Precalculus is about how quantities change together. This unit starts with rates of change, then uses them to read polynomial and rational functions: where they increase, bend, cross zero, blow up or level off. It ends with transformations and choosing a function to model real data.
A function maps each input to exactly one output. As inputs increase, a function can increase or decrease and be concave up or down. The average rate of change over an interval is the change in output over the change in input; the rate of change at a point is approximated by average rates over small intervals around it.
A linear function has a constant average rate of change. A quadratic function's average rates of change over consecutive equal-length intervals change at a constant rate (they form a linear pattern). Increasing average rates mean concave up.
A polynomial has local maxima and minima where it switches between increasing and decreasing, and inflection points where its concavity changes. A degree-n polynomial has n complex zeros counting multiplicity; non-real zeros come in conjugate pairs; even-multiplicity zeros touch the x-axis. The leading term decides end behavior.
A rational function is a quotient of polynomials. Its end behavior comes from the quotient of the leading terms (horizontal or slant asymptote, or polynomial-like). Zeros come from the numerator; vertical asymptotes from denominator zeros that don't cancel; holes from factors that cancel.
Factored form shows zeros, intercepts, asymptotes and holes; standard form shows end behavior. Polynomial long division writes f = g·q + r and finds slant asymptotes. The binomial theorem uses a row of Pascal's triangle to expand (a + b)ⁿ.
f(x) + k shifts the graph up by k; f(x + h) shifts it left by h; a·f(x) stretches it vertically by |a| (reflecting over the x-axis if a < 0); f(bx) stretches it horizontally by 1/|b| (reflecting over the y-axis if b < 0).
Linear models fit roughly constant rates of change; quadratic models fit roughly linear rates of change or symmetric data with one max or min; degree-n polynomials fit constant nth differences; piecewise functions fit data that behaves differently on different intervals; rational functions fit inverse proportion. Every model has assumptions and domain or range restrictions.