explain.
x y
1 4
2 8
3 12
4 18
The table is not proportional because 1, 2, and 3 the constant is 4. The 4 should be 16.
300
A water tank is being filled at a constant rate. The graph shows the amount of water in the tank over time. What is the slope?
m=5
300
Two students are saving money.
Student A: y=4x
Student B: y=7x
How much greater is Student B's savings after 5 weeks?
Student A: y=4(5)= $20
Student B: y=7(5)= $35
Student B saves $15 more.
300
Mason has $30 saved toward purchasing a video game. He saves $5 each week.
Write the y-intercept equation.
y=5x+30
300
A student has $15 saved. She saves an additional $8 each week toward a new pair of shoes.
The equation is: y=8x+15.
How much money will she have saved after 5 weeks?
y=8(5)+15
y=$55
400
A movie theater charges $7 for each movie ticket. The relationship between the number of tickets (x) and the total cost (y) is represented by: y=7x
How much would 6 tickets cost?
$42
400
Two students are saving money for a new video game.
Student A: Saves $6 each week.
Student B: The table shows the amount saved.
How much more money will the student with the greater slope save after 5 weeks?
Student B has the greater slope of 8. Student B saved $10 more per week than Student A.
y=8(5)
y=$40
400
A line represents the distance a student travels on a bicycle.
Write the equation of the line.
y=4x
400
A water tank contains 40 gallons of water. The amount of water decreases by 5 gallons every hour.
The line passes through (0, 40) and (8, 0).
Write the y-intercept equation.
y=-5x+40
400
A water tank contains 60 gallons of water. Water drains at a rate of 5 gallons per minute.
The equation is: y=-5x+60.
How much water remains after 6 minutes?
y= -5(6)+60
y=-30+60
y=30 gallons
500
Two students are earning money for helping their neighbors.
Student A: Earns $8 for every hour worked.
Student B: The table shows the amount of money earned.
Which student earns money at a greater rate?
Hours worked Money earned
1 $10
2 $20
3 $30
4 $40
Student B earns money at a higher rate. ($2 more per hour.)
500
A student says:
"The slope and the constant of proportionality are always the same in a proportional relationship."
Consider the equation:
y=4x
What would happen to the graph if the equation changed to y=4x+3? Would it still represent a proportional relationship? Explain.
It would not represent a proportional relationship because it doesn't pass through the origin (0,0).
500
A student is given the equation:
y=5x
The student makes the following statements:
The slope is 5.
The y-intercept is 5.
The line passes through (0, 0).
The equation represents a proportional relationship.
Which statements are correct?
Correct:
1. m=5
3. (0,0)
4. proportional relationship
500
Two students are saving money for a new bicycle.
Student A's equation is y=10x+20.
Student B's equation is y=5x+50.
Which student started with the most money?
Student B started with the most money, $50.
500
Two students are saving money for a new bicycle.
Student A: y=6x+20
Student B: y=10x+5
Which student will have more money after 5 weeks?
Student A: y=6(5)+20
y=$50
Student B: y=10(5)+5
y=$55
Student B will have more money after 5 weeks.