# PreCalculus. Curriculum (447 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 (447 topics additional topics) Algebra and Geometry Review (119 topics) Properties and Signed Numbers (12 topics) Integers and rational numbers Rational and irrational numbers Operations with absolute value Exponents and integers: Problem type 1 Exponents and integers: Problem type 2 Exponents and order of operations Evaluation of a linear expression in two variables Evaluation of a polynomial in one variable Simple addition and subtraction of signed fractions Signed fraction multiplication: Advanced Complex fractions without variables: Problem type 2 Plotting a point in the coordinate plane Linear Equations and Applications (17 topics) Solving a linear equation with several occurrences of the variable: Problem type 2 Solving a linear equation with several occurrences of the variable: Problem type 3 Solving a linear equation with several occurrences of the variable: Problem type 4 Solving a linear equation with several occurrences of the variable: Problem type 5 Solving equations with zero, one, or infinitely many solutions Simple absolute value equation Solving an equation involving absolute value: Basic Algebraic symbol manipulation: Problem type 1 Algebraic symbol manipulation: Problem type 2 Solving a word problem using a linear equation: Problem type 1 Solving a word problem using a linear equation: Problem type 2 Solving a word problem using a linear equation: Problem type 3 Word problem involving area and perimeter of a rectangle Simple interest Solving a value mixture problem using a linear equation Solving a percent mixture problem using a linear equation Solving a rate problem using a linear equation Linear Inequalities and Applications (8 topics) Solving a linear inequality: Problem type 2 Solving a linear inequality: Problem type 3 Solving a linear inequality: Problem type 4 Solving a compound linear inequality: Problem type 1 Solving an inequality involving absolute value: Basic Solving an inequality involving absolute value Copyright 2011 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 1/13

4 Finding the average rate of change of a function Linear Functions, Graphs, and Inequalities (21 topics) Solutions to a linear equation in two variables: Problem type 2 Graphing a line given its equation in slope intercept form Graphing a line given its equation in standard form Graphing a line through a given point with a given slope Graphing a vertical or horizontal line Finding x and y intercepts of a line given the equation in standard form Finding slope given the graph of a line on a grid Finding slope given two points on the line Finding the slope of a line given its equation Writing an equation of a line given the y intercept and a point Writing the equation of a line given the slope and a point on the line Writing the equation of the line through two given points Writing the equations of vertical and horizontal lines through a given point Writing equations and drawing graphs to fit a narrative Application problem with a linear function: Problem type 1 Application problem with a linear function: Problem type 2 Slopes of parallel and perpendicular lines: Problem type 1 Slopes of parallel and perpendicular lines: Problem type 2 Graphing a linear inequality in the plane: Problem type 1 Graphing a linear inequality in the plane: Problem type 2 Graphing a linear inequality in the plane: Problem type 3 Graphs and Transformations (20 topics) Finding inputs and outputs of a function from its graph Finding intercepts and zeros of a function given the graph Finding where a function is increasing, decreasing, or constant given the graph Finding local maxima and minima of a function given the graph Domain and range from the graph of a continuous function Domain and range from the graph of a piecewise function Testing an equation for symmetry about the axes and origin Even and odd functions Writing an equation for a function after a vertical translation Writing an equation for a function after a vertical and horizontal 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 Graphing a parabola: Problem type 1 Graphing a simple cubic function Graphing a function involving a square root Graphing an equation involving absolute value in the plane Graphing a piecewise defined function Combining Functions; Composite Functions; Inverse Functions (12 topics) Sum, difference, and product of two functions Quotient of two functions Combining functions: Advanced Composition of two functions: Domain and range Composition of two functions: Basic Composition of two functions: Advanced Expressing a function as a composition of two functions Horizontal line test Copyright 2011 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 4/13

6 Dividing complex numbers Simplifying a power of i Solving a quadratic equation with complex roots Multiplying expressions involving complex conjugates Finding a polynomial of a given degree with given zeros: Complex zeros Using a given zero to write a polynomial as a product of linear factors: Complex zeros Using the rational zeros theorem to find all zeros of a polynomial: Complex zeros Linear factors theorem and conjugate zeros theorem Using the conjugate zeros theorem to find all zeros of a polynomial Graphing Polynomial and Rational Functions (14 topics) Finding x and y intercepts given a polynomial function Determining the end behavior of the graph of a polynomial function Using a graphing calculator to find zeros of a polynomial function Using a graphing calculator to find local extrema of a polynomial function Inferring properties of a polynomial function from its graph Matching graphs with polynomial functions Using a graphing calculator to solve a word problem involving a local extremum of a polynomial function Finding the asymptotes of a rational function: Problem type 1 Finding the asymptotes of a rational function: Problem type 2 Sketching the graph of a rational function: Problem type 1 Sketching the graph of a rational function: Problem type 2 Graphing rational functions with holes Matching graphs with rational functions: Two vertical asymptotes Writing the equation of a rational function given its graph Polynomial and Rational Inequalities (5 topics) Solving a quadratic inequality written in factored form Solving a quadratic inequality Solving a polynomial inequality Solving a rational inequality: Problem type 1 Solving a rational inequality: Problem type 2 Exponential and Logarithmic Functions (29 topics) Properties of Logarithms (9 topics) Evaluating an exponential function that models a real world situation Converting between logarithmic and exponential equations Converting between natural logarithmic and exponential equations Evaluating a logarithmic expression Basic properties of logarithms Writing expressions as a single logarithm Expanding a logarithmic expression: Problem type 1 Expanding a logarithmic expression: Problem type 2 Change of base for logarithms: Problem type 1 Logarithmic and Exponential Equations (14 topics) Solving a logarithmic equation: Problem type 1 Solving a logarithmic equation: Problem type 2 Solving a logarithmic equation: Problem type 3 Solving a logarithmic equation: Problem type 4 Solving a logarithmic equation: Problem type 5 Solving an exponential equation: Problem type 1 Solving an exponential equation: Problem type 2 Solving an exponential equation: Problem type 3 Using a graphing calculator to solve an exponential or logarithmic equation Solving a word problem using an exponential equation: Problem type 1 Copyright 2011 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 6/13

7 Solving a word problem using an exponential equation: Problem type 2 Solving a word problem using an exponential equation: Problem type 3 Solving a word problem using an exponential equation: Problem type 4 Compound interest Graphing Logarithmic and Exponential Functions (6 topics) Sketching the graph of an exponential function: Basic The graph, domain, and range of an exponential function Sketching the graph of an exponential function: Advanced Sketching the graph of a logarithmic function: Basic The graph, domain, and range of a logarithmic function Translating the graph of a logarithmic or exponential function Trigonometry (100 topics) Angles (9 topics) Converting between degree and radian measure: Problem type 1 Converting between degree and radian measure: Problem type 2 Coterminal angles Sketching an angle in standard position Reference angles: Problem type 1 Reference angles: Problem type 2 Arc length and central angle measure Area of a sector of a circle Angular and linear speed Unit Circle and Right Triangle Trigonometry (17 topics) Finding coordinates on the unit circle for special angles Finding a point on the unit circle given one coordinate Finding trigonometric ratios from a point on the unit circle Sine, cosine, and tangent ratios Finding trigonometric ratios given a right triangle Using a trigonometric ratio to find a side length in a right triangle Using trigonometry to find distances Using a trigonometric ratio to find an angle measure in a right triangle Using trigonometry to find angles of elevation or depression Solving a right triangle Determining the location of a terminal point given the signs of trigonometric values Trigonometric functions and special angles: Problem type 1 Trigonometric functions and special angles: Problem type 2 Trigonometric functions and special angles: Problem type 3 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 Trigonometric Functions (18 topics) Amplitude and period of sine and cosine functions Amplitude, period, and phase shift of sine and cosine functions Word problem involving a sine or cosine function: Problem type 1 Word problem involving a sine or cosine function: Problem type 2 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 Sketching the graph of a sine or cosine function: Problem type 1 Sketching the graph of a sine or cosine function: Problem type 2 Sketching the graph of a sine or cosine function: Problem type 3 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 Copyright 2011 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 7/13

8 Sketching the graph of a tangent or cotangent function: Problem type 2 Matching graphs and equations for secant, cosecant, tangent, and cotangent functions Values of inverse trigonometric functions Composition of a trigonometric function and an inverse trigonometric function: Problem type 1 Composition of a trigonometric function and an inverse trigonometric function: Problem type 2 Composition of a trigonometric function and an inverse trigonometric function: Problem type 4 Trigonometric Identities (16 topics) Simplifying trigonometric expressions Using cofunction identities Sum and difference identities: Problem type 1 Sum and difference identities: Problem type 2 Sum and difference identities: Problem type 3 Double angle identities: Problem type 1 Double angle identities: Problem type 2 Half angle identities: Problem type 1 Product to sum and sum to product identities Product to sum and sum to product identities: Problem type 2 Verifying a trigonometric identity Proving trigonometric identities: Problem type 1 Proving trigonometric identities: Problem type 2 Proving trigonometric identities: Problem type 3 Proving trigonometric identities using sum and difference properties Proving trigonometric identities using double angle properties Trigonometric Equations (13 topics) Finding solutions in an interval for a basic equation involving sine or cosine Solving a basic trigonometric equation using a calculator Finding solutions in an interval for a basic tangent, cotangent, secant, or cosecant equation 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 Finding solutions in an interval for a trigonometric equation with an angle multiplied by a constant 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 Using a graphing calculator to solve a trigonometric equation Solving a basic trigonometric equation involving sine or cosine Solving a basic trigonometric equation involving tangent, cotangent, secant, or cosecant Laws of Sines and Cosines (7 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 Finding the area of a triangle using trigonometry Heron's formula Vectors (11 topics) Magnitude of a vector Multiplication of a vector by a scalar: Geometric approach Translation of a vector Vector addition and scalar multiplication Linear combination of vectors: Algebraic approach Vector addition: Geometric approach Vector subtraction: Geometric approach Copyright 2011 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 8/13

9 Calculating the magnitude and direction of a vector Finding the components of a vector Solving a force problem with vectors Dot product Polar Coordinates (9 topics) Plotting a point in polar coordinates Converting rectangular coordinates to polar coordinates Converting polar coordinates to rectangular coordinates Converting an equation written in rectangular coordinates to one written in polar form Converting an equation written in polar form to one written in rectangular coordinates Writing a complex number in trigonometric form De Moivre's theorem Finding the nth roots of a number: Problem type 1 Finding the nth roots of a number: Problem type 2 Systems of Linear Equations and Matrices (24 topics) Systems of Linear Equations (6 topics) Classifying systems of linear equations from graphs Graphically solving a system of linear equations Solving a simple system using substitution Solving a system of linear equations Solving a system that is inconsistent or consistent dependent Solving a system of 3 equations in 3 unknowns Applications (6 topics) Solving a word problem using a system of linear equations: Problem type 1 Solving a word problem using a system of linear equations: Problem type 2 Solving a word problem using a system of linear equations: Problem type 3 Solving a word problem using a system of linear equations: Problem type 4 Solving a word problem using a system of linear equations: Problem type 5 Solving a word problem using a 3 by 3 system of linear equations Systems of Linear Inequalities (1 topics) Graphing a system of linear inequalities Matrices (11 topics) Scalar multiplication of a matrix Addition and subtraction of matrices Linear combinations of matrices Multiplication of matrices: Basic Finding the determinant of a 2x2 matrix Finding the determinant of a 3x3 matrix Finding the inverse of a 2x2 matrix Finding the inverse of a 3x3 matrix Cramer's rule: Problem type 1 Using the inverse of a matrix to solve a system of linear equations Gauss Jordan elimination with a 2x2 matrix Sequences, Series, and Probability (9 topics) Sequences and Series (7 topics) Finding the first terms of a sequence Arithmetic and geometric sequences: Identifying and writing in standard form Arithmetic sequences Geometric sequences Sum of the first n terms of an arithmetic sequence Sum of the first n terms of a geometric sequence Sum of a geometric series Permutations and Combinations (2 topics) Copyright 2011 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 9/13

10 Factorial expressions Binomial formula Conic Sections (17 topics) Parabolas (2 topics) Graphing a parabola with a horizontal or a vertical axis Finding the focus of a parabola Circles and Ellipses (10 topics) Midpoint of a line segment in the plane Distance between two points in the plane Graphing a circle given its equation in standard form Graphing a circle given its equation in general form Writing an equation of a circle given its center and a point on the circle Writing an equation of a circle given the endpoints of a diameter Graph of an ellipse centered at the origin Graphing an ellipse given its equation in standard form Graphing an ellipse given its equation in general form Finding the foci of an ellipse Hyperbolas (4 topics) Graph of a hyperbola centered at the origin Graphing a hyperbola given its equation in standard form Graphing a hyperbola given its equation in general form Finding the foci of a hyperbola Classifying Conic Sections; Systems of Non linear Equations (1 topics) Classifying conics given their equations Limits and Continuity (9 topics) Introduction to Limits (2 topics) Estimating a limit numerically Finding limits from a graph Computing Limits Algebraically (4 topics) Finding limits for a piecewise defined function Finding a limit by using the limit laws: Problem type 1 Finding a limit by using the limit laws: Problem type 2 Finding a limit by using the limit laws: Problem type 3 Continuity (1 topics) Determining points of discontinuity from a graph Limits Involving Infinity (2 topics) Limits at infinity and graphs Limits at infinity and rational functions Other Topics Available(*) Algebra and Geometry Review (37 topics) Ordering integers Properties of real numbers Solving an equation involving absolute value: Advanced Solving a word problem using a linear equation: Problem type 4 Word problem on percentage: Problem type 2 Finding the value for a new score that will yield a given mean Solving a word problem with 3 unknowns using a linear equation Solving a compound linear inequality: Problem type 2 Translating sentences into inequalities Simplifying a polynomial expression Least common multiple of two monomials Copyright 2011 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 10/13

12 Finding solutions in an interval for an equation with sine and cosine using sum and difference identities Using a graphing calculator to solve a trigonometric inequality Solving a trigonometric equation involving a squared function: Problem type 1 Solving a trigonometric equation involving a squared function: Problem type 2 Solving a trigonometric equation involving more than one function Solving a trigonometric equation involving an angle multiplied by a constant Solving a trigonometric equation using sum and difference identities Solving a trigonometric equation using double angle identities Solving a trigonometric equation using half angle identities Unit vectors Finding magnitudes in a three vector problem Using the dot product to find perpendicular vectors Finding the angle between two vectors Finding the component of a vector along another vector Systems of Linear Equations and Matrices (11 topics) Creating an inconsistent system of linear equations Consistency and independence of a system of linear equations Partial fraction decomposition with distinct linear factors Partial fraction decomposition with repeated linear factors Partial fraction decomposition with an irreducible quadratic factor Solving a word problem using a system of linear inequalities Linear programming Solving a word problem using linear programming Multiplication of matrices: Advanced Cramer's rule: Problem type 2 Augmented matrix and solution set of a system of linear equations Sequences, Series, and Probability (14 topics) Introduction to permutations and combinations Permutations and combinations: Problem type 1 Permutations and combinations: Problem type 2 Permutations and combinations: Problem type 3 Probabilities of draws with replacement Probabilities of draws without replacement Venn diagrams: Two events Venn diagrams: Word problems Outcomes and event probability Die rolling Probability of intersection or union: Word problems Independent events: Basic Probability of union: Basic Intersection and conditional probability Conic Sections (9 topics) Writing an equation of a parabola given the vertex and the focus Writing an equation of an ellipse given the foci and the major axis length Writing an equation of an ellipse given the center, an endpoint of an axis, and the length of the other axis Writing an equation of a hyperbola given the foci and the vertices Writing an equation of a hyperbola given the foci and the asymptotes Solving a system of nonlinear equations Using a graphing calculator to solve a system of equations Graphing a system of nonlinear inequalities: Problem type 1 Graphing a system of nonlinear inequalities: Problem type 2 Copyright 2011 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 12/13

13 Limits and Continuity (6 topics) Squeeze Theorem Determining a parameter to make a function continuous Infinite limits and graphs Infinite limits and rational functions Finding a limit of a trigonometric function by using continuity Finding a limit by using special trigonometric limits (*) Other Topics Available By default, these topics are NOT included in the course, but can be added using the content editor in the Teacher Module. Copyright 2011 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 13/13

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