Pre-Calculus EVERYTHING Explained

Explore inverse trigonometric functions, where sine inverse, cosine inverse, and tangent inverse return angles from a numeric input, with specific ranges to guarantee a unique solution.

Identify adjacent and hypotenuse, apply cosine and inverse cosine to solve inverse trig in right triangles, with examples using 30° and 55.9°, and review unit circle concepts.

Practice solving addition and subtraction of polynomials by identifying like terms, grouping them, and applying signs to combine terms into a final expression.

Explore the multiplication and division of polynomials using the distributive property and long division, and apply power rules to simplify products and quotients.

Practice product and division of polynomials by applying the distributive property, expanding terms, identifying and combining like terms, and arranging results in descending powers.

Explore factorization of polynomials using the greatest common factor (gcf) method, extracting common factors term by term to reveal the zeros of the polynomial.

Apply binomial cube and binomial square rules to expand polynomials and work through sum and difference of cubes with concrete examples.

Practice synthetic division to solve polynomials, use the rational roots theorem to identify possible rational roots, and verify them with synthetic division, where zero remainder confirms a root.

Explore the binomial theorem and binomial expansion, using coefficients n choose k and factorials to expand (x+1)^5 and (2y-1)^4, with term-by-term applications and sign patterns.

Explore function transformations of the parent function, including vertical and horizontal shifts and stretches or shrinks, with an example y = 2(x-1)+3.

Identify the parent quadratic and its vertex at (2,3). Transform by vertical stretch 2, horizontal shift right 2, and vertical shift up 3 to obtain y = -2(x-2)^2 + 3.

Analyze vertical and horizontal asymptotes, compute the x-intercepts and y-intercept, and use point evaluations to sketch the rational function's graph.

Learn to graph rational functions using reciprocal transformations, determine horizontal and vertical asymptotes, x-intercept, y-intercept, and evaluating points to sketch accurate graphs.

Learn how composite functions combine an inner function with an outer function, using the inner output as the outer input to obtain the final result, as shown with examples.

Explore composite functions by identifying inner and outer functions, evaluating g(x) and applying f to the result, and simplifying to obtain f(g(x)) across several exercises.

Apply composite functions to prove inverse relationships by showing F(G(x)) = x and G(F(x)) = x; use examples to distinguish inverse and non-inverse functions.

Explore vectors, defined by magnitude and direction, and how they differ from scalars. See how vectors describe movement between positions and relate to velocity and force in math and physics.

Compute vector magnitude using the Pythagoras theorem and determine direction with the inverse tangent, using x and y components; illustrate with multiple vectors and discuss clockwise versus counterclockwise angles.

Graphically add vectors using the tip-to-tail method to get the resultant from the tail of A to the tip of B, and apply the parallelogram method with origin-positioned vectors.

Subtract vectors component-wise by their x and y components; the order matters due to the negative sign. For example, a(3,-4) and b(-1,6) yield a-b = (4,-10).

Compute the unit vector by dividing each component by the vector’s magnitude, as with A = (4,4), which has magnitude 4√2 and unit vector (1/√2, 1/√2).

We define matrices as rectangular arrangements of elements—numbers, letters, or symbols—whose order is given by rows and columns, including three by one and square forms, with indexing.

Explore matrix arithmetic by adding two three-by-three matrices, understanding order, rows and columns, and sums of corresponding elements, and verify that A+B equals B+A with scalar multiplication.

Learn how to multiply matrices by checking inner dimensions, determine the product order, and compute results with row by column products, illustrated with 2x2 and 2x3 examples.

Practice transposing matrices by turning each row into a column, changing dimensions from 2x3 to 3x2 and from 3x2 to 2x3, including a 3x3 case.

Introduce the determinant for square matrices and the two-by-two ad minus bc rule, and demonstrate the rain method for three-by-three matrices using the principal and secondary diagonals.

Explore how to compute determinants of 2x2 and 3x3 matrices using two methods: expansion along the first row with submatrix determinants, and the principal diagonal minus the secondary diagonal rule.

Apply the Gauss-Jordan method to find a matrix inverse by augmenting a 3×3 matrix with the identity and using row operations to turn the left side into the identity.

Compute the inverse of a matrix by calculating its determinant, forming the cofactors matrix, transposing it, and multiplying by one over the determinant.

Learn to find a matrix inverse using Gauss-Jordan method with an augmented [matrix|identity], and compare it to the cofactor matrix method while noting invertibility.

Learn to solve a two-equation, two-unknown system using matrices and Jordan elimination, forming the augmented matrix and deriving X and Y from the identity on the left.

Solve a 2x2 system of equations using Gauss-Jordan elimination on a matrix formed from -12 and 9 with constants 7 and 6 to find x and y.

Represent position vectors in matrix form and apply transformation matrices to rotate and reflect vectors; study rotation by 90, 180, 270 degrees and reflections over x, y, and y=x.

Transform polygon ABC with rotation matrices, rotating by 90, 180, and 270 degrees by multiplying 2x2 rotation matrices by the polygon's vertex coordinates and graphing the results.

The polynomial hyperbola converts to standard form x^2/16 - y^2/9 = 1 with center at (0,0). It opens horizontally, with vertices (±4,0), foci (±5,0), and eccentricity 5/4.

Apply the arithmetic sequence formula a_n = a_1 + d(n-1) to find terms like a_5, verify the common difference by subtracting consecutive terms, and solve examples 25, 20, 15.

Apply the arithmetic sequence formula to find a7 and confirm the common difference, then solve fractional sequences by separating the nominator and denominator into arithmetic progressions.

Identify geometric sequences by checking constant ratios, apply a_n = a_1 r^{n-1}, and compute terms such as the 8th with ratio 5 and the 15th with ratio 11 or 1/4.

Calculate the tenth, twelfth, and fifteenth terms using the arithmetic sequence formula. Compute the sums using the arithmetic series formula S_n = n/2 (a_1 + a_n).

Use the finite geometric series formula to find sums from 1 to 10, 4 to 15, and 2 to 20 by identifying a1, r, and n.

Solve infinite geometric series problems by identifying a1 and r from a geometric sequence, and apply the infinite sum formula a1/(1 - r) to compute totals such as 72.

Apply the multiplication rule of probability to independent events by calculating the chance of getting three sixes on three dice, using per-die probabilities and multiplying them.

Explore mutually exclusive events and how to compute probability of a or b by adding their probabilities, illustrated with a dice example showing one over six for five or one.

Explore probabilities with a standard 52-card deck, computing ace and other card chances, including ace of hearts, queen of hearts, and jack of diamonds, with 1/52 outcomes.

Explore how the multiplication rule and the counting principle determine total combinations in meals, dice outcomes, and school elections, with step-by-step probability examples.

Explore probability concepts through counting methods, calculate combinations with and without repetition for a five-character password, and distinguish mutually exclusive, overlapping, and dependent events using the multiplication rule.

Learn how to count permutations by arranging n objects in r positions using p(n, r) = n!/(n-r)!, illustrated by three letters forming two-letter orders that yield six arrangements.

Learn permutation and combination concepts with real examples, distinguishing when order matters. Calculate using nPr and nCr, such as 45 choose 5 and 20 people into four teams, with factorials.

Apply Pascal's triangle to solve probability and combinatorics problems, such as five coin flips and selecting three people from ten, by identifying the relevant coefficients and total outcomes.

Learn how to divide complex numbers by multiplying by the conjugate of the denominator, turning the denominator into a real number via a^2+b^2, then simplify to a+bi.

Explore division of complex numbers by multiplying by the conjugate, apply the difference of squares, and simplify using i squared equals minus one while practicing arithmetic operations on complex numbers.

Master the conjugate zero theorem by forming polynomials with real coefficients from given zeros, including complex conjugates, and multiplying factors to obtain cubic polynomials.

Compute rectangular form of a complex number and determine its polar coordinates using Pythagoras and inverse tangent, locating theta in third quadrant and writing r(cos theta + i sin theta).

Convert complex numbers from rectangular to polar form by computing magnitude via the Pythagorean theorem and the angle via inverse tangent, while identifying the correct quadrant.