A sentence, a table, a graph, and an equation can all show the same relationship. Learn how to connect them.
A proportional relationship can be shown with words, a table, a graph, or an equation. Each form tells you how one amount changes with the other.
The equation is y = kx. The graph is a straight line through (0, 0), the origin. In a proportional table, y ÷ x stays the same for nonzero x. The pair (0, 0) fits the equation but cannot be divided to find k. The sentence is “y is k times x.”
d = rt is that story in a car. Distance equals rate times time. Same skeleton.
Try it. Then open the steps.
Example 1 of 5
A bike moves at a constant 14 miles per hour. Write the equation for distance y after x hours, then find y when x = 3.5.
Answer: y = 14x, 49 miles
Example 2 of 5
Assume a straight proportional relationship. A graph of cost vs pounds goes through (0,0) and (4, 18). Write the equation.
Answer: y = 4.5x
Example 3 of 5
Same car, later that afternoon
You travel 90 miles in 1.5 hours at a constant speed. How far in 4 hours?
Answer: 240 miles
Example 4 of 5
The counter again
Scoops cost $2.50 each. Write y = kx for cost after x scoops, then find the cost of 6 scoops.
Answer: y = 2.50x, $15
Example 5 of 5
Concession line
A table of bottles vs dollars: 2 bottles → $5, 6 bottles → $15, 8 bottles → $20. Is it proportional? If yes, what is k?
| Bottles x | Dollars y | y ÷ x |
|---|---|---|
| 2 | 5 | 2.50 |
| 6 | 15 | 2.50 |
| 8 | 20 | 2.50 |
Answer: yes, k = 2.50
Easy to miss: Reading a graph by counting squares wrong, then inventing a k that does not fit the table.
Do this: Check what each axis label and tick mark means. Choose a point whose coordinates you can read clearly. Divide y by x, using x ≠ 0, then check another point with x ≠ 0.
Easy to miss: Writing y = x + k because addition feels friendlier than multiplication.
Do this: In y = kx, multiply x by k. Adding a nonzero fixed amount, as in y = 4x + 3, makes the graph miss (0, 0), so that relationship is not proportional.
22 problems · Camp, Trail, Summit
