Recognize when the rate changes
Compare equal input steps on a line and a curve without assuming a constant rate.
By the end: Use interval slopes to identify nonlinear change and state the limits of extrapolation.
Try the lab first · Read the explanation
Starting recap and optional refresher
A straight line has the same output change for equal input steps. A curved graph need not.
Review substitution and arithmetic, then return hereTry a concrete example
For y = x², moving from 0 to 1 changes y by 1; moving from 2 to 3 changes it by 5. Both input steps have length 1.
Make a prediction
Before changing the rule, predict one output and write down what you expect to change. Check the table after your intervention. This prediction is practice, not an assessment attempt.
Change one thing and compare
Set slope 1 and intercept 0. Switch between linear and quadratic rules. Compare the table differences from −2 to −1 and from 2 to 3. A curve can turn; one interval does not describe every input.
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Connect the experiment to the idea
Equal steps can have unequal effects
A straight line y = mx + b has constant slope. For the curve y = x², increasing x by one does not always produce the same output change. From zero to one, y changes by one. From two to three, it changes by five.
The square means x multiplied by itself. At x = −2 the output is four, and at x = −1 it is one. As x moves toward zero from the negative side, the output decreases. The same curve rises when x moves away from zero on the positive side.
An interval rate is not a universal rate
From x = 1 to x = 3, outputs change from one to nine. The average rate over that interval is (9 − 1) / (3 − 1) = 4. A line joining those two points has that slope, but other intervals on the curve have different average rates.
A local derivative describes the limiting rate near one input; this course does not require calculus. You can already detect a changing rate by comparing equal-sized steps in the table. Tiny steps in computer arithmetic also have limits, so “smaller” is not automatically “more accurate.”
Shape is evidence about a rule, not its truth
Switch the lab between y = x and y = x² with multiplier one and starting value zero. Compare outputs on both sides of zero. Then increase the multiplier and inspect the exact values that the fixed vertical scale clips.
Neither curve becomes a good real-world model merely because it is smooth. Data, units, input coverage and a stated purpose are needed. Extrapolation means using a rule beyond the range where its use was supported. A constant-rate assumption that worked on a short interval can fail on a longer one.
Try a new case
For y = x², what is the average output change per input unit from x = 1 to x = 3?
Take the idea with you
Use interval slopes to identify nonlinear change and state the limits of extrapolation.
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Original LearnAI mathematics lesson. All experiments run locally on the CPU. No refresher practice or AI achievement is changed by completing this lesson.