Function Mastery Examples

Calculus Readiness · Axiom Academy

EXAMPLE The Difference Quotient Work the calculus-readiness keystone step by step, then review the whole function toolkit For f(x) = x^2 , build the difference quotient and simplify it completely. This single computation strings together every move in the function toolkit — evaluating at an expression, expanding, factoring, and cancelling — and it is the exact object whose limit defines the derivative in calculus. Why it matters: secant slope →︎ tangent slope The difference quotient is the slope of the secant line joining (x, f(x)) and . Shrinking h toward 0 rotates the secant into the tangent — and its slope is exactly what we get below, . The rest of the function toolkit The difference quotient above leaned on evaluating, expanding, and simplifying. Here is the same toolkit applied to the other core moves — each one a fully worked mini-example you can cover and re-derive. For f(x) = 2x^2 - 3x + 1 , wrap the input in parentheses: f(-2) = 2(-2)^2 - 3(-2) + 1 = 8 + 6 + 1 . For g(x) = x^2 + 3x , substitute (x+h) everywhere: g(x+h) = (x+h)^2 + 3(x+h) . Result: g(x+h) = x^2 + 2xh + h^2 + 3x + 3h . With and g(x) = 3x + 4 , the inner output feeds the outer: . For , the radicand under a denominator must be strictly positive: . For f(x) = 3x - 7 , write y = 3x-7 , swap , and solve: . Read g(x) = -2f(x-3) + 1 inside-out: replace x with x-3 , scale by 2 , negate, then add 1 . Result: shift right 3 , stretch vertically by 2 , reflect over the x -axis, shift up 1 .

This is the written version of the interactive lesson above. See the full Calculus Readiness course.