The Complete Guide to Algebra 2

Explore the order of operations, PEMDAS, and how to correctly simplify expressions using parentheses, exponents, multiply/divide, and add/subtract, emphasizing left-to-right rules and practicing with examples.

Explore how real numbers branch into rational and irrational numbers, with examples like pi and imperfect squares, and identify integers, whole numbers, and natural numbers through nesting-doll style charts.

identify like terms with the same variables and exponents, then combine them; apply the distributive property to expand expressions and simplify.

Master two cardinal rules for solving any algebra equation: identify what's done to the variable and reverse it, then apply the same operation to both sides.

Calculate the final test score needed to reach at least a 93 percent average across four tests by solving an inequality and summing known scores.

Learn to solve harder linear equations by applying opposite operations, combining like terms, and using distributive property, with practice problems and verification by substitution.

Solve linear inequalities and graph them, applying the same rules as linear equations, with sign flips when multiplying or dividing by negatives, and differentiate between or-type and and-type compound inequalities.

Learn to solve literal equations by isolating the desired variable using inverse operations, with examples for a, c, and base two in energy–mass–speed of light and trapezoid area.

Clear away everything but the absolute value, then split into two equations for the positive and negative cases; a negative right side yields no solution.

Using an absolute-value equation, the lecture finds the record high and low temperatures for the coldest inhabited place, given a 2-degree average and 90-degree difference in degrees Fahrenheit.

Learn to solve absolute value inequalities by isolating the absolute value, clearing constants, and splitting into two cases, using examples like |x|>3 to identify distinct solution regions.

Navigate the unit 1 test by applying order of operations, exponents, distribution, and combining like terms; solve equations and absolute value inequalities, and graph solutions on the number line.

Explore the concept of functions in algebra, identifying inputs and outputs, domain and range, and testing with vertical and horizontal line tests to distinguish 1-to-1 functions.

Learn how to evaluate functions by replacing x with a given input in f(x) and computing the output, with examples like f(9) and g(-7).

Learn to identify linear functions and graph them as straight lines, using standard form ax + by = c with integers a, b, c, a positive, and no common factor.

Learn to graph linear equations in standard form by finding the x-intercept and y-intercept and connecting the two points to draw the line.

Define slope as the rate of change, rise over run, Δy/Δx, and compute m from two points; note horizontal slopes (zero) and vertical slopes (undefined) with real-world examples.

Compare Mt. Evans and Pike's Peak roads to show how slope explains climbs, compute elevations and distances, and estimate Pike's Peak would need about 34 miles to match the grade.

Learn to write the equation of a line in y=mx+b form, identify m and b from points or a graph, and verify by plugging coordinates.

Solve a two-pool word problem using linear equations to compare a 5000-gallon pool draining at 100 gpm with a 12000-gallon pool filling at 3000 gph, showing which empties first.

Plot the y intercept on the graph, then use the slope intercept form to locate a second point by rise over run, and finally connect the points.

Master the point-slope form, y − y1 = m(x − x1), and compare it to standard form and slope-intercept form. Apply a given point and slope to write the line.

Learn how to convert equations from standard form into slope-intercept form by isolating y, with examples that yield y = -13/3 x + 8 and y = -3x - 4.

Evaluate two cell phone plans with base fees of $10 and $20, break-even at 200 minutes, and show that 1800 minutes favors the $20 plan by about $80.

Write equations for lines parallel or perpendicular to a given line by using slope: parallel shares the slope, perpendicular uses the negative reciprocal; then determine the intercept from a point.

Master piecewise functions by graphing line segments under specific x-conditions and writing the corresponding F(x) expression for each piece.

Learn to graph absolute value functions by finding the vertex, building a value table, and plotting symmetric points to create the shifted or flipped V-shaped graph.

Explore transformations of parent functions, including horizontal and vertical translations (left/right, up/down), dilations, and reflections, using graphs of y = x, y = x^2, and absolute value.

Describe transformations of parent functions like the absolute value and x^2, using g(x) to adjust left-right and up-down translations, vertical stretches/shrinks, horizontal expansions/shrinkage, and reflections.

Graph linear inequalities by converting to slope-intercept form, choosing solid or dotted lines, and shading above or below to reflect the inequality.

Tackle the unit 2 challenge by constructing a proportion, cross-multiplying, and simplifying to connect rafs to erts, us, zurichs, and tanks through successive unit conversions.

analyze unit 2 test problems by identifying functions and one-to-one properties with mapping diagrams, and convert and graph lines using standard, slope-intercept, and point-slope forms, including parallel and perpendicular cases.

Demonstrate solving absolute value equations with a three-point table to form a V, sketch a three-piece g(x), apply left shift, up, stretch, reflection, and graph inequality y < -2x -3.

Learn to solve systems of linear equations by graphing: convert to slope-intercept form, graph two lines, and use their intersection as the solution, including cases of parallel or coincident lines.

Learn to solve systems by substitution, solving for x or y with coefficient 1, substitute into the other equation, and verify by substitution; cover no or infinite solutions.

Learn to solve two-variable systems by elimination, canceling a variable by equalizing coefficients, then solve for x and y using substitution.

Solve a three-variable system by elimination, selecting a variable, using two equations to eliminate, then a third, and substitute to find x, y, z.

Eliminate a variable in three-variable systems to determine one, none, or infinitely many solutions, using the 0 = 0 versus 0 = -66 check and 3D plane visuals.

Practice solving real-world word problems with substitution or elimination by translating ticket sales into two equations, solving for senior and child ticket prices, and verifying the solution.

Apply distance equals rate times time to windy plane trip; set plane speed and wind as variables and solve by substitution to obtain 180 mph plane and 12 mph wind.

Learn to graph systems of linear inequalities by converting to slope-intercept form, choosing solid or dashed lines, shading above or below, and identifying regions where the inequalities hold.

Apply linear programming to maximize bakery revenue under labor constraints by analyzing the feasible region; the optimum is 225 yellow cakes and 0 strawberry cakes, earning $5,625.

Discover matrices as grids of numbers and learn to add, subtract, and multiply them using scalar multiplication, while checking dimensions and aligning corresponding entries.

Learn how to determine when two matrices can multiply, apply the row-by-column rule, respect the order of multiplication, and compute products with both numeric and word problems.

Learn how to find the determinant of a matrix using diagonal products and subtraction, understand bars notation, and its role in solving systems and determining inverses.

Cramer's rule solves a system with one solution by computing determinants D, D_x, and D_y, replacing columns with constants, and finding x = D_x/D, y = D_y/D.

Learn how to find a matrix inverse, A^-1, verify it with A^-1 A = I, and solve systems by inverse equations when the determinant is nonzero.

Learn to solve linear systems by using inverse matrices, including finding the inverse, multiplying both sides by the inverse, and verifying with the identity matrix.

Solve the unit 3 challenge by modeling scores with variables, forming equations, and using elimination to find the probability of scoring three or higher on the AP calculus exam.

Explore solving systems of equations by graphing, substitution, and elimination; practice three-variable systems and sketch solutions with inequalities, and touch on matrices.

The lecture covers adding, subtracting, and multiplying matrices, and computing determinants via cross multiplication. It also explains solving systems using Cramer's rule and inverse matrices.

Identify and graph quadratic functions by recognizing standard form, plotting the vertex, axis of symmetry, and key points to sketch a parabola.

Identify the vertex at (4,4) and the axis of symmetry x=4; the parabola opens up with a minimum value of 4, and the y-intercept is 36.

Extend the quadratic graphing method to inequalities, find the vertex, choose dashed or solid lines for inequality signs, and shade the region with a test point.

Solve quadratic problems on a graphing calculator by locating zeros and the vertex, using a baseball trajectory h(t)=-32t^2+100t+5 to find max height and ground time.

Explore how to multiply binomials with the foil method—first, outside, inside, last—an application of the distributive property shown with numbers and variables like x^2+5x+6.

Factor quadratics with leading coefficient 1 by treating factoring as the reverse of multiplication: find numbers that multiply to c and add to b, giving x^2 + bx + c.

Learn the swing method for factoring quadratics where a not equal to one. Follow four steps to combine ac and b, divide by a, then swing back to the factors.

Learn to factor quadratics by pulling out the greatest common factor, determine the smallest power of each variable, and verify the result by distributing.

Identify a perfect square trinomial with three terms and factor it as (x+3)^2, using the swing method and noting x^2+9 is not the correct expansion.

Explore factoring the difference of squares by recognizing the pattern a^2 minus b^2 factors as (a+b)(a-b), with zero middle term and examples like 81 and 100.

Clarifies when to pull the greatest common factor and when to use the swing method in factoring quadratics, with examples showing the GCF stays front and product matches original.

Apply factoring to solve quadratic equations: convert to standard form, factor with or without a greatest common factor, and use the zero product principle, including difference of squares cases.

Compare two dog teams in an algebra 2 problem, where second starts later and travels 5 mph faster; they meet 120 miles into race, solving distance equals rate times time.

Explore imaginary numbers as the square root of negative one and see how they differ from real numbers, with I squared equals -1 and applications in physics and impedance.

Learn how to raise the imaginary unit i to any power by using a four-term cycle, determining remainders modulo 4, and applying the last two digits rule for large exponents.

simplify imaginary numbers by factoring out perfect squares under radicals and using i for the square root of minus one, with examples like sqrt(-16)=4i and sqrt(-8)=2i sqrt2.

Explore how to add, subtract, and multiply complex numbers, recognizing real and imaginary parts, using examples like 3 and 0+5i, and applying i^2 = -1 for simplification.

Divide complex numbers by multiplying the numerator and denominator by the complex conjugate to remove imaginary terms, using foiling to simplify.

Explore perfect square trinomials and the completing the square method for solving quadratic equations. Learn the pattern by taking half of the middle coefficient and squaring it.

Master solving quadratics by completing the square, using standard form, balancing both sides, and handling plus/minus square roots and imaginary results when factoring fails.

Learn to solve quadratic equations by completing the square, even when the leading coefficient isn't one, balancing the equation and forming a perfect square with fractions and imaginary numbers.

Discover the quadratic formula and how completing the square derives it to solve quadratic equations, with graphing and factoring as alternative methods.

Learn to convert to standard form, identify a, b, c, and apply the quadratic formula to solve quadratic equations, yielding one or two real solutions or identifying no real solutions.

Explore the discriminant of the quadratic formula and how it signals two, one, or no real solutions, with roots that are rational, irrational, or imaginary.

Learn to convert a quadratic from standard form to vertex form by completing the square, using four-step method, and read the vertex from y = a(x - h)^2 + k.

Examine transformations of the quadratic graph, including left-right shifts, up-down moves, vertical stretch or shrink, and reflections across the axes of the parent parabola y = x^2.

Explore patterns in polygonal numbers from triangular numbers and squares to hexagonal and octagonal, using stepwise increases and differences. Compute the tenth hexagonal and tenth octagonal numbers and their product.

Learn to graph and solve quadratics: locate vertex and axis of symmetry, find zeros with a calculator, derive standard form from roots, and master swing method plus difference of squares.

Practice solving quadratic equations by factoring and the zero-product rule, using methods like the swing method, and introduce imaginary numbers with conjugates and foil, preparing for completing the square.

The lecture applies completing the square, the quadratic formula, and discriminant analysis to solve quadratics #15–20, and converts to vertex form before sketching the parabola and shading the solution region.

Explain the laws of exponents for multiplying and dividing powers, showing how to add or subtract exponents and apply power-to-power rules. Cover zero and negative exponents, flipping the base.

Apply exponent laws to simplify complex expressions by multiplying like bases, adding exponents, and flipping negatives to the numerator or denominator, as shown in harder examples.

Explore polynomials as expressions of one or more terms separated by plus or minus signs, including monomials, binomials, and trinomials, with degrees ranging from linear to quintic, including Kordek.

Learn to add, subtract, and multiply polynomials by combining like terms, using the distributive property and the box method, with division covered in a future video.

Dividing polynomials by a monomial, this lecture demonstrates term-by-term division, exponent subtraction, and coefficient simplification, with examples using k and x to build long division skills.

Learn polynomial long division through step-by-step examples, using zero placeholders, quotient and remainder, and see how synthetic division offers an alternative.

Explore synthetic division for a polynomial by setting up coefficients in a box, adding columns, multiplying by bottom row, and comparing the remainder with long division.

Learn how polynomial graphs behave at the ends, with even powers rising to infinity and odd powers to opposite infinities, and how sign changes flip the graph.

Learn how polynomial graphs reveal relative extrema and turning points, apply the fundamental theorem of algebra to count zeros (real or imaginary), and locate zeros with a graphing calculator.

Identify a greatest common factor, factor it out, recognize perfect cubes, then apply the sum or difference of cubes formulas to factor and verify by expansion.

Learn factoring by grouping in algebra 2 by splitting four or more terms into two groups, factoring out the greatest common factor, and rearranging when needed.

Learn factoring quadratic forms, including common factors, the swing method for nonunit leading coefficients, and completing the square for higher powers like x^4 and x^8.

Apply factor by grouping to a six-term polynomial, then factor by difference of squares and sum and difference of cubes to finish factoring.

Apply synthetic division to factor polynomials when a factor is given, yielding a depressed polynomial and a zero remainder. Then factor the depressed polynomial to achieve the fully factored form.

Solve polynomial equations using quadratic form techniques, including factoring, greatest common factor, grouping, and difference of cubes; apply the swing method and substitution to find all real and complex roots.

Master polynomial equations using grouping and cubes, including factoring by grouping, recognizing sums and differences of cubes, and applying quadratic formula or completing the square when needed.

Explore Descartes' rule of signs to determine zeros by sign changes, and identify positive, negative, and imaginary zeros (which come in pairs) in polynomial examples.

Learn to build polynomial functions from given zeros, including real zeros and complex conjugate pairs. Form factors (x minus zero) and expand to a real-coefficient standard form.

Apply the rational zeros theorem to list possible zeros as p over q, where p divides the constant term and q divides the leading coefficient, then test by factoring.

This lecture walks through a challenge by factoring a multi-term polynomial using grouping, common factors, differences of squares, sums and differences of cubes, and completing the square to finalize factorization.

Identify how to name polynomials by degree, including quadratic, binomial, and trinomial cases. Apply exponent rules, distribute negatives, simplify, and perform polynomial long division for unit 5 test problems.

Practice synthetic division and factor polynomials, find zeros and conjugate zeros, and analyze end behavior and graphing of polynomial functions using calculator tools.

Derive a polynomial with zeros at -1 and 2 by forming factors x+1 and x-2, substituting a = x-2, and expanding to the final function.

Explore adding, subtracting, multiplying, and dividing polynomials with functions, using foil or box method, arranging in standard form, and determining the domain by avoiding zero denominators.

Learn function composition by substituting one function into another using f, g, and h in algebra 2, with examples like h(g(x)) and g(f(x)) to practice simplification.

Learn how to find the inverse of a function by swapping x and y, solving for y, and using inverse notation, noting reflection across y=x and domain-range swap.

determine if two functions are inverses by applying the horizontal line test and showing that composing them in both orders yields x, with examples using f, g, and h.

Derive the inverse of the Fahrenheit temperature equation and use it to convert 30 degrees Celsius to 86 degrees Fahrenheit.

Learn to graph square root functions by using a table of values from the parent function and applying left/right shifts, up/down moves, and reflections, with domain and range considerations.

Learn to simplify radicals by factoring under the radical into perfect powers and moving them outside, using square, cube, and other nth roots to cancel exponents.

Learn to rationalize the denominator by simplifying radicals and multiplying the top and bottom by the denominator's radical to eliminate square roots.

Learn how to multiply radicals by factoring under the radical, extract perfect powers such as squares, cubes, and fourths, and simplify exponents to combine coefficients and radicands.

Master adding and subtracting radicals by identifying like radicals, extracting perfect squares, and combining coefficients, while keeping unlike radicals separate.

Explore how to multiply and divide radicals using foil, including handling mixed radicals and conjugates to rationalize denominators, cancel like radicals, and simplify results.

Divide and rationalize radicals by first simplifying, then using nth powers to cancel roots. Use the rationalizing denominator method to remove radicals from the denominator and obtain a simplified expression.

Explore how rational exponents relate to roots, showing that x^(m/n) equals the nth root of x^m, with square and cube roots, and how to flip negatives and convert forms.

Explore evaluating rational exponents by taking roots before powers, using examples like the sixth root of 64 to the fifth power, and by flipping negative exponents to simplify.

Learn to solve radical equations by isolating radicals, squaring both sides, and verifying solutions to avoid extraneous roots, including understanding the principal (positive) square root.

Explains how hang time relates to vertical jump using a simple equation, showing that a 1-second hang time requires a four-foot jump, and doubling hang time quadruples the vertical leap.

Learn to solve rational exponent equations, which resemble radical equations, by isolating the exponent and applying its reciprocal power, using cube roots and power rules with guided examples.

The unit challenge models toggling 1000 lockers, revealing that only perfect squares stay open. There are 31 perfect squares up to 1000, so 31 lockers remain open.

Perform a unit 6 test review on algebra 2 topics like subtracting functions, function composition, foil, inverse functions, domain and range, and simplifying square and cube roots.

Unit 6 test problems cover simplifying and rationalizing radicals, using conjugates, FOIL, and root operations, while checking for extraneous solutions in radical equations.

Explore exponential functions, compare growth versus decay, and learn to graph transformations of the parent function, including shifts, stretches, and reflections, with emphasis on the asymptote y=0 and the y-intercept.

Practice graphing transformed exponential functions by building a 3–5 point table for 2^x, identifying growth or decay, applying shifts and vertical stretch, and deriving the equation from the graph.

Solve exponential equations by rewriting bases as powers of the same number and equating exponents, with examples like 5^x = 5^9.

Explore how exponential decay models forensic temperature changes, using a 3% per hour drop from 35°C toward 20°C to estimate homicide time around 2:54 a.m.

Discover how exponential growth makes a simple paper fold explode into a vast distance. Hypothetically folding 50 times yields about 106,619,309.4 miles, surpassing the earth-sun distance.

Learn to convert between exponential and logarithmic forms by identifying the base, exponent, and result, and practice with examples like 12^2=144 and log base 7 of 343 equals 3.

Learn to evaluate logarithms by converting to exponential form, exploiting the inverse of exponentials, and solving with same-base approach, shown with examples like log base 5 of 25 equals 2.

Graph logarithms by hand with base 2 from y = log2 x, noting domain x>0 and range all real; shift right 1 and up 2 to y = log2(x−1)+2, x=1.

Explore how logarithms underpin disaster scales, like the Fujita and Richter scales, linking amplitude to magnitude and why a 9.2 earthquake is twenty five times more intense than 7.8.

Learn the three main logarithm properties: product, quotient, and power, and apply them to expanding and condensing logs with base 2, using worked examples and practice problems.

Learn to approximate logarithms without a calculator by using product, quotient, and power rules to break numbers into factors and powers, with practical examples.

Learn to solve logarithm equations with base 10 by converting to exponential form, checking for extraneous solutions, and applying quadratic methods when needed.

Combine logs into a single log using properties A, B, and C, then use exponential form to solve for x, checking for extraneous solutions and avoiding log of negative numbers.

Learn the change of base formula to compute log base 3 of 5 via log(5)/log(3), especially when calculators lack a log base button, with a concise proof.

Learn to solve exponent equations using logarithms by taking the log of both sides, applying log properties, and isolating the exponent to solve for x.

Explore the number e, named after Euler, its origins in compound interest, and its growth behavior with monthly, weekly, and daily compounding, approximately 2.71828.

Understand the natural logarithm, the log with base e, and how it differs from base-10 logs. See how ln and exponentiation cancel and relate to interest.

Learn how natural logarithms follow the same rules as regular logs, including expanding and condensing expressions using products, quotients, and powers, and converting between ln and exponent expressions.

Learn to solve natural logarithm equations by applying log rules, isolating the variable through exponentiation with base e, and checking solutions across multiple examples.

Explore compound interest using logarithms and exponential equations to model regular and continuous compounding, solving for final amounts, time, and rates with practical examples.

Solve a uranium-238 half-life word problem in algebra 2 by deriving the decay constant k and using natural logs to determine elapsed time.

Graph an exponential function by building a table of values, then solve equations by matching bases, converting between exponential and logarithmic forms, and applying log properties.

Use logs to solve exponential equations by moving exponents in front and isolating the variable. Apply compound interest and continuous growth formulas to compute balance and time.

Explore what a rational expression is, a fraction with a variable in the denominator, and learn to simplify it by factoring techniques, canceling, and identifying excluded domain values.

Learn to simplify rational expressions by factoring and canceling, identify excluded values to determine the domain, and apply techniques like greatest common factor, swing method, and difference of squares.

Learn to multiply and divide rational expressions by factoring, canceling common factors, and using the reciprocal when dividing by fractions, with the swing method and practice problems.

Learn how to find the least common denominator for rational expressions using the least common multiple, covering all denominators—including polynomials—and adjust numerators when adding or subtracting.

Learn to add and subtract rational expressions by finding the least common denominator, converting to that denominator, and simplifying by factoring and canceling common factors.

Learn to add complex fractions by combining numerators and denominators, then multiply the top by the reciprocal of the bottom using copy dot flip flop, and simplify.

Examine the reciprocal parent function f(x)=1/x, its graph with vertical asymptote x=0 and horizontal asymptote y=0, and how the domain excludes zero and approaches infinity.

Learn to graph rational functions by factoring, identifying intercepts and vertical and horizontal asymptotes, and using a value table to fill in holes, case 1 with y=0.

Graphing a case 2 rational function reveals a vertical asymptote at x = -1 and an oblique asymptote y = 1/4 x + 1/2, with no holes.

Explore direct, inverse, joint, and combined variation, learn to set up and solve equations like y = kx, xy = k, y = kxz, and yz = kx using given values.

Learn to solve rational equations by using the least common denominator to clear fractions, factor expressions, and verify solutions to avoid extraneous roots.

master harder rational equations by factoring and using the least common denominator to clear fractions, then solve the resulting quadratic and verify no extraneous solutions.

Translate word problems into equations by modeling mixing two solutions with different concentrations. Use total amount times concentration to find each quantity and verify the mixture concentration.

Learn to solve a wind-influenced word problem by using distance equals rate times time and the quadratic formula to estimate the plane's still-air speed around 340 mph.

Explore solving two word problems on combined work rates: convert hours to hourly rates, add them, and solve a proportion to find total time when working together.

Multiply by the least common multiple and test signs on a number line to solve rational inequalities. Identify the solution regions: x < -1 or 2 < x < 6.

We solve the unit 8 challenge using direct and inverse variation, showing that food, time, and dogs relate through a constant; doubling food and dogs keeps days constant.

Practice simplifying rational expressions by factoring, canceling common factors, and using reciprocals; identify discontinuities, holes, vertical and horizontal asymptotes, and graph key points.

tackle direct and inverse variation, determine constants, and solve equations with fractions and extraneous solutions, then apply mixture and work-rate word problems to algebra 2 concepts.

Apply conic sections to model the Earth's orbit as an ellipse with semi-major axis 93 million miles, compute the closest distance 91.5 million, and write the orbit equation.

Explore conic sections by slicing a cone with a plane to produce circle, ellipse, parabola, and hyperbola, with definitions using center, focus, directrix, and foci.

Explore the midpoint formula by averaging x and y coordinates, finding midpoints of numeric pairs and endpoints, and verifying with a coordinate diagram.

Apply the distance formula to compute the straight-line distance between points on a coordinate plane, using the x- and y-differences derived from the Pythagorean theorem.

Learn how parabolas are defined by focus and directrix, locate the vertex and axis of symmetry, and determine orientation and the latouche distance from vertex form.

Explore writing equations of parabolas using transformational form and vertex form, and learn how focus, directrix, and distance relations help convert to standard forms with examples.

Derive parabola equations from focus and directrix, locate the vertex as their midpoint, and convert to vertex form and general conic form via completing the square.

Learn to write circle equations in standard form using the distance formula to determine the center (h,k) and radius r.

Write circle equations by identifying the center (h, k) and radius r to form the standard equation, and use completing the square to convert from general form to standard form.

Explore hyperbolas, the fourth conic section, and learn about centers, shifts, vertices, and how a and b determine asymptotes and opening directions.

Practice graphing hyperbolas and write standard form equations from graphs, identifying the center, a and b values, the transverse and conjugate axes, and whether the opening is horizontal or vertical.

Compute hyperbola equations in standard form from given vertices and asymptotes, identify the center and opening direction, and determine a and b from the vertex distance and the asymptotes.

Learn to write hyperbola equations in standard form from given vertices and foci, determine the center and a, b, c to form (x-h)^2/a^2 - (y-k)^2/b^2 = 1.

Convert a general conic form to the hyperbola’s standard form by completing the square, yielding a horizontal hyperbola with center (7,6) and standard form equal to 1.

Explore the ellipse as a conic section, a stretched circle, with standard form equations, major and minor axes, vertices, foci, eccentricity, and how orientation follows a under x or y.

Identify an ellipse centered at (-2, -1) with major axis along the y-direction and a = 6, b = 5. Find its vertices and foci, with eccentricity c/a ≈ 0.55.

Graph an ellipse by locating its center, vertices, and co-vertices, determine the major and minor axes, and compute foci and eccentricity for a horizontal ellipse with a^2=16 and b^2=9.

Learn to write ellipses in standard form by using vertices to determine center and orientation, then compute a^2 and b^2 and apply c^2 = a^2 − b^2.

Master writing standard form equations of ellipses by identifying center and axes, then completing the square to determine a^2 and b^2 and the ellipse’s orientation.

Master solving systems of quadratic equations with substitution or elimination, plug in and foil to simplify, and explore intersections of parabola, circle, and hyperbola.

Compute midpoints and centers using the midpoint formula and diameter endpoints, apply the distance formula to find radii, and express circles in standard form by completing the square.

Learn to find parabola vertex and axis of symmetry, derive focus and directrix from vertex form, and analyze ellipses: center, vertices, major and minor axes, and foci.

Complete unit 9 test problems by deriving standard form equations for ellipse and hyperbola, identifying centers, vertices, foci, asymptotes, and using completing the square.