Graphing Proportional Relationships
In a proportional relationship, y is always the same number times x. Its graph is a straight line through (0, 0).
What you'll be able to do
- Decide if a table, graph or equation is proportional.
- Find the unit rate, k.
- See the unit rate as the slope of the graph.
- Compare rates shown in different ways.
The big idea
The rule. In a proportional relationship, y = kx. Every y is the same number, k, times its x. The number k is the unit rate: how much y for 1 x.
The test. Divide each y by its x. If every answer is the same, the relationship is proportional. Check every pair, not just one.
The graph. The graph of y = kx is a straight line through the origin, (0, 0). A straight line that misses (0, 0) is not proportional.
Slope. For each step of 1 to the right, the line goes up k. So the unit rate is the slope. A steeper line means a bigger unit rate.
Fractions. Rates can use fractions. 1/2 mile in 1/4 hour is (1/2) ÷ (1/4) = 2 miles per hour.
Units. Before you compare two rates, put them in the same units. 1 meter per second is 3.6 kilometers per hour.
Words you need
- Proportional relationship
- y = kx: y ÷ x is the same for every pair.
- Constant of proportionality (k)
- The number y ÷ x that stays the same.
- Unit rate
- How much y for 1 x, like 30 miles per gallon.
- Slope
- How much a line goes up for each step of 1 to the right.
- Origin
- The point (0, 0), where the two axes cross.
One rate, three ways
| Shown as | Example: $3 per pound |
|---|---|
| Equation | y = 3x |
| Table | x: 1, 2, 4 and y: 3, 6, 12 |
| Graph | A line through (0, 0) and (1, 3), slope 3 |
Worked examples
Example 1
A car goes 30 miles on 1 gallon, 60 miles on 2 gallons and 120 miles on 4 gallons. Is the distance proportional to the gas? What is the unit rate?
- Divide each y by its x: 30 ÷ 1 = 30, 60 ÷ 2 = 30, 120 ÷ 4 = 30.
- Every answer is 30, so it is proportional: y = 30x.
- The unit rate is 30 miles per gallon.
Answer: Yes, y = 30x
Example 2
A line goes through (0, 0) and (5, 4). What is its slope?
- Slope = rise ÷ run = (4 − 0) ÷ (5 − 0) = 4 ÷ 5.
- So the slope is 4/5, or 0.8. The line goes up 0.8 for each step of 1.
Answer: 4/5
Example 3
Ana earns y = 12x dollars for x hours. Ben earns $26 for 2 hours and $65 for 5 hours. Who earns more per hour?
- Ana's unit rate is the 12 in y = 12x: $12 an hour.
- Ben's unit rate is 26 ÷ 2 = 13: $13 an hour.
- 13 is more than 12, so Ben earns more per hour.
Answer: Ben
Common mistakes
Calling any straight line proportional. It must also go through (0, 0). The line y = 2x + 3 is not proportional.
Dividing x by y for the unit rate. The unit rate is y ÷ x. 120 miles on 4 gallons is 120 ÷ 4 = 30 miles per gallon.
Picking the line that sits higher. The bigger rate belongs to the steeper line, not the higher one.
Checking only one pair. Divide y by x for every pair. They all have to match.
Practice: 20 questions
11 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Sofia earns $45 for 3 hours of babysitting and $75 for 5 hours. Is her pay proportional to her time? If so, what is the unit rate? Word problem
- A line goes through (0, 0) and (4, 6). What is its slope?
- A car goes 120 miles on 4 gallons and 210 miles on 7 gallons. Write an equation for miles y and gallons x. Word problem
- A taxi ride costs $10 for 3 miles and $14 for 5 miles. Is the cost proportional to the distance? Word problem
- Ben drives 150 miles in 3 hours. He says the unit rate is 3 ÷ 150 = 0.02. Find his mistake. What is the unit rate?
Tier 2 Intermediate One more step to think through
- Ana walks 3/4 of a mile in 1/2 of an hour. What is her speed in miles per hour? Word problem
- Is the table (2, 6), (4, 12), (5, 16) proportional?
- Lena runs 400 meters in 2 minutes and 1,000 meters in 5 minutes. Jamal runs y = 180x meters in x minutes. Who runs faster? Word problem
- Grapes cost $7.50 for 3 pounds and $12.50 for 5 pounds. Write an equation for the cost y of x pounds. Word problem
- Store A charges y = 1.4x dollars for x kg of apples. Store B charges $6 for 4 kg and $12 for 8 kg. Store C's price graph passes through (0, 0) and (5, 7.25). Which store charges the most per kilogram? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- A proportional graph goes through (4, 10). What is y when x = 18?
- Lia reads 2/3 of a page each minute. Max reads y = 0.65x pages in x minutes. Who reads faster? Word problem
- A tank fills at a steady rate. It has 13.5 liters after 3 minutes and 31.5 liters after 7 minutes. Write an equation for liters y after x minutes. Word problem
- A table has the points (0, 0), (2.5, 4), (5, 8) and (7.5, 12). Is it proportional? If so, what is k?
- Sam checks (2, 6) and says the table (2, 6), (4, 10), (5, 13) is proportional with k = 3. Find his mistake. Is it proportional?
Tier 4 Challenge Stretch problems, some above grade level
- Runner A goes 5 meters each second. Runner B goes y = 20x kilometers in x hours. How many kilometers per hour faster is the faster runner? Word problem
- A proportional graph goes through (6, 14). What is x when y = 35?
- In y = 4x, is x proportional to y? If so, what is its constant?
- Line A goes through (0, 0) and (5, 3). Line B is y = (2/3)x. Which has the greater unit rate?
- A gym charges a one-time sign-up fee plus a fixed monthly fee. It costs $25 for 1 month and $40 for 2 months. Write an equation for the total cost y after x months. Word problem Above grade level
Answers
Tier 1 Foundations
- Yes, y = 15x. y ÷ x is 15 for every pair, so k = 15.
- 3/2. Slope = rise ÷ run = (6 − 0) ÷ (4 − 0) = 6 ÷ 4 = 3/2.
- y = 30x. The slope is m = 30. Then b = y − mx = 0, the value of y when x = 0.
- No. y ÷ x gives 10/3, 2.8. They don't all match, so it is not proportional.
- 50. Ben divided x by y. The unit rate is y ÷ x. Slope = rise ÷ run = (150 − 0) ÷ (3 − 0) = 150 ÷ 3 = 50.
Tier 2 Intermediate
- 3/2. Speed is miles ÷ hours. (3/4) ÷ (1/2) = 3/4 × 2 = 3/2. That is 1.5 miles per hour.
- No. y ÷ x gives 3, 3, 3.2. They don't all match, so it is not proportional.
- Lena. Lena's rate of change is 200. Jamal's is 180.
- y = (5/2)x. The slope is m = 5/2. Then b = y − mx = 0, the value of y when x = 0.
- Store A < Store C < Store B, so Store B is the greatest. Find each unit rate. Store A: 1.4. Store B: 6 ÷ 4 = 1.5. Store C: 7.25 ÷ 5 = 1.45. Store B charges the most, $1.50 per kilogram.
Tier 3 Advanced
- 45. First find k = 10 ÷ 4 = 2.5. Then y = 2.5 × 18 = 45.
- Lia. 2/3 is about 0.667 pages a minute. That is more than 0.65, so Lia reads faster.
- y = (9/2)x. The slope is m = 9/2. Then b = y − mx = 0, the value of y when x = 0.
- Yes, y = 1.6x. y ÷ x is 1.6 for every pair, so k = 1.6.
- No. Sam checked only one pair. 10 ÷ 4 = 2.5, which is not 3. y ÷ x gives 3, 2.5, 2.6. They don't all match, so it is not proportional.
Tier 4 Challenge
- 2 km per hour. Put both in km per hour. 5 meters a second is 5 × 3,600 = 18,000 meters an hour, or 18 km an hour. Runner B goes 20 km an hour. So B is faster by 2 km per hour.
- 15. First find k = 14 ÷ 6 = 7/3. Then solve 35 = (7/3)x. So x = 35 × 3/7 = 15.
- Yes, x = (1/4)y. Divide both sides of y = 4x by 4. That gives x = (1/4)y. So x is y times the same number, 1/4.
- Line B. Line A's rate of change is 0.6. Line B's is 2/3.
- y = 15x + 10. The slope is m = 15. Then b = y − mx = 10, the value of y when x = 0.
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