Level 8: Advanced functions

Algebra · level 8 of 8 · rough age 17 and beyond (a guide for parents; anyone can start anywhere)

The last level of algebra is the bridge to university mathematics. It studies whole families of functions: polynomials of any degree and how to factor them, rational functions with their asymptotes, and logarithms, the inverse of exponentials, which bring the unknown down from the exponent. Then it combines functions, proves the compound angle formulas of trigonometry and solves trigonometric equations, and finally leaves the flat page to describe lines and planes in three dimensions.

Lesson	You will be able to
Polynomial functions	Predict end behaviour and the shape at each zero, divide polynomials (long and synthetic), and use the remainder and factor theorems to factor cubics
Polynomial inequalities and families	Solve inequalities like \((x - 1)(x + 2)(x - 4) > 0\) with a sign chart, and pick one member of a family of polynomials with given zeros
Rational functions	Find vertical and horizontal asymptotes and holes, and solve rational equations and inequalities without losing or inventing solutions
Logarithms	Read \(\log_b x\) as an exponent, prove the laws of logarithms from the exponent laws, change base, and graph \(\log x\) as the inverse of \(10^x\)
Exponential and logarithmic equations	Solve \(3^x = 20\) and \(\log x + \log(x - 3) = 1\), reject extraneous roots, and model half-life, pH and decibels
Combining functions	Add, subtract, multiply, divide and compose functions, find the domain of \(f(g(x))\), and break a function into simpler pieces
Trigonometric identities and equations	Prove the compound and double angle formulas, prove identities, and solve equations like \(2\sin^2 x - \sin x - 1 = 0\) on \([0, 2\pi]\)
Lines and planes in space	Write vector and parametric equations of lines, use the cross product, find the equation of a plane, and solve three equations as three planes

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