Master Pre Calculus and Trigonometry

Learn how the end behavior of polynomials depends on even or odd degree and the sign of the leading coefficient, with the graph ends moving up or down toward infinity.

Explore end behavior basics through a two-question walkthrough, identifying leading coefficient signs and even or odd degree, with examples like y = -x^2 and left-up right-down graphs.

Learn synthetic division for polynomials, using linear factors to test for factors, replace long division with faster coefficient-based steps, and handle placeholders and remainders.

Walk through the properties of exponentials quiz, evaluating negative exponents, zero exponents, and powers of fractional bases with examples like 3^x and (1/4)^x.

Explore solving basic exponential equations by aligning bases and exponents, turning expressions into the same base, solving for x, and checking results. Prepare for logarithms when base changes are impossible.

Explore solving a basic logarithms quiz: log base 3 of 243 equals 5, log base 8 of 1 equals 0, and why log base 2 of negative four is undefined.

Master common logarithms with base ten and natural logarithms with base e, including solving log expressions by converting to exponential form and recognizing ln as log base e.

Introduce matrices, explain elements, rows by columns, and perform addition, subtraction, and scalar multiplication; illustrate with 2x2 and 3x3 examples.

Learn how determinants determine whether a two by two matrix has an inverse, derive the inverse formula, and use inverses to solve matrix equations.

Learn how arithmetic series sum the terms of an arithmetic sequence using summation notation and the formula a_n = a_1 + (n-1)d, and compute sums with n/2 (a_1 + a_n).

Master geometric sequences by identifying the common ratio and using recursive and explicit formulas to find terms. See 1, 4, 16, 64 and 8, 1/2, 1/8 as illustrative examples.

Identify the vertex at the origin for y = 3x^2, compute p from a = 1/(4p) (p = 2/3), and state the focus (0, 2/3) with directrix y = -2/3.

Explore ellipses, identify major and minor axes, vertices, and foci using standard forms; learn to compute a, b, and c from given denominators.

Graph an ellipse by locating its center (h,k), identifying a and b from the denominators, marking major and minor vertices, and locating foci via c^2 = a^2 - b^2.

Explore trigonometry in right triangles, learning sine, cosine, and tangent as opposite, adjacent, and hypotenuse ratios. Use reference angles, Sokoto, and the Pythagorean theorem, including 5-12-13, to solve triangle measures.

master the law of sines for solving triangles of all types using sin A / a = sin B / b = sin C / c, and verify angle-side correspondences.

Explore the law of cosines, a triangle solving method used when two sides and included angle—or all three sides—are given; select the suitable formula and apply cosine or inverse cosine.

Explore how trig graph transformations modify amplitude, period, phase shift, and vertical shifts across sine, cosine, tangent, and cotangent, using explicit examples and order of operations.

Graph sine x and cosine x by plotting points at increments of pi, noting range from -1 to 1, and that sine starts at 0 while cosine starts at 1.

Graph sine, cosine with shifts: y = sin(bx − c) and y = cos(bx − c). Phase shift equals c/b, moving right for positive c and left for negative c.

Practice 1 demonstrates graphing sine and cosine with transformations, identifying amplitude, period, and phase shift to the right, and plotting key points for a complete transformed wave.

Explore graphing tangent and cotangent with transformations, noting amplitude, period, and phase shift. Show how phase shift moves the graph by pi and that amplitude minimally changes tangent.

Graph a tangent with transformations, including x-axis reflection, amplitude three, and period pi over two, then shift left pi over two and up one to reveal asymptotes.

Graph secant and cosecant using cosine transformations, with amplitude 1/2, period pi, and phase shift pi/4; identify asymptotes and sketch secant by lines and parabolas from cosine extrema.

Explore how to graph inverse sine and cosine by making each function one-to-one with a horizontal line test, swap x and y, and adjust domain and range accordingly.

Practice simplifying trig expressions with identities, rewriting using tangent and secant, clearing denominators, and using sec^2 x = tan^2 x + 1 to yield tan^3 x + tan x.

Explore sum and difference identities for sine, cosine, and tangent, learn a mnemonic jingle, and apply unit circle angles to evaluate expressions like sine 105, cosine 75, and tangent 15.

Solve tan x = -1 using the unit circle; quadrants II and IV with reference angle π/4. Solutions: 3π/4 and 7π/4; general x = 3π/4 + kπ.

This lecture teaches solving trig equations by factoring and converting to x. It identifies x values pi/2, 7pi/6, and 11pi/6 on the unit circle.

Use a Y substitution to solve trig equations with multiple angles, then convert back to X. From 3 tan Y + 3 = 0 on [0, pi], X = 3π/2.