College Algebra with Trigonometry - ALEKS

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College Algebra with TrigonometryThis course covers the topics shown below, new topics have been highlighted.Students navigate learning paths based on their level of readiness.Inst itut ional users may customize the scope and sequence to meet curricular needs.

Curriculum (597 topics + 646 addit ional topics)

Algebra and Geometry Review (126 topics)

Real Numbers and Algebraic Expressions (14 topics)Signed f ract ion addit ion or subtract ion: BasicSigned f ract ion subtract ion involving double negationSigned f ract ion mult iplicat ion: BasicSigned f ract ion divisionComputing the distance between two integers on a number lineExponents and integers: Problem type 1Exponents and signed f ract ionsOrder of operat ions with integersEvaluat ing a linear expression: Integer mult iplicat ion with addit ion or subtract ionEvaluat ing a quadrat ic expression: IntegersEvaluat ing a linear expression: Signed f ract ion mult iplicat ion with addit ion or subtract ionDistribut ive property: Integer coefficientsUsing distribut ion and combining like terms to simplify: UnivariateUsing distribut ion with double negation and combining like terms to simplify: Mult ivariate

Exponents (20 topics)Introduct ion to the product rule of exponentsProduct rule with posit ive exponents: UnivariateProduct rule with posit ive exponents: Mult ivariateIntroduct ion to the power of a power rule of exponentsIntroduct ion to the power of a product rule of exponentsPower rules with posit ive exponents: Mult ivariate productsPower rules with posit ive exponents: Mult ivariate quotientsSimplifying a rat io of mult ivariate monomials: BasicIntroduct ion to the quotient rule of exponentsSimplifying a rat io of univariate monomialsQuotient of expressions involving exponentsEvaluat ing expressions with exponents of zeroEvaluat ing an expression with a negative exponent: Whole number baseEvaluat ing an expression with a negative exponent: Posit ive f ract ion baseEvaluat ing an expression with a negative exponent: Negative integer baseRewrit ing an algebraic expression without a negative exponentIntroduct ion to the product rule with negative exponentsQuotient rule with negative exponents: Problem type 1Power of a power rule with negative exponentsPower rules with negative exponents

Polynomial Expressions (14 topics)Degree and leading coefficient of a univariate polynomialSimplifying a sum or difference of two univariate polynomialsMult iplying a univariate polynomial by a monomial with a posit ive coefficientMult iplying a univariate polynomial by a monomial with a negative coefficientMult iplying a mult ivariate polynomial by a monomialMult iplying binomials with leading coefficients of 1Mult iplying binomials with leading coefficients greater than 1Mult iplying binomials in two variablesMult iplying conjugate binomials: UnivariateSquaring a binomial: UnivariateSquaring a binomial: Mult ivariateMult iplying binomials with negative coefficientsMult iplicat ion involving binomials and trinomials in one variableMult iplicat ion involving binomials and trinomials in two variables

Factoring Polynomials (16 topics)Greatest common factor of 2 numbersFactoring a linear binomialIntroduct ion to the GCF of two monomialsGreatest common factor of two mult ivariate monomialsFactoring out a monomial f rom a polynomial: Univariate


Factoring out a monomial f rom a polynomial: Mult ivariateFactoring out a binomial f rom a polynomial: GCF factoring, basicFactoring a univariate polynomial by grouping: Problem type 1Factoring a quadrat ic with leading coefficient 1Factoring out a constant before factoring a quadrat icFactoring a quadrat ic with leading coefficient greater than 1: Problem type 1Factoring a quadrat ic with leading coefficient greater than 1: Problem type 2Factoring a quadrat ic with a negative leading coefficientFactoring a perfect square trinomial with leading coefficient 1Factoring a difference of squares in one variable: BasicFactoring a difference of squares in one variable: Advanced

Rational Expressions (28 topics)Restrict ion on a variable in a denominator: LinearSimplifying a rat io of factored polynomials: Linear factorsSimplifying a rat io of polynomials using GCF factoringSimplifying a rat io of polynomials by factoring a quadrat ic with leading coefficient 1Simplifying a rat io of polynomials: Problem type 1Mult iplying rat ional expressions involving linear expressionsMult iplying rat ional expressions involving quadrat ics with leading coefficients of 1Dividing rat ional expressions involving linear expressionsDividing rat ional expressions involving quadrat ics with leading coefficients of 1Least common mult iple of 2 numbersLeast common mult iple of 3 numbersIntroduct ion to the LCM of two monomialsFinding the LCD of rat ional expressions with linear denominators: Relat ively primeWrit ing equivalent rat ional expressions with polynomial denominatorsIntroduct ion to adding f ract ions with variables and common denominatorsAdding rat ional expressions with common denominators and monomial numeratorsAdding rat ional expressions with common denominators and binomial numeratorsAdding rat ional expressions with common denominators and GCF factoringAdding rat ional expressions with common denominators and quadrat ic factoringAdding rat ional expressions with different denominators and a single occurrence of a variableAdding rat ional expressions with denominators ax and bx: BasicAdding rat ional expressions with denominators ax and bx: AdvancedAdding rat ional expressions with linear denominators without common factors: BasicComplex f ract ion without variables: Problem type 1Complex f ract ion without variables: Problem type 2Complex f ract ion involving univariate monomialsComplex f ract ion: GCF factoringComplex f ract ion made of sums involving rat ional expressions: Problem type 1

Perfect Squares and nth Roots (7 topics)Square root of a rat ional perfect squareSquare roots of perfect squares with signsIntroduct ion to simplifying a radical expression with an even exponentSquare root of a perfect square monomialIntroduct ion to solving an absolute value equationCube root of an integerFinding nth roots of perfect nth powers with signs

Rational Exponents (4 topics)Convert ing between radical form and exponent formRational exponents: Unit f ract ion exponents and whole number basesRational exponents: Non-unit f ract ion exponent with a whole number baseRational exponents: Negative exponents and f ract ional bases

Radical Expressions (19 topics)Simplifying the square root of a whole number less than 100Simplifying a radical expression with an even exponentIntroduct ion to simplifying a radical expression with an odd exponentSimplifying a radical expression with an odd exponentSimplifying a higher root of a whole numberIntroduct ion to square root addit ion or subtract ionSquare root addit ion or subtract ionIntroduct ion to square root mult iplicat ionSquare root mult iplicat ion: BasicSquare root mult iplicat ion: AdvancedIntroduct ion to simplifying a product of radical expressions: UnivariateIntroduct ion to simplifying a product involving square roots using the distribut ive propertySimplifying a product involving square roots using the distribut ive property: BasicSimplifying a product involving square roots using the distribut ive property: AdvancedSimplifying a quotient of square rootsSimplifying a quotient involving a sum or difference with a square root


Rationalizing a denominator: Quotient involving square rootsRationalizing a denominator: Square root of a f ract ionRationalizing a denominator using conjugates: Integer numerator

Geometry (4 topics)Circumference of a circleVolume of a rectangular prismIntroduct ion to the Pythagorean TheoremPythagorean Theorem

Equations and Inequalit ies (83 topics)

Linear Equations and Applicat ions (27 topics)Addit ive property of equality with signed f ract ionsMult iplicat ive property of equality with signed f ract ionsSolving a mult i-step equation given in f ract ional formSolving a linear equation with several occurrences of the variable: Variables on the same side and distribut ionSolving a linear equation with several occurrences of the variable: Variables on both sides and distribut ionSolving a linear equation with several occurrences of the variable: Variables on both sides and two distribut ionsSolving a linear equation with several occurrences of the variable: Fract ional forms with monomial numeratorsSolving a two-step equation with signed f ract ionsSolving a linear equation with several occurrences of the variable: Variables on both sides and f ract ional coefficientsSolving a linear equation with several occurrences of the variable: Fract ional forms with binomial numeratorsSolving a proport ion of the form (x+a)/b = c/dSolving for a variable in terms of other variables using addit ion or subtract ion: BasicSolving for a variable in terms of other variables using addit ion or subtract ion: AdvancedSolving for a variable in terms of other variables using mult iplicat ion or division: BasicSolving for a variable in terms of other variables using mult iplicat ion or division: AdvancedSolving for a variable in terms of other variables using addit ion or subtract ion with divisionSolving for a variable inside parentheses in terms of other variablesSolving for a variable in terms of other variables in a linear equation with f ract ionsTranslat ing a sentence into a one-step equationTranslat ing a sentence into a mult i-step equationSolving a word problem with two unknowns using a linear equationSolving a decimal word problem using a linear equation of the form Ax + B = CSolving a word problem with three unknowns using a linear equationSolving a one-step word problem using the formula d = rtSolving a distance, rate, t ime problem using a linear equationFinding the perimeter or area of a rectangle given one of these valuesFinding the sale price given the original price and percent discount

Absolute Value Equations (2 topics)Solving an absolute value equation: Problem type 1Solving an absolute value equation: Problem type 2

Linear Inequalit ies and Applicat ions (7 topics)Graphing a linear inequality on the number lineGraphing a compound inequality on the number lineSet-builder and interval notat ionIdentifying solut ions to a two-step linear inequality in one variableSolving a two-step linear inequality: Problem type 1Solving a two-step linear inequality: Problem type 2Solving a linear inequality with mult iple occurrences of the variable: Problem type 1

Rat ional Equations that Simplify to Linear (8 topics)Solving a rat ional equation that simplifies to linear: Denominator xSolving a rat ional equation that simplifies to linear: Denominator x+aSolving a rat ional equation that simplifies to linear: Denominators a, x, or axSolving a rat ional equation that simplifies to linear: Denominators ax and bxSolving a rat ional equation that simplifies to linear: Like binomial denominatorsSolving a rat ional equation that simplifies to linear: Unlike binomial denominatorsSolving for a variable in terms of other variables in a rat ional equation: Problem type 1Solving for a variable in terms of other variables in a rat ional equation: Problem type 2

Complex Numbers (4 topics)Using i to rewrite square roots of negative numbersAdding or subtract ing complex numbersMult iplying complex numbersDividing complex numbers

Quadrat ic Equations (20 topics)Solving an equation writ ten in factored formFinding the roots of a quadrat ic equation of the form ax2 + bx = 0Finding the roots of a quadrat ic equation with leading coefficient 1


Finding the roots of a quadrat ic equation with leading coefficient greater than 1Solving a quadrat ic equation needing simplificat ionRoots of a product of polynomialsWrit ing a quadrat ic equation given the roots and the leading coefficientSolving a word problem using a quadrat ic equation with rat ional rootsSolving an equation of the form x2 = a using the square root propertySolving a quadrat ic equation using the square root property: Exact answers, basicSolving a quadrat ic equation using the square root property: Exact answers, advancedComplet ing the squareSolving a quadrat ic equation by complet ing the square: Exact answersApplying the quadrat ic formula: Exact answersApplying the quadrat ic formula: Decimal answersSolving a quadrat ic equation with complex rootsDiscriminant of a quadrat ic equationSolving a word problem using a quadrat ic equation with irrat ional rootsSolving an equation using the odd-root property: Problem type 1Solving an equation using the odd-root property: Problem type 2

Rational Equations that Simplify to Quadrat ic (5 topics)Restrict ion on a variable in a denominator: Quadrat icSolving a rat ional equation that simplifies to linear: Factorable quadrat ic denominatorSolving a rat ional equation that simplifies to quadrat ic: Denominator xSolving a rat ional equation that simplifies to quadrat ic: Binomial denominators, constant numeratorsSolving a rat ional equation that simplifies to quadrat ic: Binomial denominators and numerators

Radical Equations (10 topics)Introduct ion to solving a radical equationSolving a radical equation that simplifies to a linear equation: One radical, basicSolving a radical equation that simplifies to a linear equation: One radical, advancedSolving a radical equation that simplifies to a linear equation: Two radicalsSolving a radical equation that simplifies to a quadrat ic equation: One radical, basicSolving a radical equation that simplifies to a quadrat ic equation: One radical, advancedSolving for a variable in terms of other variables in an equation involving radicalsSolving an equation with a root index greater than 2: Problem type 1Solving an equation with a root index greater than 2: Problem type 2Solving an equation that can be writ ten in quadrat ic form: Problem type 1

Graphs and Funct ions (139 topics)

The Coordinate Plane, Distance, and Midpoint (9 topics)Reading a point in the coordinate planePlott ing a point in the coordinate planeNaming the quadrant or axis of a point given its coordinatesNaming the quadrant or axis of a point given the signs of its coordinatesTable for a linear equationDistance between two points in the plane: Exact answersMidpoint of a line segment in the planeIdentifying solut ions to a linear equation in two variablesFinding a solut ion to a linear equation in two variables

Graphs of Equations (16 topics)Graphing a linear equation of the form y = mxGraphing a line given its equation in slope-intercept form: Integer slopeGraphing a line given its equation in slope-intercept form: Fract ional slopeGraphing a line given its equation in standard formGraphing a vert ical or horizontal lineFinding x- and y-intercepts given the graph of a line on a gridFinding x- and y-intercepts of a line given the equation: BasicFinding x- and y-intercepts of a line given the equation: AdvancedGraphing a line by first finding its x- and y-interceptsFinding intercepts of a nonlinear funct ion given its graphFinding x- and y-intercepts of the graph of a nonlinear equationGraphing an absolute value equation of the form y = A|x|Graphing a parabola of the form y = ax2

Graphing a parabola of the form y = ax2 + cGraphing a cubic funct ion of the form y = ax3Determining if graphs have symmetry with respect to the x-axis, y-axis, or origin

Slope and Equations of Lines (17 topics)Finding slope given the graph of a line on a gridFinding slope given two points on a lineFinding the slopes of horizontal and vert ical linesGraphing a line given its slope and y-intercept


Finding the slope and y-intercept of a line given its equation in the form y = mx + bFinding the slope and y-intercept of a line given its equation in the form Ax + By = CGraphing a line by first finding its slope and y-interceptWrit ing an equation of a line given its slope and y-interceptWrit ing an equation in slope-intercept form given the slope and a pointFinding the slope and a point on a line given its equation in point-slope formWrit ing the equation of a line in point-slope form given the slope and a pointWrit ing the equation of a line given the y-intercept and another pointWrit ing the equation of a line through two given pointsWrit ing the equations of vert ical and horizontal lines through a given pointFinding slopes of lines parallel and perpendicular to a line given in slope-intercept formFinding slopes of lines parallel and perpendicular to a line given in the form Ax + By = CWrit ing equations of lines parallel and perpendicular to a given line through a point

Linear Applicat ions (5 topics)Writ ing and evaluat ing a funct ion that models a real-world situat ion: AdvancedWrit ing an equation and drawing its graph to model a real-world situat ion: AdvancedFinding the intercepts and rate of change given a graph of a linear funct ionInterpret ing the parameters of a linear funct ion that models a real-world situat ionApplicat ion problem with a linear funct ion: Finding a coordinate given two points

Circles (6 topics)Identifying the center and radius to graph a circle given its equation in standard formIdentifying the center and radius to graph a circle given its equation in general form: BasicWrit ing the equation of a circle centered at the origin given its radius or a point on the circleWrit ing an equation of a circle given its center and radius or diameterWrit ing an equation of a circle given its center and a point on the circleWrit ing an equation of a circle given the endpoints of a diameter

Funct ions (26 topics)Identifying funct ions f rom relat ionsVert ical line testTable for a linear funct ionEvaluat ing funct ions: Linear and quadrat ic or cubicEvaluat ing a rat ional funct ion: Problem type 1Evaluat ing a rat ional funct ion: Problem type 2Table for a square root funct ionEvaluat ing a cube root funct ionEvaluat ing funct ions: Absolute value, rat ional, radicalEvaluat ing a piecewise-defined funct ionVariable expressions as inputs of funct ions: Problem type 1Variable expressions as inputs of funct ions: Problem type 2Variable expressions as inputs of funct ions: Problem type 3Domain and range f rom ordered pairsDomain of a rat ional funct ion: Excluded valuesDomain of a rat ional funct ion: Interval notat ionDomain of a square root funct ion: BasicDomain of a square root funct ion: AdvancedFinding the domain of a f ract ional funct ion involving radicalsDetermining whether an equation defines a funct ion: BasicDetermining whether an equation defines a funct ion: AdvancedFinding outputs of a one-step funct ion that models a real-world situat ion: Funct ion notat ionFinding outputs of a two-step funct ion with decimals that models a real-world situat ion: Funct ion notat ionFinding inputs and outputs of a two-step funct ion that models a real-world situat ion: Funct ion notat ionFinding a difference quotient for a linear or quadrat ic funct ionFinding a difference quotient for a rat ional funct ion

Graphs of Funct ions (28 topics)Finding an output of a funct ion f rom its graphFinding inputs and outputs of a funct ion f rom its graphDomain and range f rom the graph of a continuous funct ionDomain and range f rom the graph of a piecewise funct ionFinding where a funct ion is increasing, decreasing, or constant given the graphFinding where a funct ion is increasing, decreasing, or constant given the graph: Interval notat ionFinding local maxima and minima of a funct ion given the graphFinding the absolute maximum and minimum of a funct ion given the graphFinding values and intervals where the graph of a funct ion is zero, posit ive, or negativeGraphing a funct ion of the form f (x) = ax + b: Integer slopeGraphing a funct ion of the form f (x) = ax + b: Fract ional slopeGraphing an absolute value equation in the plane: BasicGraphing an absolute value equation in the plane: AdvancedGraphing a funct ion of the form f (x) = ax2

Graphing a funct ion of the form f (x) = ax2 + cGraphing a parabola of the form y = (x-h)2 + k


Graphing a square root funct ion: Problem type 1Graphing a square root funct ion: Problem type 2Matching parent graphs with their equationsGraphing a piecewise-defined funct ion: Problem type 1Introduct ion to graphing a piecewise-defined funct ion involving lines with non-zero slopeGraphing a piecewise-defined funct ion: Problem type 2Graphing a piecewise-defined funct ion: Problem type 3Even and odd funct ions: Problem type 1Even and odd funct ions: Problem type 2Finding the average rate of change of a funct ionFinding the average rate of change of a funct ion given its graphWord problem involving average rate of change

Transformations (13 topics)Translat ing the graph of a parabola: One stepTranslat ing the graph of a parabola: Two stepsHow the leading coefficient affects the shape of a parabolaTranslat ing the graph of an absolute value funct ion: One stepTranslat ing the graph of an absolute value funct ion: Two stepsWrit ing an equation for a funct ion af ter a vert ical t ranslat ionTranslat ing the graph of a funct ion: One stepTranslat ing the graph of a funct ion: Two stepsTransforming the graph of a funct ion by reflect ing over an axisTransforming the graph of a funct ion by shrinking or stretchingTransforming the graph of a funct ion using more than one transformationTransforming the graph of a quadrat ic, cubic, square root, or absolute value funct ionWrit ing an equation for a funct ion af ter a vert ical and horizontal t ranslat ion

Combining Funct ions; Composite Funct ions; Inverse Funct ions (19 topics)Sum, difference, and product of two funct ionsQuotient of two funct ions: BasicQuotient of two funct ions: AdvancedCombining funct ions: AdvancedIntroduct ion to the composit ion of two funct ionsComposit ion of two funct ions: BasicComposit ion of a funct ion with itselfExpressing a funct ion as a composit ion of two funct ionsComposit ion of two funct ions: AdvancedComposit ion of two rat ional funct ionsWord problem involving composit ion of two funct ionsHorizontal line testDetermining whether two funct ions are inverses of each otherInverse funct ions: Linear, discreteInverse funct ions: Quadrat ic, square rootInverse funct ions: Cubic, cube rootInverse funct ions: Rat ionalGraphing the inverse of a funct ion given its graphFinding, evaluat ing, and interpret ing an inverse funct ion for a given linear relat ionship

Polynomial and Rat ional Funct ions (64 topics)

Quadrat ic Funct ions (16 topics)Finding the vertex, intercepts, and axis of symmetry f rom the graph of a parabolaGraphing a parabola of the form y = x2 + bx + cGraphing a parabola of the form y = a(x-h)2 + kGraphing a parabola of the form y = ax2 + bx + c: Integer coefficientsFinding the zeros of a quadrat ic funct ion given its equationUsing a graphing calculator to find the zeros of a quadrat ic funct ionWrit ing a quadrat ic funct ion given its zerosFinding the x-intercept(s) and the vertex of a parabolaUsing a graphing calculator to find the x-intercept(s) and vertex of a quadrat ic funct ionRewrit ing a quadrat ic funct ion to find its vertex and sketch its graphFinding the maximum or minimum of a quadrat ic funct ionWord problem involving the maximum or minimum of a quadrat ic funct ionWord problem involving optimizing area by using a quadrat ic funct ionDomain and range f rom the graph of a quadrat ic funct ionRange of a quadrat ic funct ionWrit ing the equation of a quadrat ic funct ion given its graph

Polynomial Funct ions (10 topics)Finding zeros of a polynomial funct ion writ ten in factored formFinding zeros and their mult iplicit ies given a polynomial funct ion writ ten in factored formFinding a polynomial of a given degree with given zeros: Real zerosFinding x- and y-intercepts given a polynomial funct ion


Determining the end behavior of the graph of a polynomial funct ionDetermining end behavior and intercepts to graph a polynomial funct ionMatching graphs with polynomial funct ionsInferring propert ies of a polynomial funct ion f rom its graphUsing a graphing calculator to find local extrema of a polynomial funct ionUsing a graphing calculator to solve a word problem involving a local extremum of a polynomial funct ion

Division of Polynomials; Remainder and Factor Theorems (6 topics)Polynomial long division: Problem type 1Polynomial long division: Problem type 2Polynomial long division: Problem type 3Synthetic divisionUsing the remainder theorem to evaluate a polynomialThe Factor Theorem

Real Zeros of Polynomial Funct ions (7 topics)Using a given zero to write a polynomial as a product of linear factors: Real zerosFinding all possible rat ional zeros using the rat ional zeros theorem: Problem type 1Finding all possible rat ional zeros using the rat ional zeros theorem: Problem type 2Using the rat ional zeros theorem to find all zeros of a polynomial: Rat ional zerosUsing the rat ional zeros theorem to find all zeros of a polynomial: Irrat ional zerosUsing a graphing calculator to find zeros of a polynomial funct ionUsing a graphing calculator to solve a word problem involving a polynomial of degree 3

Complex Zeros of Polynomials Funct ions (4 topics)Mult iplying expressions involving complex conjugatesFinding a polynomial of a given degree with given zeros: Complex zerosUsing a given zero to write a polynomial as a product of linear factors: Complex zerosUsing the rat ional zeros theorem to find all zeros of a polynomial: Complex zeros

Rational Funct ions (13 topics)Finding the intercepts, asymptotes, domain, and range f rom the graph of a rat ional funct ionFinding the asymptotes of a rat ional funct ion: Constant over linearFinding the asymptotes of a rat ional funct ion: Linear over linearFinding horizontal and vert ical asymptotes of a rat ional funct ion: Quadrat ic numerator or denominatorFinding the asymptotes of a rat ional funct ion: Quadrat ic over linearGraphing a rat ional funct ion: Constant over linearGraphing a rat ional funct ion: Linear over linearTransforming the graph of a rat ional funct ionGraphing a rat ional funct ion: Quadrat ic over linearGraphing rat ional funct ions with holesMatching graphs with rat ional funct ions: Two vert ical asymptotesGraphing a rat ional funct ion with more than one vert ical asymptoteUsing a graphing calculator to solve a word problem involving a local extremum of a rat ional funct ion

Polynomial and Rat ional Inequalit ies (8 topics)Solving a quadrat ic inequality writ ten in factored formSolving a quadrat ic inequalitySolving a polynomial inequality: Problem type 1Solving a polynomial inequality: Problem type 2Solving a polynomial inequality: Problem type 3Solving a polynomial inequality: Problem type 4Solving a rat ional inequality: Problem type 1Solving a rat ional inequality: Problem type 2

Exponential and Logarithmic Funct ions (50 topics)

Graphing Exponential Funct ions (8 topics)Table for an exponential funct ionGraphing an exponential funct ion and its asymptote: f (x)=bx

Graphing an exponential funct ion and its asymptote: f (x) = a(b)xGraphing an exponential funct ion and its asymptote: f (x) = b-x or f (x) = -bx or f (x) = -b-xTranslat ing the graph of an exponential funct ionGraphing an exponential funct ion and finding its domain and rangeTransforming the graph of a natural exponential funct ion and finding its domain and rangeGraphing an exponential funct ion and its asymptote: f (x) = a(e)x-b + c

Applicat ions of Exponential Funct ions (7 topics)Using a calculator to evaluate exponential expressionsEvaluat ing an exponential funct ion that models a real-world situat ionUsing a calculator to evaluate exponential expressions involving base eEvaluat ing an exponential funct ion with base e that models a real-world situat ionIntroduct ion to compound interest


Finding a final amount in a word problem on exponential growth or decayFinding the final amount in a word problem on compound interest

Logarithmic Funct ions (9 topics)Using a calculator to evaluate natural and common logarithmic expressionsConvert ing between logarithmic and exponential equationsConvert ing between natural logarithmic and exponential equationsEvaluat ing logarithmic expressionsSolving an equation of the form logba = cTranslat ing the graph of a logarithmic funct ionGraphing a logarithmic funct ion: BasicGraphing a logarithmic funct ion and finding its domain and rangeDomain of a logarithmic funct ion: Advanced

Propert ies of Logarithms (6 topics)Basic propert ies of logarithmsUsing propert ies of logarithms to evaluate expressionsExpanding a logarithmic expression: Problem type 1Expanding a logarithmic expression: Problem type 2Writ ing an expression as a single logarithmChange of base for logarithms: Problem type 1

Logarithmic and Exponential Equations (10 topics)Solving a mult i-step equation involving a single logarithm: Problem type 1Solving a mult i-step equation involving a single logarithm: Problem type 2Solving a mult i-step equation involving natural logarithmsSolving an equation involving logarithms on both sides: Problem type 1Solving an equation involving logarithms on both sides: Problem type 2Solving an exponential equation by finding common bases: Linear exponentsSolving an exponential equation by using logarithms: Decimal answers, basicSolving an exponential equation by using natural logarithms: Decimal answersSolving an exponential equation by using logarithms: Decimal answers, advancedSolving an exponential equation by using logarithms: Exact answers in logarithmic form

Applicat ions (10 topics)Finding the t ime to reach a limit in a word problem on exponential growth or decayFinding the t ime in a word problem on compound interestFinding the t ime given an exponential funct ion with base e that models a real-world situat ionFinding the final amount in a word problem on continuous compound interestFinding the init ial amount in a word problem on continuous compound interestFinding the final amount in a word problem on continuous exponential growth or decayFinding the rate or t ime in a word problem on continuous exponential growth or decayFinding half -life or doubling t imeWrit ing and evaluat ing a funct ion modeling continuous exponential growth or decay given doubling t ime or half -lifeWrit ing and evaluat ing a funct ion modeling continuous exponential growth or decay given two outputs

Trigonometric Funct ions (80 topics)

Angles and Their Measure (6 topics)Convert ing degrees to radians and radians to degrees: Problem type 1Convert ing degrees to radians and radians to degrees: Problem type 2Sketching an angle with absolute value less than 360 degrees in standard posit ionSketching an angle with absolute value less than 2π radians in standard posit ionCoterminal anglesArc length and central angle measure

The Unit Circle and Evaluat ing Trigonometric Funct ions (15 topics)Finding coordinates on the unit circle for special anglesUsing the coordinates of points on the unit circle to define sine and cosine for all real numbersSpecial t riangles with a hypotenuse of length 1Drawing a reference triangle on the unit circle and using it to derive values of trigonometric funct ions: RadiansTrigonometric funct ions and special angles: Problem type 1: DegreesTrigonometric funct ions and special angles: Problem type 1: RadiansFinding values of trigonometric funct ions f rom a point on the unit circleTrigonometric funct ions and special angles: Problem type 2Using the coordinates of points on the unit circle to define trigonometric funct ions for all real numbersTrigonometric funct ions and special angles: Problem type 3Using the unit circle to understand the odd and even identit ies for sine and cosineEvaluat ing expressions involving sine or cosineOdd and even identit ies for trigonometric funct ionsUsing a calculator to approximate sine, cosine, and tangent valuesEvaluat ing a sinusoidal funct ion that models a real-world situat ion

Right Triangle Trigonometry (10 topics)


Sine, cosine, and tangent rat ios: Variables for side lengthsUsing the Pythagorean Theorem to find a sine, cosine, or tangent rat io in a right triangleUsing the Pythagorean Theorem to find several t rigonometric rat ios in a right triangleUsing a trigonometric rat io to find a side length in a right triangleUsing trigonometry to find a length in a word problem with one right triangleUsing trigonometric funct ions and the formula d = rt in a real-world situat ionUsing a trigonometric rat io to find an angle measure in a right triangleUsing trigonometry to find angles of elevat ion or depression in a word problemSolving a right triangleUsing trigonometry to find a length in a word problem with two right triangles

Trigonometric Funct ions of Angles (12 topics)Sketching an angle with absolute value less than 360 degrees, and also its reference angleReference angles in degrees: Problem type 1Reference angles in degrees: Problem type 2Sketching an angle with absolute value less than 2π radians, and also its reference angleReference angles in radians: Problem type 1Sketching an angle with absolute value greater than 2π radians, and also its reference angleReference angles in radians: Problem type 2Determining the locat ion of a terminal point given the signs of trigonometric valuesFinding values of trigonometric funct ions given information about an angle: Problem type 1Finding values of trigonometric funct ions given information about an angle: Problem type 2Finding values of trigonometric funct ions given information about an angle: Problem type 3Finding values of trigonometric funct ions given information about an angle: Problem type 4

Graphs of Sine and Cosine Funct ions (23 topics)Sketching the graph of y = a sin(x) or y = a cos(x)Sketching the graph of y = sin(bx) or y = cos(bx)Using transformations to graph y = ±sin(bx) or y = ±cos(bx)Sketching the graph of y = sin(x) + d or y = cos(x) + dUsing transformations to graph y = ±sin(x) + d or y = ±cos(x) + dUsing transformations to graph y = a sin(x) + d or y = a cos(x) + dUsing transformations to graph y = ±sin(bx) + d or y = ±cos(bx) + dSketching the graph of y = sin(x+c) or y = cos(x+c)Sketching the graph of y = a sin(x+c) or y = a cos(x+c)Using transformations to graph y = a sin(x+c) + d or y = a cos(x+c) + dSketching the graph of y = a sin(bx) or y = a cos(bx)Using transformations to graph y = sin(bx+c) or y = cos(bx+c)Sketching the graph of y = a sin(bx+c) or y = a cos(bx+c)Sketching the graph of y = a sin(bx) + d or y = a cos(bx) + dUsing transformations to graph y = a sin(bx+c) + d or y = a cos(bx+c) + dAmplitude and period of a sine or cosine funct ionAmplitude, period, and phase shif t of a sine or cosine funct ionInterpret ing the graph of a sinusoidal funct ion that models a real-world situat ionWrit ing the equation of a sine or cosine funct ion given its graph: Problem type 1Writ ing the equation of a sine or cosine funct ion given its graph: Problem type 2Word problem involving a sine or cosine funct ion: Problem type 1Developing a sinusoidal model for a real-world situat ionWord problem involving a sine or cosine funct ion: Problem type 2

Graphs of Other Trigonometric Funct ions (6 topics)Domains and ranges of trigonometric funct ionsMatching graphs and equations for secant, cosecant, tangent, and cotangent funct ionsSketching the graph of a secant or cosecant funct ion: Problem type 1Sketching the graph of a secant or cosecant funct ion: Problem type 2Sketching the graph of a tangent or cotangent funct ion: Problem type 2Sketching the graph of a tangent or cotangent funct ion: Problem type 1

Inverse Trigonometric Funct ions (8 topics)Values of inverse trigonometric funct ionsComposit ion of a trigonometric funct ion with its inverse trigonometric funct ion: Problem type 1Composit ion of a trigonometric funct ion with its inverse trigonometric funct ion: Problem type 2Composit ion of a trigonometric funct ion with the inverse of another trigonometric funct ion: Problem type 1Composit ion of a trigonometric funct ion with the inverse of another trigonometric funct ion: Problem type 2Composit ion of a trigonometric funct ion with the inverse of another trigonometric funct ion: Problem type 3Composit ion of trigonometric funct ions with variable expressions as inputs: Problem type 1Using a calculator to approximate inverse trigonometric values

Trigonometric Identit ies and Equations (49 topics)

Verifying Trigonometric Identit ies (13 topics)Using reciprocal and quotient identit ies to simplify a trigonometric expressionUsing Pythagorean identit ies to simplify a trigonometric expressionUsing cofunct ion identit ies


Verifying a trigonometric identity: Problem type 1Verifying a trigonometric identity: Problem type 2Verifying a trigonometric identity: Problem type 3Proving an identity using fundamental t rigonometric identit ies: Problem type 1Proving an identity using fundamental t rigonometric identit ies: Problem type 2Proving an identity using fundamental t rigonometric identit ies: Problem type 3Proving an identity using fundamental t rigonometric identit ies: Problem type 4Proving an identity using fundamental t rigonometric identit ies: Problem type 5Proving an identity using fundamental t rigonometric identit ies: Problem type 6Proving an identity using fundamental t rigonometric identit ies: Problem type 7

Sum and Difference Formulas (9 topics)Sum and difference identit ies: Problem type 1: DegreesSum and difference identit ies: Problem type 1: RadiansSum and difference identit ies: Problem type 2: DegreesSum and difference identit ies: Problem type 2: RadiansSum and difference identit ies: Problem type 3Sum and difference identit ies: Problem type 4Proving trigonometric identit ies using sum and difference identit ies: Problem type 1Proving trigonometric identit ies using sum and difference identit ies: Problem type 2Proving trigonometric identit ies using sum and difference identit ies: Problem type 3

Double-Angle, Half -Angle, Product-to-Sum, and Power Reducing Formulas (11 topics)Double-angle identit ies: Problem type 1Double-angle identit ies: Problem type 2Power-reducing identit iesHalf -angle identit ies: Problem type 1: DegreesHalf -angle identit ies: Problem type 1: RadiansHalf -angle identit ies: Problem type 2Product-to-sum and sum-to-product identit ies: Problem type 1: DegreesProduct-to-sum and sum-to-product identit ies: Problem type 1: RadiansProduct-to-sum and sum-to-product identit ies: Problem type 2Proving trigonometric identit ies using double-angle identit ies: Problem type 1Proving trigonometric identit ies using double-angle identit ies: Problem type 2

Trigonometric Equations (16 topics)Finding solut ions in an interval for a basic trigonometric equation involving sine or cosineFinding solut ions in an interval for a basic trigonometric equation involving tangent, cotangent, secant, or cosecantFinding solut ions in an interval for a basic trigonometric equation using a calculatorSolving a basic trigonometric equation involving sine or cosineSolving a basic trigonometric equation involving tangent, cotangent, secant, or cosecantFinding solut ions in an interval for a trigonometric equation involving sine and cosine and writ ten in factored formFinding solut ions in an interval for a trigonometric equation writ ten in factored formFinding solut ions in an interval for a trigonometric equation involving a squared funct ion: Problem type 1Factoring to find solut ions in an interval for a trigonometric equation involving sine or cosineFactoring to find solut ions in an interval for a trigonometric equationUsing a Pythagorean identity to find solut ions in an interval for a trigonometric equation involving sine and cosine:Problem type 1Using a Pythagorean identity to find solut ions in an interval for a trigonometric equation: Problem type 1Finding solut ions in an interval for a trigonometric equation involving sine and/or cosine using double-angle identit iesSolving a trigonometric equation modeling a real-world situat ionFinding solut ions in an interval for a trigonometric equation involving sine or cosine and an angle mult iplied by aconstantFinding solut ions in an interval for a trigonometric equation involving an angle mult iplied by a constant

Addit ional Topics in Trigonometry (6 topics)

Laws of Sines and Cosines (6 topics)Solving a triangle with the law of sines: Problem type 1Solving a triangle with the law of sines: Problem type 2Solving a word problem using the law of sinesSolving a triangle with the law of cosinesSolving a word problem using the law of cosinesSolving a word problem using the law of sines and the law of cosines

Other Topics Available(*) (646 addit ional topics)

Algebra and Geometry Review (160 topics)Fract ional posit ion on a number linePlott ing rat ional numbers on a number lineOrdering integersEst imating a square rootOrdering real numbers


Ident ifying numbers as integers or non-integersIdentifying numbers as rat ional or irrat ionalSigned f ract ion addit ion or subtract ion: AdvancedAddit ion and subtract ion of 3 f ract ions involving signsSigned f ract ion mult iplicat ion: AdvancedOperat ions with absolute value: Problem type 2Exponents and integers: Problem type 2Order of operat ions with integers and exponentsConvert ing between temperatures in Fahrenheit and CelsiusPropert ies of addit ionPropert ies of real numbersIdentifying propert ies used to simplify an algebraic expressionUnderstanding the product rule of exponentsOrdering numbers with posit ive exponentsUnderstanding the power rules of exponentsPower and product rules with posit ive exponentsSimplifying a rat io of mult ivariate monomials: AdvancedPower and quotient rules with posit ive exponentsOrdering numbers with negative exponentsProduct rule with negative exponentsQuotient rule with negative exponents: Problem type 2Power and quotient rules with negative exponents: Problem type 1Power and quotient rules with negative exponents: Problem type 2Power, product, and quotient rules with negative exponentsScient ific notat ion with a posit ive exponentScient ific notat ion with a negative exponentConvert ing between scient ific notat ion and standard form in a real-world situat ionMult iplying numbers writ ten in scient ific notat ion: BasicMult iplying numbers writ ten in scient ific notat ion: AdvancedMult iplying numbers writ ten in decimal form or scient ific notat ion in a real-world situat ionDividing numbers writ ten in scient ific notat ion: BasicDividing numbers writ ten in scient ific notat ion: AdvancedFinding the scale factor between numbers given in scient ific notat ion in a real-world situat ionDegree of a mult ivariate polynomialSimplifying a sum or difference of three univariate polynomialsSimplifying a sum or difference of mult ivariate polynomialsMult iplying conjugate binomials: Mult ivariatePrime numbersPrime factorizat ionGreatest common factor of three univariate monomialsFactoring a univariate polynomial by grouping: Problem type 2Factoring a mult ivariate polynomial by grouping: Problem type 1Factoring a mult ivariate polynomial by grouping: Problem type 2Factoring a quadrat ic in two variables with leading coefficient 1Factoring a quadrat ic with leading coefficient greater than 1: Problem type 3Factoring a quadrat ic by the ac-methodFactoring a quadrat ic in two variables with leading coefficient greater than 1Factoring a perfect square trinomial with leading coefficient greater than 1Factoring a perfect square trinomial in two variablesFactoring a difference of squares in two variablesFactoring a polynomial involving a GCF and a difference of squares: UnivariateFactoring a polynomial involving a GCF and a difference of squares: Mult ivariateFactoring a product of a quadrat ic trinomial and a monomialFactoring with repeated use of the difference of squares formulaFactoring a sum or difference of two cubesFactoring out binomials f rom a polynomial: GCF factoring, advancedUsing subst itut ion to factor polynomialsSimplifying a rat io of factored polynomials: Factors with exponentsSimplifying a rat io of linear polynomials: 1, -1, and no simplificat ionSimplifying a rat io of polynomials: Problem type 2Simplifying a rat io of polynomials: Problem type 3Simplifying a rat io of mult ivariate polynomialsMult iplying rat ional expressions involving mult ivariate monomialsMult iplying rat ional expressions involving quadrat ics with leading coefficients greater than 1Mult iplying rat ional expressions involving mult ivariate quadrat icsDividing rat ional expressions involving mult ivariate monomialsDividing rat ional expressions involving quadrat ics with leading coefficients greater than 1Dividing rat ional expressions involving mult ivariate quadrat icsMult iplicat ion and division of 3 rat ional expressionsLeast common mult iple of two monomialsFinding the LCD of rat ional expressions with linear denominators: Common factorsFinding the LCD of rat ional expressions with quadrat ic denominatorsWrit ing equivalent rat ional expressions with monomial denominatorsWrit ing equivalent rat ional expressions involving opposite factors


Adding rat ional expressions with denominators axn and bxmAdding rat ional expressions with mult ivariate monomial denominators: BasicAdding rat ional expressions with mult ivariate monomial denominators: AdvancedAdding rat ional expressions with linear denominators without common factors: AdvancedAdding rat ional expressions with linear denominators with common factors: BasicAdding rat ional expressions with linear denominators with common factors: AdvancedAdding rat ional expressions with denominators ax-b and b-axAdding rat ional expressions involving different quadrat ic denominatorsAdding 3 rat ional expressions with different quadrat ic denominatorsComplex f ract ion involving mult ivariate monomialsComplex f ract ion: Quadrat ic factoringComplex f ract ion made of sums involving rat ional expressions: Problem type 2Complex f ract ion made of sums involving rat ional expressions: Problem type 3Complex f ract ion made of sums involving rat ional expressions: Problem type 4Complex f ract ion made of sums involving rat ional expressions: Problem type 5Complex f ract ion made of sums involving rat ional expressions: Problem type 6Complex f ract ion made of sums involving rat ional expressions: Mult ivariateComplex f ract ion with negative exponents: Problem type 1Complex f ract ion with negative exponents: Problem type 2Complex f ract ion that contains a complex f ract ionFinding all square roots of a numberSquare roots of integers raised to even exponentsUsing absolute value to simplify square roots of perfect square monomialsFinding the nth root of a perfect nth power f ract ionFinding the nth root of a perfect nth power monomialUsing absolute value to simplify higher radical expressionsRational exponents: Unit f ract ion exponents and bases involving signsRational exponents: Product ruleRational exponents: Quotient ruleRational exponents: Products and quotients with negative exponentsRational exponents: Power of a power ruleRational exponents: Powers of powers with negative exponentsSimplifying the square root of a whole number greater than 100Simplifying a radical expression with two variablesIntroduct ion to simplifying a higher radical expressionSimplifying a higher radical expression: UnivariateSimplifying a higher radical expression: Mult ivariateSquare root addit ion or subtract ion with three termsIntroduct ion to simplifying a sum or difference of radical expressions: UnivariateSimplifying a sum or difference of radical expressions: UnivariateSimplifying a sum or difference of radical expressions: Mult ivariateSimplifying a sum or difference of higher rootsSimplifying a sum or difference of higher radical expressionsSimplifying a product of radical expressions: UnivariateSimplifying a product of radical expressions: Mult ivariateSimplifying a product of radical expressions: Mult ivariate, f ract ional expressionsIntroduct ion to simplifying a product of higher rootsSimplifying a product of higher radical expressionsSpecial products of radical expressions: Conjugates and squaringClassifying sums and products as rat ional or irrat ionalRat ionalizing a denominator: Quotient involving a monomialRat ionalizing a denominator using conjugates: Square root in numeratorRat ionalizing a denominator using conjugates: Variable in denominatorRat ionalizing a denominator: Quotient involving a higher radicalRat ionalizing a denominator: Quotient involving higher radicals and monomialsSimplifying products or quotients of higher radicals with different indices: UnivariateSimplifying products or quotients of higher radicals with different indices: Mult ivariateArea of a piecewise rectangular figureWord problem involving the area between two rectanglesArea of a triangleArea of a parallelogramArea of a trapezoidPerimeter involving rectangles and circlesCircumference and area of a circleCircumference and area of a circle: Exact answers in terms of piArea involving rectangles and circlesWord problem involving the area between two concentric circlesArea involving inscribed figuresVolume of a triangular prismVolume of a pyramidVolume of a cylinderWord problem involving the rate of filling or emptying a cylinderVolume of a coneVolume of a cone: Exact answers in terms of pi


Volume of a sphereSurface area of a cube or a rectangular prismSurface area of a triangular prismSurface area of a cylinderSurface area of a cylinder: Exact answers in terms of piSurface area of a sphereWord problem involving the Pythagorean Theorem

Equations and Inequalit ies (76 topics)Identifying propert ies used to solve a linear equationSolving equations with zero, one, or infinitely many solut ionsSolving a decimal word problem using a linear equation with the variable on both sidesSolving a f ract ion word problem using a linear equation with the variable on both sidesSolving a word problem involving consecutive integersWrit ing a mult i-step equation for a real-world situat ionSolving a value mixture problem using a linear equationFinding a side length given the perimeter and side lengths with variablesCircumference rat iosSolving equations involving vert ical anglesFinding angle measures of a triangle given angles with variablesFinding the value for a new score that will yield a given meanFinding the mult iplier to give a final amount af ter a percentage increase or decreaseFinding the total cost including tax or markupFinding the original price given the sale price and percent discountComputing a percent mixtureSolving a percent mixture problem using a linear equationFinding simple interest without a calculatorConvert ing a repeating decimal to a f ract ionSolving an absolute value equation: Problem type 3Solving an absolute value equation: Problem type 4Solving an absolute value equation of the form |ax+b| = |cx+d|Translat ing a sentence into a one-step inequalityTranslat ing a sentence into a mult i-step inequalityWrit ing an inequality for a real-world situat ionWrit ing an inequality given a graph on the number lineTranslat ing a sentence into a compound inequalityWrit ing a compound inequality given a graph on the number lineWrit ing sets of integers using set-builder and roster formsUnion and intersect ion of finite setsUnion and intersect ion of intervalsAddit ive property of inequality with signed f ract ionsMult iplicat ive property of inequality with signed f ract ionsSolving a two-step linear inequality with a f ract ional coefficientSolving a linear inequality with mult iple occurrences of the variable: Problem type 2Solving a linear inequality with mult iple occurrences of the variable: Problem type 3Solving inequalit ies with no solut ion or all real numbers as solut ionsSolving a compound linear inequality: Graph solut ion, basicSolving a compound linear inequality: Interval notat ionSolving a decimal word problem using a two-step linear inequalitySolving a decimal word problem using a linear inequality with the variable on both sidesSolving an absolute value inequality: Problem type 1Writ ing an absolute value inequality given a graph on the number lineSolving an absolute value inequality: Problem type 2Solving an absolute value inequality: Problem type 3Solving an absolute value inequality: Problem type 4Solving an absolute value inequality: Problem type 5Solving a proport ion of the form a/(x+b) = c/xSolving for a variable in terms of other variables in a rat ional equation: Problem type 3Word problem on proport ions: Problem type 1Word problem on proport ions: Problem type 2Similar polygonsSimilar right trianglesIndirect measurementRatio of volumesWord problem involving mult iple ratesSolving a work problem using a rat ional equationSolving a distance, rate, t ime problem using a rat ional equationOrdering f ract ions with variablesSimplifying a product and quotient involving square roots of negative numbersSimplifying a power of iUsing the Pythagorean Theorem and a quadrat ic equation to find side lengths of a right triangleDiscriminant of a quadrat ic equation with a parameterSolving a rat ional equation that simplifies to quadrat ic: Proport ional form, basicSolving a rat ional equation that simplifies to quadrat ic: Factorable quadrat ic denominator


Solving a rat ional equation that simplifies to quadrat ic: Proport ional form, advancedSolving a radical equation with a quadrat ic expression under the radicalSolving a radical equation with two radicals that simplifies to sqrt (x) = aSolving a radical equation that simplifies to a quadrat ic equation: Two radicalsWord problem involving radical equations: BasicWord problem involving radical equations: AdvancedSolving an equation with exponent 1/a: Problem type 1Solving an equation with exponent 1/a: Problem type 2Solving an equation with a posit ive rat ional exponentSolving an equation with a negative rat ional exponentSolving an equation that can be writ ten in quadrat ic form: Problem type 2

Graphs and Funct ions (61 topics)Finding the area of a triangle or parallelogram in the coordinate planeDistance between two points in the plane: Decimal answersIdentifying scalene, isosceles, and equilateral t riangles given coordinates of their vert icesFinding an endpoint of a line segment given the other endpoint and the midpointGraphing a line given its x- and y-interceptsTest ing an equation for symmetry about the axes and originClassifying slopes given graphs of linesFinding the coordinate that yields a given slopeGraphing a line through a given point with a given slopeIdentifying linear equations: AdvancedIdentifying linear funct ions given ordered pairsRewrit ing a linear equation in the form Ax + By = CWrit ing an equation and graphing a line given its slope and y-interceptFinding the slope, y-intercept, and equation for a linear funct ion given a table of valuesGraphing a line given its equation in point-slope formWrit ing the equation of a line in standard form given the slope and a pointComparing linear funct ions to the parent funct ion y = xIdentifying parallel and perpendicular lines f rom equationsIdentifying parallel and perpendicular lines f rom coordinatesIdentifying coordinates that give right trianglesGraphing ordered pairs and writ ing an equation f rom a table of values in contextFinding the init ial amount and rate of change given a table for a linear funct ionCombining funct ions to write a new funct ion that models a real-world situat ionComparing propert ies of linear funct ions given in different formsApplicat ion problem with a linear funct ion: Finding a coordinate given the slope and a pointSolving a linear equation by graphingConstruct ing a scatter plotSketching the line of best fitScatter plots and correlat ionPredict ions f rom the line of best fitApproximating the equation of a line of best fit and making predict ionsComputing residualsInterpret ing residual plotsClassifying linear and nonlinear relat ionships f rom scatter plotsLinear relat ionship and the correlat ion coefficientIdentifying out liers and clustering in scatter plotsFinding out liers in a data setIdentifying solut ions to a system of linear equationsGraphically solving a system of linear equationsUsing a graphing calculator to solve a system of linear equations: BasicUsing a graphing calculator to solve a system of linear equations: AdvancedWrit ing a system of linear equations given its graphSolving a system of linear equations using subst itut ionSolving a system of linear equations using eliminat ion with addit ionSolving a system of linear equations using eliminat ion with mult iplicat ion and addit ionIdentifying the center and radius to graph a circle given its equation in general form: AdvancedWrit ing an equation of a circle and identifying points that lie on the circleDeriving the equation of a circle using the Pythagorean TheoremDomains of higher root funct ionsDomain and range of a linear funct ion that models a real-world situat ionRewrit ing a mult ivariate funct ion as a univariate funct ion given a relat ionship between its variablesDomain and range f rom the graph of a discrete relat ionFinding domain and range f rom a linear graph in contextChoosing a graph to fit a narrat ive: BasicChoosing a graph to fit a narrat ive: AdvancedGraphing an integer funct ion and finding its range for a given domainGraphing a square root funct ion: Problem type 3Graphing a cube root funct ionWrit ing the equation of a secant lineHow the leading coefficient affects the graph of an absolute value funct ionComposit ion of two funct ions: Domain and range


Polynomial and Rat ional Funct ions (27 topics)Graphing a parabola of the form y = ax2 + bx + c: Rat ional coefficientsRewrit ing a quadrat ic funct ion in standard formSolving a quadrat ic equation by graphingComparing propert ies of quadrat ic funct ions given in different formsClassifying the graph of a funct ionChoosing a quadrat ic model and using it to make a predict ionIdentifying polynomial funct ionsDividing a polynomial by a monomial: UnivariateDividing a polynomial by a monomial: Mult ivariateRemainder theorem: AdvancedClosure propert ies of integers and polynomialsDescartes' Rule of SignsUsing the conjugate zeros theorem to find all zeros of a polynomialLinear factors theorem and conjugate zeros theoremWrit ing the equation of a rat ional funct ion given its graphIdentifying direct variat ion equationsIdentifying direct variat ion f rom ordered pairs and writ ing equationsWrit ing a direct variat ion equationWord problem on direct variat ionInterpret ing direct variat ion f rom a graphWrit ing an inverse variat ion equationIdentifying direct and inverse variat ion equationsIdentifying direct and inverse variat ion f rom ordered pairs and writ ing equationsWord problem on inverse variat ionWord problem on inverse variat ion involving the complet ion of a taskWrit ing an equation that models variat ionWord problem on combined variat ion

Exponential and Logarithmic Funct ions (12 topics)Finding domain and range f rom the graph of an exponential funct ionCalculat ing and comparing simple interest and compound interestFinding the init ial amount and rate of change given an exponential funct ionWrit ing an equation that models exponential growth or decayWrit ing an exponential funct ion rule given a table of ordered pairsChoosing an exponential model and using it to make a predict ionComparing linear, polynomial, and exponential funct ionsGraphing a logarithmic funct ion: AdvancedExpanding a logarithmic expression: Problem type 3Change of base for logarithms: Problem type 2Solving an exponential equation by finding common bases: Linear and quadrat ic exponentsSolving an exponential equation by using subst itut ion and quadrat ic factoring

Trigonometric Funct ions (38 topics)Convert ing degrees-minutes-seconds to decimal degreesConvert ing decimal degrees to degrees-minutes-secondsSketching an approximation of an angle given in radiansSketching an angle in standard posit ion given in degrees and finding a coterminal angleSketching an angle in standard posit ion given in radians and finding a coterminal angleSketching an angle with absolute value greater than 360 degrees in standard posit ionSketching an angle with absolute value greater than 2π radians in standard posit ionSketching an angle with absolute value greater than 360 degrees and finding coterminal anglesSketching an angle with absolute value greater than 2π radians and finding coterminal anglesDrawing an arc to find a central angle or an arc length on the unit circleDrawing an arc to find a central angle or an arc length on a non-unit circleRelat ing an angle and an arc length in a real-world situat ionRelat ing two angle measures in a real-world situat ion that involves interlocking gearsArea of a sector of a circleUsing the area formula for a sector of a circle in a real-world situat ionAngular and linear speedFinding a point on the unit circle given one coordinate and the quadrantDrawing a reference triangle on the unit circle and using it to derive values of trigonometric funct ions: DegreesUsing symmetries on the unit circle to understand trigonometric identit ies: Problem type 1Using the unit circle to understand that sine and cosine are periodicUsing symmetries on the unit circle to understand trigonometric identit ies: Problem type 2Simplifying a trigonometric expression: Rat ionalizing the denominator using conjugatesUsing a calculator to approximate cosecant, secant, and cotangent valuesSpecial right triangles: Exact answersSine, cosine, and tangent rat ios: Numbers for side lengthsUnderstanding trigonometric rat ios through similar right trianglesRelat ionship between the sines and cosines of complementary anglesUsing similar right triangles to find trigonometric rat iosWord problem involving a triangle whose side lengths change with t ime: Problem type 1Word problem involving a triangle whose side lengths change with t ime: Problem type 2


Using trigonometry to find lengths in a figure involving two right trianglesSketching an angle with absolute value greater than 360 degrees, and also its reference angleUsing graphing to solve a trigonometric equation involving sine or cosineAverage rate of change involving a sinusoidal funct ionUnderstanding how changes to the amplitude, period, phase shif t , and vert ical shif t affect a sinusoidal graphSketching the graph of a sinusoidal funct ion that models a real-world situat ion and using the graph to approximatesolut ions to an equation

Sketching a graph of a damped sine or cosine funct ionComposit ion of trigonometric funct ions with variable expressions as inputs: Problem type 2

Trigonometric Identit ies and Equations (20 topics)Proving an identity using fundamental t rigonometric identit ies: Problem type 8Proving trigonometric identit ies using odd and even identit iesProving identit ies involving trigonometric funct ions and logarithmic funct ionsUsing a double-angle identity to find the exact value of a composit ion of trigonometric funct ionsDouble-angle identit ies: Problem type 3Proving trigonometric identit ies using sum-to-product identit ies: Problem type 1Proving trigonometric identit ies using sum-to-product identit ies: Problem type 2Finding solut ions in an interval for an equation involving a trigonometric expression and either exponentials orlogarithms

Using a Pythagorean identity to find solut ions in an interval for a trigonometric equation involving sine and cosine:Problem type 2Using a Pythagorean identity to find solut ions in an interval for a trigonometric equation: Problem type 2Using a graphing calculator to solve a trigonometric equationUsing a graphing calculator to solve a trigonometric inequalitySolving a trigonometric equation involving a squared funct ion: Problem type 1Solving a trigonometric equation involving a squared funct ion: Problem type 2Solving a trigonometric equation involving more than one funct ionSolving a trigonometric equation involving an angle mult iplied by a constantFinding solut ions in an interval for a trigonometric equation involving sine and cosine using sum and difference identit iesSolving a trigonometric equation using sum and difference identit iesSolving a trigonometric equation using double-angle identit iesSolving a trigonometric equation using half -angle identit ies

Addit ional Topics in Trigonometry (60 topics)Proving the law of sinesProving the law of cosinesUsing trigonometry to find the area of a right triangleUsing trigonometry to find the area of a triangleExpressing the area of a triangle in terms of the sine of one of its anglesHeron's formulaWrit ing a posit ion vector in ai+bj form given its graphWrit ing a vector in ai+bj form given its init ial and terminal pointsWrit ing a vector in component form given its init ial and terminal pointsMagnitude of a vector given in ai+bj formMagnitude of a vector given in component formVector addit ion and scalar mult iplicat ion: ai+bj formLinear combinat ion of vectors: ai+bj formVector addit ion and scalar mult iplicat ion: Component formLinear combinat ion of vectors: Component formUnit vectorsMult iplicat ion of a vector by a scalar: Geometric approachVector addit ion: Geometric approachVector subtract ion: Geometric approachFinding the magnitude and direct ion of a vector given its graphFinding the components of a vector given its graphFinding the direct ion angle of a vector given in ai+bj formWrit ing a vector given its magnitude and direct ion angleWrit ing a vector to represent a force pushing or pulling an objectFinding the magnitude and direct ion angle of the resultant force of two vectorsFinding magnitudes of forces related to a sum of three vectorsFinding magnitudes of forces related to an object suspended by cablesDot product of vectors given in ai+bj formDot product of vectors given in component formFinding the angle between two vectors given in component formClassifying vector relat ionships by finding the angle between two vectors given in ai + bj formUsing the dot product to find perpendicular vectorsFinding the component of a vector along another vectorDecomposing a vector into two orthogonal vectorsFinding the amount of work done given a force vector and a distanceFinding magnitudes of forces related to an object on a rampPlott ing points in polar coordinates


Mult iple representat ions of polar coordinatesConvert ing rectangular coordinates to polar coordinates: Special anglesConvert ing rectangular coordinates to polar coordinates: Decimal answersConvert ing polar coordinates to rectangular coordinatesConvert ing an equation writ ten in rectangular form to one writ ten in polar formConvert ing an equation writ ten in polar form to one writ ten in rectangular form: Problem type 1Convert ing an equation writ ten in polar form to one writ ten in rectangular form: Problem type 2Graphing a polar equation: BasicGraphing a polar equation: CircleGraphing a polar equation: LimaconGraphing a polar equation: RoseGraphing a polar equation: LemniscateMatching polar equations with their graphsIdentifying symmetries of graphs given their polar equationsPlott ing complex numbersWrit ing a complex number in standard form given its trigonometric formWrit ing a complex number in trigonometric form: Special anglesWrit ing a complex number in trigonometric form: Decimal answersMult iplying and dividing complex numbers in trigonometric formDe Moivre's Theorem: Answers in trigonometric formDe Moivre's Theorem: Answers in standard formFinding the nth roots of a number: Problem type 1Finding the nth roots of a number: Problem type 2

Systems of Equations and Matrices (72 topics)Classifying systems of linear equations f rom graphsSolving a system of linear equations with f ract ional coefficientsSolving a system of linear equations with decimal coefficientsSolving a 2x2 system of linear equations that is inconsistent or consistent dependentCreat ing an inconsistent system of linear equationsIdentifying the operat ions used to create equivalent systems of equationsConsistency and independence of a system of linear equationsInterpret ing the graphs of two funct ionsSolving a word problem involving a sum and another basic relat ionship using a system of linear equationsSolving a word problem using a system of linear equations of the form Ax + By = CSolving a word problem using a system of linear equations of the form y = mx + bSolving a value mixture problem using a system of linear equationsSolving a percent mixture problem using a system of linear equationsSolving a distance, rate, t ime problem using a system of linear equationsSolving a tax rate or interest rate problem using a system of linear equationsIntroduct ion to solving a 3x3 system of linear equationsSolving a 3x3 system of linear equations: Problem type 1Solving a 3x3 system of linear equations: Problem type 2Solving a 3x3 system of linear equations that is inconsistent or consistent dependentSolving a word problem using a 3x3 system of linear equations: Problem type 1Solving a word problem using a 3x3 system of linear equations: Problem type 2Scalar mult iplicat ion of a matrixAddit ion or subtract ion of matricesLinear combinat ion of matricesSquaring and mult iplying 2x2 matricesMult iplicat ion of matrices: BasicMult iplicat ion of matrices: AdvancedWord problem involving mult iplicat ion of matricesFinding the inverse of a 2x2 matrixFinding the inverse of a 3x3 matrixFinding the determinant of a 2x2 matrixFinding the determinant of a 3x3 matrixComplet ing Gauss-Jordan eliminat ion with a 2x2 matrixGauss-Jordan eliminat ion with a 2x2 matrixWrit ing solut ions to 3x3 systems of linear equations f rom augmented matricesComplet ing Gauss-Jordan eliminat ion with a 3x3 matrixSolving a system of linear equations given its augmented matrixFinding the inverse of a matrix to solve a 2x2 system of linear equationsUsing the inverse of a matrix to solve a 3x3 system of linear equationsUsing Cramer's rule to solve a 2x2 system of linear equationsUsing Cramer's rule to solve a 3x3 system of linear equationsIntroduct ion to part ial f ract ion decomposit ion with dist inct linear factorsPart ial f ract ion decomposit ion with dist inct linear factorsPart ial f ract ion decomposit ion with repeated linear factorsPart ial f ract ion decomposit ion with an irreducible quadrat ic factorPart ial f ract ion decomposit ion with repeated, irreducible quadrat ic factorsGraphically solving a system of linear and quadrat ic equationsUsing a graphing calculator to solve a nonlinear system of equations: BasicUsing a graphing calculator to solve a nonlinear system of equations: Advanced


Using a graphing calculator to solve an exponential or logarithmic equationSolving a system of linear and quadrat ic equationsSolving a system of nonlinear equations: Problem type 1Solving a system of nonlinear equations: Problem type 2Solving a word problem involving geometry using a system of nonlinear equationsIdentifying solut ions to a linear inequality in two variablesGraphing a linear inequality in the plane: Vert ical or horizontal lineGraphing a linear inequality in the plane: Slope-intercept formGraphing a linear inequality in the plane: Standard formWrit ing an inequality given its graph in the plane: Horizontal or vert ical boundary lineWrit ing an inequality given its graph in the plane: Slanted boundary lineGraphing a quadrat ic inequality: Problem type 1Graphing a quadrat ic inequality: Problem type 2Graphing an inequality involving a circleGraphing a system of two linear inequalit ies: BasicGraphing a system of two linear inequalit ies: AdvancedGraphing a system of three linear inequalit iesGraphing a system of nonlinear inequalit ies: Problem type 1Writ ing a mult i-step inequality for a real-world situat ionSolving a word problem using a system of linear inequalit ies: Problem type 1Solving a word problem using a system of linear inequalit ies: Problem type 2Linear programmingSolving a word problem using linear programming

Conic Sect ions (38 topics)Graphing a parabola of the form y2 = ax or x2 = ayGraphing a parabola of the form x = a(y-k)2 + h or y = a(x-h)2 + kGraphing a parabola of the form ay2 + by + cx + d = 0 or ax2 + bx + cy + d = 0Writ ing an equation of a parabola given the vertex and the focusWrit ing an equation of a parabola given the focus and the directrixDeriving the equation of a parabola given its focus and directrixFinding the vertex, focus, directrix, and axis of symmetry of a parabolaFinding the focus of a parabola of the form ay2 + by + cx + d = 0 or ax2 + bx + cy + d = 0Writ ing an equation of a parabola given its graphWord problem involving a parabolaGraphing an ellipse given its equation in standard formGraphing an ellipse centered at the origin: Ax2 + By2 = CGraphing an ellipse given its equation in general formFinding the center, vert ices, and foci of an ellipseFinding the foci of an ellipse given its equation in general formWrit ing an equation of an ellipse given the center, an endpoint of an axis, and the length of the other axisWrit ing an equation of an ellipse given the foci and the major axis lengthGraphing a system of nonlinear inequalit ies: Problem type 2Word problem involving an ellipseGraphing a hyperbola given its equation in standard formGraphing a hyperbola centered at the origin: Ax2 + By2 = CGraphing a hyperbola given its equation in general formFinding the center, vert ices, foci, and asymptotes of a hyperbolaFinding the foci of a hyperbola given its equation in general formWrit ing an equation of a hyperbola given the foci and the vert icesWrit ing an equation of a hyperbola given the foci and the asymptotes: BasicWrit ing an equation of a hyperbola given the foci and the asymptotes: AdvancedClassifying conics given their equationsComplet ing a table and choosing a graph given a pair of parametric equationsWrit ing the equation of a line and sketching its graph given its parametric equationsWrit ing the equation of a parabola and sketching its graph given its parametric equationsWrit ing the equation of a circle or ellipse and sketching its graph given its parametric equationsGraphing a pair of parametric equations with a restricted domain: Line or parabolaGraphing a pair of parametric equations with a restricted domain: CircleGraphing a pair of parametric equations with a restricted domain: EllipseComplet ing pairs of parametric equationsWord problem involving parametric equations for project ile motion: Problem type 1Word problem involving parametric equations for project ile motion: Problem type 2

Sequences, Series, and Probability (67 topics)Finding the first terms of an arithmetic sequence using an explicit ruleFinding the first terms of a geometric sequence using an explicit ruleFinding the first terms of a sequence using an explicit rule with mult iple occurrences of nFinding the next terms of an arithmetic sequence with integersFinding the first terms of a sequence using a recursive ruleIdentifying arithmetic sequences and finding the common differenceFinding a specified term of an arithmetic sequence given the first termsFinding a specified term of an arithmetic sequence given the common difference and first term


Finding a specified term of an arithmetic sequence given two terms of the sequenceWrit ing an explicit rule for an arithmetic sequenceWrit ing a recursive rule for an arithmetic sequenceSum of the first n terms of an arithmetic sequenceFinding the next terms of a geometric sequence with signed numbersIdentifying arithmetic and geometric sequencesIdentifying geometric sequences and finding the common rat ioFinding a specified term of a geometric sequence given the first termsFinding a specified term of a geometric sequence given the common rat io and first termFinding a specified term of a geometric sequence given two terms of the sequenceArithmetic and geometric sequences: Identifying and writ ing an explicit ruleWrit ing recursive rules for arithmetic and geometric sequencesSum of the first n terms of a geometric sequenceSum of an infinite geometric seriesIdentifying linear, quadrat ic, and exponential funct ions given ordered pairsFactorial expressionsInterpret ing a tree diagramIntroduct ion to the counting principleCounting principleComputing permutat ions and combinat ionsIntroduct ion to permutat ions and combinat ionsPermutat ions and combinat ions: Problem type 1Permutat ions and combinat ions: Problem type 2Permutat ions and combinat ions: Problem type 3Binomial formulaDetermining a sample space and outcomes for an event: Experiment involving a single select ionDetermining a sample space and outcomes for an event: Experiment involving mult iple select ionsProbability involving one die or choosing f rom n dist inct objectsProbability involving choosing f rom objects that are not dist inctExperimental and theoret ical probabilityOutcomes and event probabilityProbabilit ies of a permutat ion and a combinat ionArea as probabilityProbability of independent events: Decimal answersProbability of dependent events: Decimal answersProbabilit ies of draws with replacementProbabilit ies of draws without replacementInterpret ing a Venn diagram of 2 setsInterpret ing a Venn diagram of 3 setsIntroduct ion to shading a Venn diagram with 2 eventsShading a Venn diagram with 2 events: Unions, intersect ions, and complementsShading a Venn diagram with 3 sets to represent a groupProbabilit ies involving two rolls of a dieDetermining outcomes for unions, intersect ions, and complements of eventsUsing a Venn diagram to understand the addit ion rule for probabilityOutcomes and event probability: Addit ion ruleWord problem involving the probability of a union or an intersect ionIdentifying independent events given values of probabilit iesProbability of the union and intersect ion of independent eventsProbability of the union of mutually exclusive events and independent eventsUsing a Venn diagram to understand the mult iplicat ion rule for probabilityOutcomes and event probability: Condit ional probabilityComputing condit ional probability using a two-way f requency tableComputing condit ional probability to make an inference using a two-way f requency tableCondit ional probability: BasicIntersect ion and condit ional probabilityBinomial problems: BasicBinomial problems: AdvancedUsing a random number table to make a fair decision

Limits and Continuity (15 topics)Est imating a limit numericallyFinding limits f rom a graphFinding a limit by using the limit laws: Problem type 1Finding limits for a piecewise-defined funct ionFinding a limit by using the limit laws: Problem type 2Finding a limit by using the limit laws: Problem type 3Squeeze TheoremDetermining points of discontinuity f rom a graphDetermining a parameter to make a funct ion continuousInfinite limits and graphsLimits at infinity and graphsLimits at infinity and rat ional funct ionsInfinite limits and rat ional funct ions


Finding a limit of a trigonometric funct ion by using continuityFinding a limit by using special t rigonometric limits

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College Algebra with Trigonometry - ALEKS

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