Linear and affine functions
A constant rate gives a straight-line graph; an affine rule may include a nonzero starting value.
Separate a starting value from a rate
Change slope and intercept independently and calculate a rate from two points.
By the end: Explain slope as output change per input unit and intercept as the output at zero.
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Starting recap and optional refresher
For y = mx + b, m multiplies the input and b is added afterward. A change is final value minus initial value.
Review substitution and arithmetic, then return hereTry a concrete example
A tank starts with 6 litres and loses 2 litres each minute. y = −2x + 6 is useful only until the tank empties at 3 minutes.
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 −2 and intercept 6. Compare x = 0, 1 and 3. Why is the displayed negative output at x = 4 not a physical prediction for this tank?
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Starting amount and change per unit
In y = mx + b, b is the output when x = 0. It is called the intercept because the graph meets the vertical axis there. The number m is the slope: output change divided by input change. A negative slope means output falls as input increases.
For a tank, y = −2x + 6 uses x in minutes and y in litres. The intercept is six litres, and the slope is minus two litres per minute. At one minute, four litres remain; at three minutes, zero remain.
Calculate slope from two points
For (1, 5) and (3, 11), the output change is 11 − 5 = 6 and the input change is 3 − 1 = 2. The slope is 6 / 2 = 3 output units per input unit. Dividing 11 by 3 instead would ignore the starting value.
If the two input coordinates are equal, this division has a zero denominator and does not give a finite slope. Two different outputs for the same input also cannot belong to a single-valued function of that input.
Separate two interventions
Changing b shifts every output by the same amount. Changing m keeps the output at x = 0 fixed, but changes other outputs by an amount that depends on x. For negative x, increasing m can lower the output. Test both sides of zero before making a universal claim.
The tank rule predicts negative litres after three minutes. That is a limitation of extending a draining rule past an empty tank. Keep the valid domain visible rather than pretending every plotted value is physically possible. Likewise, the apparent steepness changes if a graph stretches one axis; calculate slope using labelled units.
Try a new case
Check your understanding
A line passes through (1, 5) and (3, 11). What is its slope?
Take the idea with you
Explain slope as output change per input unit and intercept as the output at zero.
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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.
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Functions
Linear and affine functions extends Functions.
Straight-line rules provide a concrete family of functions with constant rate and an optional starting value; not every function is linear or affine.
Scope and source review
Affine y = mx + b rules in the introductory function lessons; strictly linear rules have zero intercept.
Source-audited 2026-10-31. Independent subject-expert and learner review remain pending.
A single neuron
A single neuron can review Linear and affine functions.
The first neuron begins with input times weight plus bias. Comparing weight with slope and bias with a starting value makes that calculation visible.
Scope and source review
The initial scalar affine neuron example, not a claim that every neural activation is affine.
Source-audited 2026-10-31. Independent subject-expert and learner review remain pending.
11 concepts in this cross-subject pilot, not a complete curriculum graph. The existing AI lesson map and progress records remain separate and available below or through AI discovery.