Knowledge MountainA PLACE FOR EVERY CURIOUS MINDORIGINAL STARTER TRAIL / Foundations preview
Make the invisible pattern visible.
FIELD NOTE / Mathematics
A function assigns exactly one output to each input in its domain. For f(x) = 2x + 3, input 4 produces 11. Different inputs may share an output; what is forbidden is one input producing two different outputs under the same rule.
A way to rememberInput → rule → one output.
f(−1) = 2(−1) + 3 = 1. Substitute the input everywhere x appears.
FIELD NOTE / Mathematics
In y = mx + b, m is the change in y for each increase of 1 in x, and b is the value of y when x = 0. A negative slope falls as x increases. A zero slope makes a horizontal line.
A way to rememberm moves; b begins at the y-axis.
y decreases by 3. The outputs are −1 and −4, so Δy/Δx = −3/1.
FIELD NOTE / Mathematics
To solve 2x + 3 = 11, subtract 3 from both sides to get 2x = 8, then divide both sides by 2 to get x = 4. Doing the same valid operation to both sides preserves the equality. Check by substituting your answer into the original equation.
A way to rememberUndo addition, then undo multiplication.
Add 4 to both sides: 3x = 12. Divide by 3: x = 4. Check: 3(4) − 4 = 8.
THE LIVE GRAPH BENCH
Move the rule, then the input. The line shows y = mx + b; the dot marks your chosen input.
Try m = 0. Why does changing x stop changing the output?
A practice check, not an exam. Pick an answer for each question, then read the reasoning.
Lesson completion is self-reported. Checkpoint scores reflect these practice questions, not a qualification or certified mastery.