Advanced Functions: A Complete Course on Precalculus

Explore polynomial functions by understanding their form, coefficients, and whole-number degrees, and distinguish them from non-polynomial expressions through clear examples.

Explore characteristics of polynomial functions, including degree, leading coefficient, even and odd behavior, end behavior, turning points, and possible x-intercepts up to degree five.

Explore the family of polynomial functions and their characteristics in factored form, including degree, leading coefficient, order of factors, X-intercepts, zeros, and turning points.

Learn to divide polynomials using long division and synthetic division, identifying the dividend, divisor, quotient, and remainder, with an example dividing by x minus three.

Learn to factor polynomials by applying the remainder theorem and the factor theorem, using division and synthetic division to find zeros and complete factorization.

Learn to factor cubic expressions using sum and difference of cubes, and verify with synthetic division through examples such as x^3 minus eight and 27x^3 plus 125.

Learn to solve linear equations and inequalities by isolating the variable and moving terms across the equals sign, with sign changes for negatives.

Learn how to solve polynomial equations and inequalities by factoring, using the factor theorem and synthetic division, and analyzing sign changes with a number line to identify solution intervals.

Explore reciprocal functions as numbers over a function, starting from the parent function 1/x. Learn to identify vertical asymptotes at x=0 and horizontal asymptotes at y=0 with examples like 1/x.

Explore rational functions and distinguish them from reciprocal functions, identify vertical and horizontal asymptotes, and analyze degrees, leading coefficients, holes, and cancellations with examples.

Graph rational functions by identifying vertical asymptotes from the denominator, horizontal asymptotes from leading coefficients, and locating x-intercepts and y-intercepts to sketch the graph.

This correction clarifies that for the function (2x - 1)/(x - 4), the x-intercept is at x = 1/2, not x = 2, correcting the prior miscalculation.

Learn to solve rational equations by bringing all terms to one side, using a common denominator, and checking signs, with a salt-concentration word problem.

Explore rate of change by showing how a dependent variable in a function y=f(x) changes with an independent variable like height or elevation, with examples such as volume and temperature.

Learn the two kinds of rate of change: the average rate over an interval and the instantaneous rate at a specific instant, using y = f(x) and travel examples.

Learn to determine the average rate of change using a secant line through points such as (-4,-5) and (1,0) and apply the slope formula (y2 - y1)/(x2 - x1) over a given interval.

Explore how the slope of a tangent line equals the instantaneous rate of change, using the quotient rule with f(x) and f(x+h) and an x^2 example.