Linear functions: from a constant step to a straight line

Where we are on the map

A rule between variables now becomes a function and its graph.

Begin with a familiar image

When a fare or water level changes by the same amount for every equal input step, a constant table difference produces a straight graph.

Precise meaning

In y=kx+b, k is the change in y for one unit of x, and b is the starting value when x=0.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

start b → constant step k → table → points → line → context check

Worked example

For y=2x+1, x=−1,0,2 gives y=−1,1,5; all three points lie on one line.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Calling b=1 the x-intercept is wrong: b is the y-intercept because it is the value at x=0.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. Build a five-value table for y=−3x+4, draw its graph, and explain k and b.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

When two linear conditions hold at once, their intersection solves a system of equations.

Course contents