# PreCalculus. Curriculum (637 topics additional topics)

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1 PreCalculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs. Curriculum (637 topics additional topics) Algebra and Geometry Review (104 topics) Real Numbers and Algebraic Expressions (8 topics) Signed fraction subtraction involving double negation Signed fraction multiplication: Basic Signed fraction division Exponents and signed fractions Order of operations with integers Evaluating a quadratic expression: Integers Using distribution and combining like terms to simplify: Univariate Using distribution with double negation and combining like terms to simplify: Multivariate Exponents (19 topics) Introduction to the product rule of exponents Product rule with positive exponents: Univariate Product rule with positive exponents: Multivariate Introduction to the power of a power rule of exponents Introduction to the power of a product rule of exponents Power rules with positive exponents: Multivariate products Power rules with positive exponents: Multivariate quotients Introduction to the quotient rule of exponents Simplifying a ratio of univariate monomials Quotient of expressions involving exponents Evaluating expressions with exponents of zero Evaluating an expression with a negative exponent: Whole number base Evaluating an expression with a negative exponent: Positive fraction base Evaluating an expression with a negative exponent: Negative integer base Rewriting an algebraic expression without a negative exponent Introduction to the product rule with negative exponents Quotient rule with negative exponents: Problem type 1 Power of a power rule with negative exponents Power rules with negative exponents Polynomial Expressions (13 topics) Degree and leading coefficient of a univariate polynomial Simplifying a sum or difference of two univariate polynomials Multiplying a univariate polynomial by a monomial with a positive coefficient Multiplying a multivariate polynomial by a monomial Multiplying binomials with leading coefficients of 1 Multiplying binomials with leading coefficients greater than 1 Multiplying binomials in two variables Multiplying conjugate binomials: Univariate Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 1/25

5 Algebraic symbol manipulation with radicals Solving an equation with a root index greater than 2: Problem type 1 Solving an equation with a root index greater than 2: Problem type 2 Solving an equation that can be written in quadratic form: Problem type 1 Graphs and Functions (136 topics) The Coordinate Plane, Distance, and Midpoint (5 topics) Naming the quadrant or axis of a point given its coordinates Naming the quadrant or axis of a point given the signs of its coordinates Distance between two points in the plane: Exact answers Midpoint of a line segment in the plane Finding a solution to a linear equation in two variables Graphs of Equations (14 topics) Graphing a linear equation of the form y = mx Graphing a line given its equation in slope intercept form: Integer slope Graphing a line given its equation in slope intercept form: Fractional slope Graphing a line given its equation in standard form Graphing a vertical or horizontal line Finding x and y intercepts of a line given the equation: Basic Graphing a line by first finding its x and y intercepts Finding intercepts of a nonlinear function given its graph Finding x and y intercepts of the graph of a nonlinear equation Graphing an absolute value equation of the form y = A x Graphing a parabola of the form y = ax 2 Graphing a parabola of the form y = ax 2 + c Graphing a cubic function of the form y = ax 3 Determining if graphs have symmetry with respect to the x axis, y axis, or origin Slope and Equations of Lines (16 topics) Finding slope given the graph of a line on a grid Finding slope given two points on the line Finding the slope of horizontal and vertical lines Graphing 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 + b Finding the slope and y intercept of a line given its equation in the form Ax + By = C Writing an equation of a line given its slope and y intercept Writing an equation in slope intercept form given the slope and a point Finding the slope and a point on a line given its equation in point slope form Writing an equation in point slope form given the slope and a point Writing an equation of a line given the y intercept and another point Writing the equation of the line through two given points Writing the equations of vertical and horizontal lines through a given point Finding slopes of lines parallel and perpendicular to a line given in slope intercept form Finding slopes of lines parallel and perpendicular to a line given in the form Ax + By = C Writing equations of lines parallel and perpendicular to a given line through a point Linear Applications (5 topics) Writing and evaluating a function that models a real world situation: Advanced Writing an equation and drawing its graph to model a real world situation: Advanced Finding the initial amount and rate of change given a graph of a linear function Interpreting the parameters of a linear function that models a real world situation Application problem with a linear function: Finding a coordinate given two points Review of Systems of Linear Equations (5 topics) Graphically solving a system of linear equations Using a graphing calculator to solve a system of linear equations: Basic Solving a system of linear equations using substitution Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 5/25

7 Graphing a function of the form f(x) = ax 2 + c Graphing a parabola of the form y = (x h) 2 + k Graphing a square root function: Problem type 1 Graphing a square root function: Problem type 2 Matching parent graphs with their equations Graphing a piecewise defined function: Problem type 1 Graphing a piecewise defined function: Problem type 2 Graphing a piecewise defined function: Problem type 3 Even and odd functions: Problem type 1 Even and odd functions: Problem type 2 Finding the average rate of change of a function Finding the average rate of change of a function given its graph Word problem involving average rate of change Transformations (13 topics) Translating the graph of a parabola: One step Translating the graph of a parabola: Two steps How the leading coefficient affects the shape of a parabola Translating the graph of an absolute value function: One step Translating the graph of an absolute value function: Two steps Writing an equation for a function after a vertical translation Translating the graph of a function: One step Translating the graph of a function: Two steps Transforming the graph of a function by reflecting over an axis Transforming the graph of a function by shrinking or stretching Transforming the graph of a function using more than one transformation Transforming the graph of a quadratic, cubic, square root, or absolute value function Writing an equation for a function after a vertical and horizontal translation Combining Functions; Composite Functions; Inverse Functions (19 topics) Sum, difference, and product of two functions Quotient of two functions: Basic Quotient of two functions: Advanced Combining functions: Advanced Introduction to the composition of two functions Composition of two functions: Basic Composition of a function with itself Expressing a function as a composition of two functions Composition of two functions: Advanced Composition of two rational functions Word problem involving composition of two functions Horizontal line test Determining whether two functions are inverses of each other Inverse functions: Linear, discrete Inverse functions: Quadratic, square root Inverse functions: Cubic, cube root Inverse functions: Rational Graphing the inverse of a function given its graph Finding, evaluating, and interpreting an inverse function for a given linear relationship Polynomial and Rational Functions (64 topics) Quadratic Functions (16 topics) Finding the vertex, intercepts, and axis of symmetry from the graph of a parabola Graphing a parabola of the form y = x 2 + bx + c Graphing a parabola of the form y = a(x h) 2 + k Graphing a parabola of the form y = ax 2 + bx + c: Integer coefficients Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 7/25

9 Transforming the graph of a rational function Graphing a rational function: Quadratic over linear Graphing rational functions with holes Matching graphs with rational functions: Two vertical asymptotes Graphing a rational function with more than one vertical asymptote Using a graphing calculator to solve a word problem involving a local extremum of a rational function Polynomial and Rational Inequalities (8 topics) Solving a quadratic inequality written in factored form Solving a quadratic inequality Solving a polynomial inequality: Problem type 1 Solving a polynomial inequality: Problem type 2 Solving a polynomial inequality: Problem type 3 Solving a polynomial inequality: Problem type 4 Solving a rational inequality: Problem type 1 Solving a rational inequality: Problem type 2 Exponential and Logarithmic Functions (50 topics) Graphing Exponential Functions (8 topics) Table for an exponential function Graphing an exponential function and its asymptote: f(x)=b x Graphing an exponential function and its asymptote: f(x) = a(b) x Graphing an exponential function and its asymptote: f(x)=b x or f(x)= b ax Translating the graph of an exponential function The graph, domain, and range of an exponential function Transforming the graph of a natural exponential function Graphing an exponential function and its asymptote: f(x) = a(e) x b + c Applications of Exponential Functions (7 topics) Using a calculator to evaluate exponential expressions Evaluating an exponential function that models a real world situation Using a calculator to evaluate exponential expressions involving base e Evaluating an exponential function with base e that models a real world situation Introduction to compound interest Finding a final amount in a word problem on exponential growth or decay Finding the final amount in a word problem on compound interest Logarithmic Functions (9 topics) Using a calculator to evaluate natural and common logarithmic expressions Converting between logarithmic and exponential equations Converting between natural logarithmic and exponential equations Evaluating logarithmic expressions Solving an equation of the form log b a = c Translating the graph of a logarithmic function Graphing a logarithmic function: Basic The graph, domain, and range of a logarithmic function Domain of a logarithmic function: Advanced Properties of Logarithms (6 topics) Basic properties of logarithms Using properties of logarithms to evaluate expressions Expanding a logarithmic expression: Problem type 1 Expanding a logarithmic expression: Problem type 2 Writing an expression as a single logarithm Change of base for logarithms: Problem type 1 Logarithmic and Exponential Equations (10 topics) Solving a multi step equation involving a single logarithm: Problem type 1 Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 9/25

10 Solving a multi step equation involving a single logarithm: Problem type 2 Solving a multi step equation involving natural logarithms Solving an equation involving logarithms on both sides: Problem type 1 Solving an equation involving logarithms on both sides: Problem type 2 Solving an exponential equation by finding common bases: Linear exponents Solving an exponential equation by using logarithms: Decimal answers, basic Solving an exponential equation by using natural logarithms: Decimal answers Solving an exponential equation by using logarithms: Decimal answers, advanced Solving an exponential equation by using logarithms: Exact answers in logarithmic form Applications (10 topics) Finding the time to reach a limit in a word problem on exponential growth or decay Finding the time in a word problem on compound interest Finding the time given an exponential function with base e that models a real world situation Finding the final amount in a word problem on continuous compound interest Finding the initial amount in a word problem on continuous compound interest Finding the final amount in a word problem on continuous exponential growth or decay Finding the rate or time in a word problem on continuous exponential growth or decay Finding half life or doubling time Writing and evaluating a function modeling continuous exponential growth or decay given doubling time or half life Writing and evaluating a function modeling continuous exponential growth or decay given two outputs Trigonometric Functions (58 topics) Angles and Their Measure (5 topics) Converting between degree and radian measure: Problem type 1 Converting between degree and radian measure: Problem type 2 Sketching an angle in standard position Coterminal angles Arc length and central angle measure The Unit Circle and Evaluating Trigonometric Functions (9 topics) Finding coordinates on the unit circle for special angles Trigonometric functions and special angles: Problem type 1 Finding trigonometric ratios from a point on the unit circle Trigonometric functions and special angles: Problem type 2 Trigonometric functions and special angles: Problem type 3 Evaluating expressions involving sine and cosine Even and odd properties of trigonometric functions Using a calculator to approximate sine, cosine, and tangent values Evaluating a sinusoidal function that models a real world situation Right Triangle Trigonometry (9 topics) Sine, cosine, and tangent ratios: Variables for side lengths Using the Pythagorean Theorem to find a trigonometric ratio Finding trigonometric ratios given a right triangle Using a trigonometric ratio to find a side length in a right triangle Using trigonometry to find a length in a word problem with one right triangle Using a trigonometric ratio to find an angle measure in a right triangle Using trigonometry to find angles of elevation or depression in a word problem Solving a right triangle Using trigonometry to find a length in a word problem with two right triangles Trigonometric Functions of Angles (7 topics) Reference angles: Problem type 1 Reference angles: Problem type 2 Determining the location of a terminal point given the signs of trigonometric values Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 10/25

11 Finding values of trigonometric functions given information about an angle: Problem type 1 Finding values of trigonometric functions given information about an angle: Problem type 2 Finding values of trigonometric functions given information about an angle: Problem type 3 Finding values of trigonometric functions given information about an angle: Problem type 4 Graphs of Sine and Cosine Functions (14 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) Sketching the graph of y= sin(x)+ d or y= cos(x)+ d Sketching 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) Sketching the graph of y= a sin(bx) or y= a cos(bx) 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)+ d Amplitude and period of sine and cosine functions Amplitude, period, and phase shift of sine and cosine functions Writing the equation of a sine or cosine function given its graph: Problem type 1 Writing the equation of a sine or cosine function given its graph: Problem type 2 Word problem involving a sine or cosine function: Problem type 1 Word problem involving a sine or cosine function: Problem type 2 Graphs of Other Trigonometric Functions (6 topics) Domains and ranges of trigonometric functions Matching graphs and equations for secant, cosecant, tangent, and cotangent functions Sketching the graph of a secant or cosecant function: Problem type 1 Sketching the graph of a secant or cosecant function: Problem type 2 Sketching the graph of a tangent or cotangent function: Problem type 1 Sketching the graph of a tangent or cotangent function: Problem type 2 Inverse Trigonometric Functions (8 topics) Values of inverse trigonometric functions Composition of a trigonometric function with its inverse trigonometric function: Problem type 1 Composition of a trigonometric function with its inverse trigonometric function: Problem type 2 Composition of a trigonometric function with the inverse of another trigonometric function: Problem type 1 Composition of a trigonometric function with the inverse of another trigonometric function: Problem type 2 Composition of a trigonometric function with the inverse of another trigonometric function: Problem type 3 Composition of trigonometric functions with variable expressions as inputs: Problem type 1 Using a calculator to approximate inverse trigonometric values Trigonometric Identities and Equations (32 topics) Verifying Trigonometric Identities (6 topics) Simplifying trigonometric expressions Using cofunction identities Verifying a trigonometric identity Proving trigonometric identities: Problem type 1 Proving trigonometric identities: Problem type 2 Proving trigonometric identities: Problem type 3 Sum and Difference Formulas (6 topics) Sum and difference identities: Problem type 1 Sum and difference identities: Problem type 2 Sum and difference identities: Problem type 3 Sum and difference identities: Problem type 4 Proving trigonometric identities using sum and difference properties: Problem type 1 Proving trigonometric identities using sum and difference properties: Problem type 2 Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 11/25

12 Double Angle, Half Angle, Product to Sum, and Power Reducing Formulas (8 topics) Double angle identities: Problem type 1 Double angle identities: Problem type 2 Power reducing identities Half angle identities: Problem type 1 Half angle identities: Problem type 2 Product to sum and sum to product identities: Problem type 1 Product to sum and sum to product identities: Problem type 2 Proving trigonometric identities using double angle properties Trigonometric Equations (12 topics) Finding solutions in an interval for a basic equation involving sine or cosine Finding solutions in an interval for a basic tangent, cotangent, secant, or cosecant equation Solving a basic trigonometric equation using a calculator Solving a basic trigonometric equation involving sine or cosine Solving a basic trigonometric equation involving tangent, cotangent, secant, or cosecant Finding solutions in an interval for a trigonometric equation in factored form Finding solutions in an interval for a trigonometric equation with a squared function: Problem type 1 Finding solutions in an interval for a trigonometric equation with a squared function: Problem type 2 Finding solutions in an interval for a trigonometric equation using Pythagorean identities: Problem type 1 Finding solutions in an interval for an equation with sine and cosine using double angle identities Solving a trigonometric equation modeling a real world situation Finding solutions in an interval for a trigonometric equation with an angle multiplied by a constant Additional Topics in Trigonometry (53 topics) Laws of Sines and Cosines (5 topics) Solving a triangle with the law of sines: Problem type 1 Solving a triangle with the law of sines: Problem type 2 Solving a word problem using the law of sines Solving a triangle with the law of cosines Solving a word problem using the law of cosines Vectors (19 topics) Writing a position vector in ai+bj form given its graph Writing a vector in ai+bj form given its initial and terminal points Writing a vector in component form given its initial and terminal points Magnitude of a vector given in ai+bj form Magnitude of a vector given in component form Vector addition and scalar multiplication: ai+bj form Linear combination of vectors: ai+bj form Vector addition and scalar multiplication: Component form Linear combination of vectors: Component form Unit vectors Multiplication of a vector by a scalar: Geometric approach Vector addition: Geometric approach Vector subtraction: Geometric approach Finding the magnitude and direction of a vector given its graph Finding the components of a vector given its graph Finding the direction angle of a vector given in ai+bj form Writing a vector given its magnitude and direction angle Writing a vector to represent a force pushing or pulling an object Finding the magnitude and direction angle of the resultant force of two vectors The Dot Product (7 topics) Dot product of vectors given in ai+bj form Dot product of vectors given in component form Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 12/25

13 Finding the angle between two vectors given in component form Classifying vector relationships by finding the angle between two vectors given in ai + bj form Decomposing a vector into two orthogonal vectors Finding the amount of work done given a force vector and a distance Finding magnitudes of forces related to an object on a ramp Polar Coordinates and Equations (7 topics) Plotting points in polar coordinates Multiple representations of polar coordinates Converting rectangular coordinates to polar coordinates: Special angles Converting polar coordinates to rectangular coordinates Converting an equation written in rectangular form to one written in polar form Converting an equation written in polar form to one written in rectangular form: Problem type 1 Converting an equation written in polar form to one written in rectangular form: Problem type 2 Graphs of Polar Equations (6 topics) Graphing a polar equation: Basic Graphing a polar equation: Circle Graphing a polar equation: Limacon Graphing a polar equation: Rose Graphing a polar equation: Lemniscate Matching polar equations with their graphs Complex Numbers and De Moivre's Theorem (9 topics) Plotting complex numbers Writing a complex number in standard form given its trigonometric form Writing a complex number in trigonometric form: Special angles Writing a complex number in trigonometric form: Decimal answers Multiplying and dividing complex numbers in trigonometric form De Moivre's Theorem: Answers in trigonometric form De Moivre's Theorem: Answers in standard form Finding the nth roots of a number: Problem type 1 Finding the nth roots of a number: Problem type 2 Systems of Equations and Matrices (32 topics) Systems of Linear Equations in Two Variables (2 topics) Solving a system of linear equations with fractional coefficients Solving a 2x2 system of linear equations that is inconsistent or consistent dependent Applications of Systems of Linear Equations in Two Variables (4 topics) Interpreting the graphs of two functions Solving a word problem involving a sum and another basic relationship using a system of linear equations Solving a word problem using a system of linear equations of the form Ax + By = C Solving a value mixture problem using a system of linear equations Systems of Linear Equations in Three Variables (4 topics) Introduction to solving a 3x3 system of linear equations Solving a 3x3 system of linear equations: Problem type 1 Solving a 3x3 system of linear equations: Problem type 2 Solving a word problem using a 3x3 system of linear equations: Problem type 1 Operations with Matrices (10 topics) Scalar multiplication of a matrix Addition or subtraction of matrices Linear combination of matrices Squaring and multiplying 2x2 matrices Multiplication of matrices: Basic Word problem involving multiplication of matrices Finding the inverse of a 2x2 matrix Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 13/25

14 Finding the inverse of a 3x3 matrix Finding the determinant of a 2x2 matrix Finding the determinant of a 3x3 matrix Using Matrices to Solve Systems of Equations (7 topics) Completing Gauss Jordan elimination with a 2x2 matrix Gauss Jordan elimination with a 2x2 matrix Writing solutions to 3x3 systems of linear equations from augmented matrices Completing Gauss Jordan elimination with a 3x3 matrix Finding the inverse of a matrix to solve a 2x2 system of linear equations Using Cramer's rule to solve a 2x2 system of linear equations Using Cramer's rule to solve a 3x3 system of linear equations Partial Fraction Decomposition (5 topics) Introduction to partial fraction decomposition with distinct linear factors Partial fraction decomposition with distinct linear factors Partial fraction decomposition with repeated linear factors Partial fraction decomposition with an irreducible quadratic factor Partial fraction decomposition with repeated, irreducible quadratic factors Conic Sections (34 topics) Parabolas (9 topics) Graphing a parabola of the form y 2 = ax or x 2 = ay Graphing a parabola of the form x=a(y k) 2 +h or y=a(x h) 2 +k Graphing a parabola of the form ay 2 + by + cx + d = 0 or ax 2 + bx + cy + d = 0 Writing an equation of a parabola given the vertex and the focus Writing an equation of a parabola given the focus and the directrix Finding the vertex, focus, directrix, and axis of symmetry of a parabola Finding the focus of a parabola of the form ay 2 + by + cx + d = 0 or ax 2 + bx + cy + d = 0 Writing an equation of a parabola given its graph Word problem involving a parabola Ellipses (7 topics) Graphing an ellipse given its equation in standard form Graphing an ellipse centered at the origin: Ax 2 + By 2 = C Graphing an ellipse given its equation in general form Finding the center, vertices, and foci of an ellipse Finding the foci of an ellipse given its equation in general form Writing an equation of an ellipse given the center, an endpoint of an axis, and the length of the other axis Word problem involving an ellipse Hyperbolas (8 topics) Graphing a hyperbola given its equation in standard form Graphing a hyperbola centered at the origin: Ax 2 + By 2 = C Graphing a hyperbola given its equation in general form Finding the center, vertices, foci, and asymptotes of a hyperbola Finding the foci of a hyperbola given its equation in general form Writing an equation of a hyperbola given the foci and the vertices Writing an equation of a hyperbola given the foci and the asymptotes: Basic Classifying conics given their equations Parametric Equations (10 topics) Completing a table and choosing a graph given a pair of parametric equations Writing the equation of a line and sketching its graph given its parametric equations Writing the equation of a parabola and sketching its graph given its parametric equations Writing the equation of a circle or ellipse and sketching its graph given its parametric equations Graphing a pair of parametric equations with a restricted domain: Line or parabola Graphing a pair of parametric equations with a restricted domain: Circle Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 14/25

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