Pre-Calculus Video Series

Review factoring techniques: extract GCF, factor differences of squares, sums and differences of cubes, and group terms; apply product-sum and trial-and-error for three- and four-term polynomials.

In this pre-calculus lesson, learn to simplify rational expressions by factoring, canceling common factors, and applying keep-change-flip for division, plus adding and subtracting with common denominators.

Explore exponent properties, including the product and power rules, negative exponents, and fractional exponents, through explicit examples and step-by-step simplification.

Explore the composition of functions by evaluating f(g(x)) and g(f(x)) from inside out, with concrete examples and results f(g(x))=4x^2+19 and g(f(x))=16x^2-8x+6.

Explores transformations of functions, including translations (left-right and up-down), dilations (vertical and horizontal), and reflections, with practical examples using linear, quadratic, absolute value, and exponential functions.

Sketch functions by transforming parent graphs, applying horizontal and vertical shifts, reflections, stretches, and compressions, and determine zeros, domain, range, and symmetry for polynomials, reciprocals, exponentials, absolutes, logs, and roots.

Explore solving equations via inverse operations, identify inverse relations and functions, apply vertical and horizontal line tests, and derive inverse functions using switch of x and y.

Analyze and graph piecewise functions by domain rules, ensuring non overlapping pieces and a valid vertical line test. Practice evaluating pieces and plotting with open and closed endpoints.

Explore step functions, including greatest integer and ceiling functions, with evaluation and graphing techniques, and apply horizontal and vertical transformations through real-world examples like parking costs and car rental.

Find domain algebraically by identifying restrictions from even radicals and denominators, solve for zeros, test intervals, and exclude restricted values to produce the domain.

Explore graphing polynomial functions by identifying degree and leading coefficient, and analyzing end behavior. Locate zeros with multiplicities and sketch from factored form.

Explore dividing polynomials using exponent rules and coefficients, compute remainders and quotients with polynomial long division and synthetic division, and apply the factor theorem to factor polynomials.

Learn to factor polynomials using the rational root theorem, identify P and Q factors, test rational roots with synthetic division, and factor by grouping, including quadratics and difference-of-squares.

Learn to find complex zeros of polynomials by factoring to quadratics, using rational root theorem, quadratic formula, and square-root methods, and note that complex roots come in conjugate pairs.

Explore rational functions built from polynomials, including reciprocal forms, and learn to analyze asymptotes, intercepts, holes, domains, ranges, and graph transformations.

Solve non-linear inequalities using polynomials and rational functions, including factoring, zeros, and vertical asymptotes. Graph intervals on a number line, test points, and apply end behavior for sign analysis.

Explore function modeling to optimize real-world problems, including maximizing area and volume of geometric shapes and minimizing time or energy in river crossing and bird flight scenarios.

Explore exponential functions, including growth and decay with horizontal asymptotes and the 0 1 intercept, then learn logarithms as the inverse and how to solve with change-of-base.

Explore the properties of logarithms and their relationship to exponentials, applying product, quotient, and power rules to condense and expand logs for simplifying and graphing.

Learn to graph logarithms as the inverse of exponential functions, reflect them over the line y = x, and identify their asymptotes, domain, and intercepts.

Learn to solve exponential and logarithmic equations algebraically by selecting methods: matching bases, applying logarithms and power properties, factoring, and checking for extraneous solutions in log-based domains.

Explore applications of exponential and logarithmic functions, including growth and decay with initial values, rates, and time. Learn to model compounding interest, continuous growth with e, and half-life scenarios.

Explore conic sections by visualizing circles, ellipses, parabolas, and hyperbolas, and learn to convert general form to standard form using completing the square, identifying centers, radii, and axes.

Explore the circle equation in standard form, identify centers and radii, and convert from general to standard form by completing the square, with graphing techniques for center-radius visualization.

Learn the properties of ellipses, including center, foci, major and minor axes, and vertices, and how to derive and graph standard form equations.

Explore hyperbolas and compare them to ellipses. Learn standard forms, center, vertices, foci, and asymptotes, then practice graphing with guided examples.

Explore parabolas as conics with vertex, focus, directrix, axis of symmetry, horizontal or vertical opens via x squared equals 4 p y and y squared equals 4 p x.

Explore the foundations of right triangle trigonometry by mastering sine, cosine, tangent, and reciprocal identities using soh cah toa, and apply them to 30-60-90 and 45-45-90 triangles.

Solve right triangles using sine, cosine, and tangent with Sokoto; apply inverse trig and angles of elevation and depression to real-world distance and height problems.

Explore how angles are measured in degrees and radians, using standard position, positive and negative angles, and how to convert between degrees and radians while finding co-terminal and reference angles.

Learn the unit circle foundations for trigonometry, including cosine and sine as x and y coordinates, special right triangles, quadrant signs, radians, and applying arc length and sector area formulas.

Explore graphing sine and cosine from the unit circle, understanding amplitude, period, and phase shifts. Learn to apply horizontal and vertical transformations to sketch accurate trigonometric graphs.

Explore how secant and cosecant arise as reciprocals of sine and cosine, identify vertical asymptotes where sine or cosine vanishes, and use reference graphs to analyze domain, range, and transformations.

Learn tangent and cotangent graphs as sine over cosine and cosine over sine, identify vertical asymptotes from zeros, note period pi, and apply basic transformations.

Examine how to make trig functions 1-to-1 by restricting domains to pass the horizontal line test, then evaluate sine, cosine, and tangent inverses with unit-circle examples.

Master the law of sines for solving non-right triangles using sine ratios, exploring angle-angle side and side-angle relationships, the ambiguous case, and setting up the equations to find missing sides.

Apply the law of cosines to solve triangles in side–side–side and side–angle–side cases, find sides and angles, and compute areas using Heron’s formula and 1/2 b c sin A method.

Explore the law of sines and the law of cosines through four real-world triangle problems, including computing angles, side lengths, and areas.