Functions Summary

Algebra 1 · Axiom Academy

SUMMARY Functions & Linear Relationships Functions create predictable relationships between inputs and outputs. Linear functions have a constant rate of change and can be written several equivalent ways. A function assigns each input exactly one output — and you can spot one on a graph with the Vertical Line Test . Function notation f(x) names the output of the rule f at the input x — it is not multiplication. A linear function has a constant rate of change (the slope m ); the y -intercept b is its output when x = 0 . Slope-intercept y = mx + b and point-slope y - y_1 = m(x - x_1) are equivalent forms of the same line. Two distinct lines are parallel when their slopes are equal, and perpendicular when their slopes are negative reciprocals. Core Concept The Idea of a Function A function is a rule where each input value has exactly one output value. We write f(x) for the output of the function f at input x . Identify it: a graph is a function when it passes the Vertical Line Test — any vertical line meets it at most once. Watch out for: f(x) is one output, not f times x . Core Concept Slope & Y-Intercept A linear function's defining feature is its constant slope m — the rate of change. The y -intercept b is the output when the input is zero. Read the sign: m is positive (rising), negative (falling), zero (horizontal), or undefined (vertical). Watch out for: keep the order of points consistent in top and bottom of the slope ratio.

This is the written version of the interactive lesson above. See the full Algebra 1 course.