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Learn to distinguish between proportional and non-proportional relationships using tables and graphs.
If a recipe for 4 people costs 40 or $50? Knowing how to spot a proportional relationship helps you decide if you're getting a fair deal or being overcharged!
A proportional relationship is a link between two quantities where the ratio of one to the other is always constant. In a table, we check this by calculating the ratio for every pair of numbers. This constant value is called , the constant of proportionality. If every single pair results in the same , the relationship is proportional. If even one pair results in a different number, the relationship is non-proportional.
A snack bar sells apples. Is the cost proportional to the number of apples?
1. Table: (2 apples, 6), (6 apples, \frac{y}{x}\frac{3}{2} = 1.5\frac{6}{4} = 1.5\frac{9}{6} = 1.51.5$, the relationship is proportional.
Quick Check
If a table has the pairs (2, 10) and (3, 12), is it proportional?
Answer
No, because but . The ratios are not equivalent.
Graphs provide a visual way to spot proportions instantly. For a relationship to be proportional, its graph must meet two strict rules: 1. It must be a straight line (linear). 2. It must pass through the origin, which is the point . If a line is straight but starts at , it is not proportional. If it passes through but curves, it is not proportional. Both conditions must be met!
A gym charges 50 sign-up fee. Is this proportional? 1. At 0 months, you have already paid (0, 50)60. Point: . 3. Because the graph starts at and not the origin , it is non-proportional, even though the monthly increase is steady.
Quick Check
A graph shows a straight line passing through (0,0) and (4, 8). Is it proportional?
Answer
Yes, it is a straight line and it passes through the origin.
On a proportional graph, every point tells a story. The -coordinate usually represents the 'input' (like time), and the -coordinate represents the 'output' (like distance). A very special point is , where is the unit rate. This tells you how much you get for exactly one .
A faucet leaks 3 cups of water every 2 hours. 1. Find the unit rate: cups per hour. 2. The point represents this unit rate. 3. The point means that in 4 hours, 6 cups of water have leaked. 4. Since 0 hours equals 0 cups, the line is straight through , making it proportional.
Which set of points represents a proportional relationship?
If a proportional graph passes through the point , what is the constant of proportionality ?
A straight line that passes through represents a proportional relationship.
Review Tomorrow
In 24 hours, try to sketch two graphs: one that is proportional and one that is linear but NOT proportional. What is the difference?
Practice Activity
Look at a grocery receipt. Pick an item where you bought multiple units (like 3 yogurts). Create a table and check if the total cost is proportional to the number of items.