What you’ll learn
Recognize proportional relationships.
Key idea
If y represents the output and x the input, the relationship has the form y = kx. That means y/x is constant whenever x is not zero, and the graph includes the origin.
A constant difference is not the same as a constant ratio. The pairs (2, 5), (4, 10), and (6, 15) all have y/x = 5/2, so they fit y = (5/2)x. By contrast, y = 2x + 3 has a starting value of 3 and is not proportional, even though its rate of change is constant.
Worked example
Classify examples and non-examples for the concept 'Recognize proportional relationships'.
For nonnegative x, classify y=5x+3 as a proportional example or non-example using the defining conditions.
- At x=0 the rule gives y=3; proportionality requires zero output.
- The nonzero ratios are 8 and 13/2; they differ, so the rule is not proportional.
Answer: Non-example — the nonzero intercept prevents origin passage and constant y/x.
Quick Check
Classify examples and non-examples for the concept 'Recognize proportional relationships'.
For nonnegative x, classify y=3x as a proportional example or non-example using the defining conditions.