# College Algebra with Trigonometry

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1 College Algebra with Trigonometry This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (556 topics additional topics) Algebra and Geometry Review (126 topics) Real Numbers and Algebraic Expressions (14 topics) Signed fraction addition or subtraction: Basic Signed fraction subtraction involving double negation Signed fraction multiplication: Basic Signed fraction division Computing the distance between two integers on a number line Exponents and integers: Problem type 1 Exponents and signed fractions Order of operations with integers Evaluating a linear expression: Integer multiplication with addition or subtraction Evaluating a quadratic expression: Integers Evaluating a linear expression: Signed fraction multiplication with addition or subtraction Distributive property: Integer coefficients Using distribution and combining like terms to simplify: Univariate Using distribution with double negation and combining like terms to simplify: Multivariate Exponents (20 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 Simplifying a ratio of multivariate monomials: Basic 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 (14 topics) Degree and leading coefficient of a univariate polynomial Simplifying a sum or difference of two univariate polynomials Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 1/25

4 Solving a linear equation with several occurrences of the variable: Variables on both sides and two distributions Solving a linear equation with several occurrences of the variable: Fractional forms with monomial numerators Solving a two step equation with signed fractions Solving a linear equation with several occurrences of the variable: Variables on both sides and fractional coefficients Solving a linear equation with several occurrences of the variable: Fractional forms with binomial numerators Solving a proportion of the form (x+a)/b = c/d Solving for a variable in terms of other variables using addition or subtraction: Basic Solving for a variable in terms of other variables using addition or subtraction: Advanced Solving for a variable in terms of other variables using multiplication or division: Basic Solving for a variable in terms of other variables using multiplication or division: Advanced Solving for a variable in terms of other variables using addition or subtraction with division Solving for a variable inside parentheses in terms of other variables Solving for a variable in terms of other variables in a linear equation with fractions Translating a sentence into a one step equation Translating a sentence into a multi step equation Solving a word problem with two unknowns using a linear equation Solving a decimal word problem using a linear equation of the form Ax + B = C Solving a word problem with three unknowns using a linear equation Solving a one step word problem using the formula d = rt Solving a distance, rate, time problem using a linear equation Finding the perimeter or area of a rectangle given one of these values Finding the sale price given the original price and percent discount Absolute Value Equations (2 topics) Solving an absolute value equation: Problem type 1 Solving an absolute value equation: Problem type 2 Linear Inequalities and Applications (7 topics) Graphing a linear inequality on the number line Graphing a compound inequality on the number line Set builder and interval notation Identifying solutions to a two step linear inequality in one variable Solving a two step linear inequality: Problem type 1 Solving a two step linear inequality: Problem type 2 Solving a linear inequality with multiple occurrences of the variable: Problem type 1 Rational Equations that Simplify to Linear (8 topics) Solving a rational equation that simplifies to linear: Denominator x Solving a rational equation that simplifies to linear: Denominator x+a Solving a rational equation that simplifies to linear: Denominators a, x, or ax Solving a rational equation that simplifies to linear: Denominators ax and bx Solving a rational equation that simplifies to linear: Like binomial denominators Solving a rational equation that simplifies to linear: Unlike binomial denominators Solving for a variable in terms of other variables in a rational equation: Problem type 1 Solving for a variable in terms of other variables in a rational equation: Problem type 2 Complex Numbers (4 topics) Using i to rewrite square roots of negative numbers Adding or subtracting complex numbers Multiplying complex numbers Dividing complex numbers Quadratic Equations (20 topics) Solving an equation written in factored form Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 4/25

7 Variable expressions as inputs of functions: Problem type 1 Variable expressions as inputs of functions: Problem type 2 Variable expressions as inputs of functions: Problem type 3 Domain and range from ordered pairs Domain of a rational function: Excluded values Domain of a rational function: Interval notation Domain of a square root function: Basic Domain of a square root function: Advanced Finding the domain of a fractional function involving radicals Determining whether an equation defines a function: Basic Determining whether an equation defines a function: Advanced Finding outputs of a one step function that models a real world situation: Function notation Finding outputs of a two step function with decimals that models a real world situation: Function notation Finding inputs and outputs of a two step function that models a real world situation: Function notation Finding a difference quotient for a linear or quadratic function Finding a difference quotient for a rational function Graphs of Functions (27 topics) Finding an output of a function from its graph Finding inputs and outputs of a function from its graph Domain and range from the graph of a continuous function Domain and range from the graph of a piecewise function Finding where a function is increasing, decreasing, or constant given the graph Finding where a function is increasing, decreasing, or constant given the graph: Interval notation Finding local maxima and minima of a function given the graph Finding the absolute maximum and minimum of a function given the graph Finding values and intervals where the graph of a function is zero, positive, or negative Graphing a function of the form f(x) = ax + b: Integer slope Graphing a function of the form f(x) = ax + b: Fractional slope Graphing an absolute value equation in the plane: Basic Graphing an absolute value equation in the plane: Advanced Graphing a function of the form f(x) = ax 2 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 Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 7/25

9 Inferring properties of a polynomial function from its graph Using a graphing calculator to find local extrema of a polynomial function Using a graphing calculator to solve a word problem involving a local extremum of a polynomial function Division of Polynomials; Remainder and Factor Theorems (6 topics) Polynomial long division: Problem type 1 Polynomial long division: Problem type 2 Polynomial long division: Problem type 3 Synthetic division Using the remainder theorem to evaluate a polynomial The Factor Theorem Real Zeros of Polynomial Functions (7 topics) Using a given zero to write a polynomial as a product of linear factors: Real zeros Finding all possible rational zeros using the rational zeros theorem: Problem type 1 Finding all possible rational zeros using the rational zeros theorem: Problem type 2 Using the rational zeros theorem to find all zeros of a polynomial: Rational zeros Using the rational zeros theorem to find all zeros of a polynomial: Irrational zeros Using a graphing calculator to find zeros of a polynomial function Using a graphing calculator to solve a word problem involving a polynomial of degree 3 Complex Zeros of Polynomials Functions (4 topics) 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 Rational Functions (13 topics) Finding the intercepts, asymptotes, domain, and range from the graph of a rational function Finding the asymptotes of a rational function: Constant over linear Finding the asymptotes of a rational function: Linear over linear Finding horizontal and vertical asymptotes of a rational function: Quadratic numerator or denominator Finding the asymptotes of a rational function: Quadratic over linear Graphing a rational function: Constant over linear Graphing a rational function: Linear over linear 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 Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 9/25

10 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 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 Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 10/25

11 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 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) Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 11/25

12 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 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 Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 12/25

16 Word problem involving the area between two concentric circles Area involving inscribed figures Volume of a triangular prism Volume of a pyramid Volume of a cylinder Word problem involving the rate of filling or emptying a cylinder Volume of a cone Volume of a cone: Exact answers in terms of pi Volume of a sphere Surface area of a cube or a rectangular prism Surface area of a triangular prism Surface area of a cylinder Surface area of a cylinder: Exact answers in terms of pi Surface area of a sphere Word problem involving the Pythagorean Theorem Equations and Inequalities (76 topics) Identifying properties used to solve a linear equation Solving equations with zero, one, or infinitely many solutions Solving a decimal word problem using a linear equation with the variable on both sides Solving a fraction word problem using a linear equation with the variable on both sides Solving a word problem involving consecutive integers Writing a multi step equation for a real world situation Solving a value mixture problem using a linear equation Finding a side length given the perimeter and side lengths with variables Circumference ratios Solving equations involving vertical angles Finding angle measures of a triangle given angles with variables Finding the value for a new score that will yield a given mean Finding the multiplier to give a final amount after a percentage increase or decrease Finding the total cost including tax or markup Finding the original price given the sale price and percent discount Computing a percent mixture Solving a percent mixture problem using a linear equation Finding simple interest without a calculator Converting a repeating decimal to a fraction Solving an absolute value equation: Problem type 3 Solving an absolute value equation: Problem type 4 Solving an absolute value equation of the form ax+b = cx+d Translating a sentence into a one step inequality Translating a sentence into a multi step inequality Writing an inequality for a real world situation Writing an inequality given a graph on the number line Translating a sentence into a compound inequality Writing a compound inequality given a graph on the number line Set builder notation Union and intersection of finite sets Union and intersection of intervals Additive property of inequality with signed fractions Multiplicative property of inequality with signed fractions Solving a two step linear inequality with a fractional coefficient Solving a linear inequality with multiple occurrences of the variable: Problem type 2 Solving a linear inequality with multiple occurrences of the variable: Problem type 3 Solving inequalities with no solution or all real numbers as solutions Copyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 16/25

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