Lesson 5: Analyzing Proportional and Non-Proportional Relationships in Graphs
- A proportional relationship can be shown in a graph.
- The graph of a proportional relationship is always a straight line through the origin (0, 0).
- If the graph is not straight or does not go through the origin, it is not proportional.
Part B
Opening Exercise. Isaiah sells candy bars.
Isaiah receives money for each candy bar he sells.
| Candy Bars Sold | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Money ($) | 2 | 4 | 6 | 8 |
| Ratio | 2÷1=2 | 4÷2=2 | 6÷3=2 | 8÷4=2 |
Step 1: The ratios are all equal to 2.
Step 2: This means Isaiah gets $2 for each candy bar.
Step 3: If we plot the points on a graph, they will line up and pass through (0, 0).
Answer: The relationship is proportional. The constant is $2 per candy bar.
Example 1. A toy company ships boxes.
Each box holds 24 toy cars.
| Boxes | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cars | 24 | 48 | 72 | 96 |
| Ratio | 24÷1=24 | 48÷2=24 | 72÷3=24 | 96÷4=24 |
Step 1: Each ratio = 24.
Step 2: The graph will be a straight line through (0, 0).
Answer: Proportional, constant = 24 cars per box.
Example 2. Comparing two graphs.
Graph A shows y = 3x (a straight line through the origin).
Graph B shows y = 3x + 2 (a straight line but not through the origin).
- Graph A is proportional because it passes through the origin.
- Graph B is not proportional because it crosses the y-axis at (0, 2), not at (0, 0).
Answer: Graph A is proportional, Graph B is not.
Example 3. A strawberry farm.
| Hours | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Pounds Picked | 8 | 16 | 24 | 30 |
| Ratio | 8÷1=8 | 16÷2=8 | 24÷3=8 | 30÷4=7.5 |
Step 1: The first three ratios equal 8, but the last is 7.5.
Step 2: The graph will not be a perfect straight line through (0, 0).
Answer: Not proportional.
Part C. Extra Practice Problems
5. A car rental company charges $50 per day.
| Days | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost ($) | 50 | 100 | 150 | 200 |
| Ratio | 50÷1=50 | 100÷2=50 | 150÷3=50 | 200÷4=50 |
Answer: Proportional, constant = $50 per day
6. A water tank fills at 5 liters per minute.
| Minutes | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Liters | 5 | 10 | 15 | 20 |
| Ratio | 5÷1=5 | 10÷2=5 | 15÷3=5 | 20÷4=5 |
Answer: Proportional, constant = 5 liters/minute
7. Taxi fare: $3 starting fee, plus $2 per mile.
| Miles | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost ($) | 5 | 7 | 9 | 11 |
| Ratio | 5÷1=5 | 7÷2=3.5 | 9÷3=3 | 11÷4=2.75 |
Answer: Not proportional (line does not go through origin)
8. A baker uses 2 cups of sugar for every 3 cups of flour.
| Flour | 3 | 6 | 9 | 12 |
|---|---|---|---|---|
| Sugar | 2 | 4 | 6 | 8 |
| Ratio | 2÷3=0.67 | 4÷6=0.67 | 6÷9=0.67 | 8÷12=0.67 |
It is a straight line through the origin, showing a proportional relationship: 2 cups of sugar are always used for every 3 cups of flour.
Answer: Proportional, constant = 2/3 = 0.67
9. A plumber charges $40 per hour.
| Hours | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost | 40 | 80 | 120 | 160 |
| Ratio | 40÷1=40 | 80÷2=40 | 120÷3=40 | 160÷4=40 |
It is a straight line through the origin, showing a proportional relationship: the cost increases by $40 for each additional hour.
Answer: Proportional, constant = $40/hour
10. A babysitter charges $30 plus $10 per hour.
| Hours | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost | 40 | 50 | 60 | 70 |
| Ratio | 40÷1=40 | 50÷2=25 | 60÷3=20 | 70÷4=17.5 |
It is a straight line but does not pass through the origin, because there is a $30 starting fee. This makes it a linear but non-proportional relationship.
Answer: Not proportional
11. A runner jogs 6 miles each hour.
| Hours | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Miles | 6 | 12 | 18 | 24 |
| Ratio | 6÷1=6 | 12÷2=6 | 18÷3=6 | 24÷4=6 |
It is a straight line through the origin and proportional: the runner jogs 6 miles every hour at a constant rate.
Answer: Proportional, constant = 6 miles/hour
12. A pool company charges $200 plus $50 per hour.
| Hours | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost | 250 | 300 | 350 | 400 |
| Ratio | 250÷1=250 | 300÷2=150 | 350÷3≈116.7 | 400÷4=100 |
It is a straight line but shifted upward, starting at $200 when hours = 0. Therefore, it is linear but non-proportional.
Answer: Not proportional
13. A fruit stand sells 5 bananas for $1.
| Bananas | 5 | 10 | 15 | 20 |
|---|---|---|---|---|
| Cost $ | 1 | 2 | 3 | 4 |
| Ratio | 1÷5=0.2 | 2÷10=0.2 | 3÷15=0.2 | 4÷20=0.2 |
It is a straight line through the origin, and proportional: 5 bananas always cost $1, so the cost increases at a constant rate.
Answer: Proportional, constant = $0.20 per banana
14. Movie tickets cost $12 each, plus a $5 service fee.
| Tickets | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost $ | 17 | 29 | 41 | 53 |
| Ratio | 17÷1=17 | 29÷2=14.5 | 41÷3≈13.67 | 53÷4≈13.25 |
It is a straight line, but it does not pass through the origin because of the $5 service fee. So, it is linear but not proportional.
Answer: Not proportional
15. A painter uses 4 gallons of paint for every 2 rooms.
| Rooms | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Gallons | 4 | 8 | 12 | 16 |
| Ratio | 4÷2=2 | 8÷4=2 | 12÷6=2 | 16÷8=2 |
It forms a straight line through the origin and is proportional: the painter uses 2 gallons of paint for each room.
Answer: Proportional, constant = 2 gallons/room
16. A music subscription costs $10 per month.
| Months | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost $ | 10 | 20 | 30 | 40 |
| Ratio | 10÷1=10 | 20÷2=10 | 30÷3=10 | 40÷4=10 |
It is a straight line through the origin, and it is proportional: the cost increases by $10 for each month.
Answer: Proportional, constant = $10/month
17. A theme park charges $25 entry + $3 per ride.
| Rides | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost $ | 28 | 31 | 34 | 37 |
| Ratio | 28÷1=28 | 31÷2=15.5 | 34÷3≈11.3 | 37÷4≈9.25 |
It is a straight line but does not pass through the origin, since the $25 entry fee adds a fixed starting cost. Therefore, it is linear but not proportional.
Answer: Not proportional
18. A gym charges $30 per month.
| Months | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost $ | 30 | 60 | 90 | 120 |
| Ratio | 30÷1=30 | 60÷2=30 | 90÷3=30 | 120÷4=30 |
It forms a straight line through the origin, so it is proportional: the cost increases by $30 for each month.
Answer: Proportional, constant = $30/month
19. A parking garage charges $5 per hour plus $2 entry fee.
| Hours | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost $ | 7 | 12 | 17 | 22 |
| Ratio | 7÷1=7 | 12÷2=6 | 17÷3≈5.67 | 22÷4=5.5 |
It is a straight line but shifted upward, starting at $7 for the first hour because of the $2 entry fee. Therefore, it is linear but not proportional.
Answer: Not proportional
20. A bicycle rental is $8 per hour.
| Hours | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost $ | 8 | 16 | 24 | 32 |
| Ratio | 8÷1=8 | 16÷2=8 | 24÷3=8 | 32÷4=8 |
It forms a straight line through the origin, and it is proportional: the cost rises by $8 for every hour rented.
Answer: Proportional, constant = $8/hour