Pre-Calculus EVERYTHING Explained

Explore six trigonometric ratios—sine, cosine, tangent, cosecant, secant, cotangent—and their relations to opposite, adjacent, and hypotenuse in right triangles, with reference angles, unit circle, and inverse trig functions.

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.

Practice applying the unit circle to evaluate trigonometric functions at various angles, determine reference angles, quadrant signs, and use 30–60–90 and 45–45–90 values to find cosine, sine, and tangent.

Apply inverse trigonometric functions to determine angles in right triangles by identifying opposite and adjacent sides, using tangent, sine, and cosine ratios.

Identify the triangle sides relative to the angle, use sine, cosine, and tangent definitions, then apply the inverse trig functions with a calculator when values aren’t on the unit circle.

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.

Explores polynomials by defining terms, constants, variables, and exponents, then shows how to add and subtract polynomials by combining similar terms.

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.

Learn to factor quadratics using the difference of squares by decomposing c into two numbers that multiply to c and sum to b, yielding the factorized form and the zeros.

Apply the ac method to factor quadratics by using a, b, c; multiply a and c, decompose into two numbers, and apply the greatest common factor.

Master the square and cube of binomials, including the sum and difference of squares and cubes, memorize the expansion formulas, identify a and b, and practice polynomial expansion.

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

Distinguish between binomial cubes and difference of cubes, and apply the appropriate factoring formulas to solve exercises using A^3 and B^3 values.

Identify a, b, and c and apply the quadratic formula to find roots. Examples show -1 ± square root of 2 and 2/3 and -3/2 as solutions.

Perform synthetic division on a polynomial by a binomial, using coefficients in leading-term order, identify the root, compute the quotient, and verify zero remainder.

Explore the rational root theorem and synthetic division to identify possible and actual roots of polynomials, using coefficients, constants, and leading terms to factor and verify.

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.

Learn to expand binomials using Pascal's triangle and the binomial theorem, identify rows as coefficients, assign decreasing powers, and handle alternating signs in negative cases.

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 the binomial theorem and binomial expansion through practical exercises, learn to compute coefficients with Pascal's triangle, and apply these techniques to expand polynomials.

Define the domain and range of piecewise functions from graphs using interval notation and union, and reinforce function notation and the concepts of domain and range.

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.

Identify the parent function x^2, apply a horizontal stretch by 1/3, and shift the graph to x = -6 and y = -2; verify by plotting points like (0,6).

Identify the parent function square root of x, apply a vertical shift of -1 and a horizontal effect from multiplying x by 1/2, then plot points to sketch the graph.

Explore rational functions with polynomial numerators and denominators, graphing them by locating vertical and horizontal asymptotes, intercepts, reciprocal functions, and using long division.

Identify vertical asymptotes at x = ±3 for 1/(x^2-9) and the horizontal asymptote y = 0; note no x-intercept, y-intercept -1/9, and how factoring reveals these features.

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.

Explore real-life modelling with composite functions by linking hours studied to grades and then to job performance in percentage, using the first function's output as the second's input.

Apply composite functions to model ticket sales: hours of daily practice determine wins, which determine season tickets; with seven point five hours, roughly 47 wins yield 220,900 tickets.

Explore composite functions using table-based evaluations of F(G(x)) with inputs like minus five and minus three, and learn to map inner outputs through F to obtain final results.

Explore inverse functions by swapping x and y, proving with composite functions, and practicing with exercises to solidify the concept.

Learn to find inverse functions by swapping x and y and solving for y, shown through linear, quadratic, and cubic examples to master the inverse function.

Demonstrate proving two functions are inverse via composites, by showing F(G(x)) = x and G(F(x)) = x using the example F(x)=4/(x−1) and G(x)=(4+x)/x.

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.

Explore how position vectors use x and y components, compute magnitude with the Euclidean norm via Pythagoras, and find direction as the angle with the x axis using inverse tangent.

Explore the difference between vectors and scalars by examining magnitude and direction, including the angle with the x axis, and how multiplying by a scalar changes length but preserves direction.

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.

Explore scalar multiplication of vectors and graph the results, including doubling, halving, and tripling. Learn vector magnitude via the pythagorean theorem and direction via arctan(y/x).

Learn how to add vectors by summing their x and y components, explore the commutative property A+B = B+A, and apply this to example vectors to find the resultant vector.

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).

Explore graphical subtraction of vectors by subtracting A minus B using a graphic join, showing the result as a vector from B to A in position vectors, contrasting with addition.

Graph the vectors a and b from the origin, scale b by two, and use the parallelogram method to find a + 2b = (5,0) in the coordinate system.

Explore graphical vector operations, including scalar multiplication by fractions, addition and subtraction of vectors, and the commutative property A+B=B+A, all demonstrated with components and graphs.

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).

Compute unit vectors from component vectors by finding magnitudes with Pythagoras and dividing components by the magnitude. Practice with vectors A, B, C, and D reinforces i and j directions.

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 how to determine the order of a matrix by counting rows and columns, identify each element by its row–column position, and compute the total elements as rows times columns.

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.

Identify the order and inner-dimension feasibility of matrix multiplication, then multiply rows by columns to obtain a 2 by 3 product matrix, illustrated with 2x2, 2x3, and 2x1 examples.

Practice multiplying matrices by verifying inner dimensions, determining the result order from the outer numbers, and computing products for 3x3 and 2x3 cases using rows-by-columns.

Identify and apply the identity matrix in matrix multiplication, noting its square shape and diagonal ones with zeros elsewhere, and prove it commutes with any same-size matrix.

Learn how to transpose matrices by converting rows to columns, flipping dimensions from 3x2 to 2x3 and from 2x3 to 3x2, with example matrices A and B.

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.

Compute the inverse of a 3×3 matrix by finding its determinant, constructing the cofactors matrix, transposing to form the adjugate, and applying one over the determinant.

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.

Explore how to find the inverse of a 2x2 matrix by computing its determinant, swapping the diagonal elements, negating the off-diagonal terms, and multiplying by the reciprocal of the determinant.

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 system of equations using the jordan elimination method by forming the augmented matrix and applying row operations to find x and y.

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.

Explore conic sections formed by plane cuts of a cone, covering circles, parabolas, ellipses, and hyperbolas, and practice the circle's equation and graphing.

Learn the parabola’s key parts—focus, directrix, and vertex—and its equations, including (x−h)^2=4p(y−k) and (y−k)^2=4p(x−h), with axis of symmetry and concavity explained.

Identify the parabola’s vertex, determine p from the coefficient of y, compute the focus and directrix, and sketch the parabola using the vertex, focus, and directrix as guides.

Determine the vertex (-2, 1) and p = 1/2 for the parabola x - h squared = 4 p (y - k). The focus is (-2, 3) and the directrix is y = 1/2.

Using completing the square, this lecture converts a quadratic into the standard form to identify a parabola that opens upwards, with vertex (-1, -3) and focus (-1, -71/24).

Derive the equation of the parabola by completing the square, identify the vertex (-4,2) and focus (-4,7/4), and graph the parabola opening downward.

Identify and graph the vertically oriented ellipse with center at the origin, its foci at (0, ±√7), vertices at (0, ±4), major axis 8, minor axis 6, and eccentricity √7/4.

Identify the ellipse centered at 3, -2 with vertical major axis; a = 1, b = 5. Plot vertices 3, 3 and 3, -7; foci 3, -2 ± 2√6.

Finds and graphs an ellipse with center (-1,2), major axis 8, minor axis 4, vertices (3,2) and (-5,2); compute foci, eccentricity, and equation.

Explore the hyperbola's parts, derive its standard equation, and practice problems, including center coordinates (h,k), a and b, vertices, asymptotes, and orientation along the x or y axis.

Identify the origin as center, a = 4, b = 3, vertices at ±4,0, foci at ±5,0; box method draws asymptotes y = ±(3/4)x and e = 5/4.

Identify the hyperbola with center (-2,3) and that opens along the y-axis. Compute a=4, b=√2, and C=3√2; asymptotes are diagonals of box, with vertices (-2,3±√2) and foci (-2,3±3√2), eccentricity 3.

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.

Explore sequences in pre-calculus, focusing on arithmetic sequences, the common difference, and solving related problems with examples and practice.

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.

Explore arithmetic sequences through word problems, identify the common difference, and apply the formula a_n = a_1 + (n-1)d to solve day-by-day studying, seating, and bounce scenarios.

Identify geometric sequences by using a common ratio r and the formula a_n = a1 r^(n-1), and illustrate with examples like 2, 4, 8 where a10 = 1024.

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.

Learn to solve geometric sequence problems by forming a system of equations from two known terms, then apply a_n = a_1 r^{n-1} to find missing values such as a_9.

Understand arithmetic series by contrasting sequences and series, using sigma notation to sum arithmetic sequences. Apply the formula n/2 (a1 + an) to total from 1 to 100.

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.

Compute the sum of an infinite geometric series when the ratio is less than one, using a1/(1-r).

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 dependent events and the multiplication rule, using a two-draw marble example (without replacement) to show how the probability of blue then red is computed.

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.

Explore how Pascal's triangle explains probability and binomial-like expansion, using a three-child scenario to count outcomes and compute two girls and one boy as three-eighths.

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.

Explore imaginary numbers by defining i as the square root of negative one, and follow the powers i, i^2, i^3, and i^4 to show patterns: -1, -i, and 1.

Learn to compute square roots of negative numbers using imaginary numbers, with i = sqrt(-1), factoring negatives and applying sqrt rules to get expressions like sqrt(-25)=5i and sqrt(-8)=2i sqrt(2).

Explore the power of imaginary numbers by decomposing exponents into multiples of four, using i^4=1 and exponent rules to simplify i^n with worked examples.

Define and identify the conjugate of a complex number by changing the sign of the imaginary part while keeping the real part unchanged, illustrated with examples.

Explore adding and subtracting complex numbers by separately adding real parts and imaginary parts, using the distributive property. A complex number equals a real part plus an imaginary part.

Apply the distributive property to multiply complex numbers, treating i as the imaginary unit and using i^2 = -1 to combine real and imaginary parts.

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.

Explore the complex plane by graphing complex numbers on the real and imaginary axes, learn their polar form, and practice polar coordinates using examples like 4+3i.

Explore conjugates of complex numbers in the complex plane by flipping the imaginary part, revealing reflections across the real axis with examples such as 3+2i and 3-2i.

Explore polar coordinates for complex numbers by using magnitude and angle to express z as r(cos theta + i sin theta), with r from pythagorean theorem and theta from arctan.

Convert complex numbers between rectangular and polar forms on the complex plane by computing magnitude and angle, then express in form using r cos theta + i r sin theta.

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.

Learn to divide complex numbers in polar form by subtracting exponents, using e^{i theta}, and express results in polar form with cos and sin.

Learn to compute powers of complex numbers in polar form using De Moivre's formula, convert between forms, and evaluate angles with e^{i theta} = cos theta + i sin theta.