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.

Master polynomial long division by computing quotients and remainders, identifying factors when the remainder is zero, and applying these steps to higher-degree cases.

Learn when to use synthetic division for linear divisors and when to use long division; identify zero-remainder factors, as x^2-3x-4 factors into (x-4)(x+1), while x^2+3 does not.

Explore exponential functions, where a base to the x power drives growth. Learn negative exponent rules and graph basics with y = 2^x and y = 5^x, including (0,1).

Explore exponential functions, including y = 2^x and negative outside and negative exponents, and see reflections over the x-axis and y-axis while noting the point (0,1).

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.

Learn to solve exponential equations by converting to a common base, solving exponents, and verifying results, illustrated with base 3 to 27 and base 4 to 64.

Learn how logarithms answer how many times to multiply a base to reach a number, the inverse of exponential, and the roll-out method for converting between forms.

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.

Master the rules for expanding and condensing logarithms, including the product rule, quotient rule, and handling exponents inside logs with same-base inputs, using practical examples.

Explore expanding and condensing logarithms to solve equations, using common and natural logs, base rules, and the inverse relationship between logarithms and exponentials.

Master advanced logarithms through a quiz walkthrough, solving for x using exponential reasoning. Tackle base five and base two logs, natural log with e, and base one third cases.

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.

Walks through the determinants and inverses quiz, defining the identity matrix and inverse, applying the 2 by 2 determinant ad - bc, and identifying invertibility via det ≠ 0.

Master arithmetic sequences by identifying the common difference and applying both recursive and explicit formulas to find any term, using examples like a_n and a_1.

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.

Learn how geometric series differ from arithmetic series, determine convergence for |R|<1 and divergence for |R|>1, and apply finite and infinite sum formulas.

Explore geometric series by identifying the common ratio, computing S10 and S15, and noting the infinite sum of a1 = 100, r = 1/4 equals 100/3.

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 hyperbola formulas, vertices, foci, and asymptotes for x-first and y-first orientations, centered at the origin.

Graph hyperbolas from general equations by identifying the center (h, k), locating vertices and foci with a and c, and drawing asymptotes using the c^2 = a^2 + b^2 relation.

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 right triangle trigonometry by solving for sides and angles using sine, cosine, and tangent with opposite, adjacent, and hypotenuse, plus Sokoto mnemonics.

Apply heron's formula to compute triangle area from three side lengths for not a right triangle by calculating the half perimeter s, then area equals sqrt(s(s-a)(s-b)(s-c)).

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.

Learn to convert between degrees and radians using the unit circle, with formulas degrees to radians multiply by pi/180 and radians to degrees multiply by 180/pi.

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.

Explore how sine and cosine graphs change with periods using period = 2π/|B| for y = sin(Bx) or y = cos(Bx); smaller periods stretch graph along x-axis without changing amplitude.

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 sine and cosine with transformations using y equals a cos(bx) minus c plus d, detailing amplitude, period, phase shift, vertical shift, and reflection over x axis.

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 tangent and cotangent with transformations, including reflection over the x-axis, a period of pi/2, and a phase shift of negative pi/4 for y = -cot(2x+2), showing asymptotes and intercepts.

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 by starting from sine and cosine, apply asymptotes where the base functions cross, and transform to obtain y = sec x and y = csc x.

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.

Master half-angle identities to evaluate expressions quickly, understanding quadrant-based signs and applying sine, cosine, and tangent half-angle formulas.

solve basic trigonometric equations by isolating sine and using the unit circle to find where sine is negative one half, yielding two solutions in 0 to 2π: 7π/6 and 11π/6.

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.

Solve an equation by factoring a quadratic in sin x, substitute x for sin x, and show sin x = -1 yields solutions while sin x = 2 has none.

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.

Practice solving trig equations with multiple angles by substituting y for 3x, solving sin y equals -root three over two, and converting back to x by y/3.