Math 147: PreCalculus

Chapter 1: Fundamentals

Section 1.1: Real Numbers


Set builder and interval notation
Union and intersection of intervals

Section 1.2 Exponents and Radicals


Product rule of exponents
Quotients of expressions involving exponents
Power rule: Positive exponents
Power rule: Negative exponents
Product rule of exponents in a multivariate monomial
Simplifying a radical expression: Problem type 1
Simplifying a sum of radical expressions
Simplifying a product of radical expressions
Rationalizing the denominator of a radical expression
Rational exponents: Basic
Rational exponents: Negative exponents and fractional bases
Rational exponents: Powers of powers

Section 1.3 Algebraic Expressions


Simplifying a polynomial expression
Multiplying monomials
Multiplying polynomials
Factoring a quadratic with leading coefficient 1
Factoring a quadratic with leading coefficient greater than 1
Factoring a product of a quadratic trinomial and a monomial
Factoring a difference of squares
Factoring with repeated use of the difference of squares formula
Factoring a sum or difference of two cubes
Factoring a multivariate polynomial by grouping: Problem type 1

Section 1.4 Rational Expressions


Rationalizing the denominator of a radical expression using conjugates

Section 1.5 Equations


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 an equation involving absolute value: Basic
Solving a rational equation that simplifies to a linear equation: Problem type 1
Solving a rational equation that simplifies to a linear equation: Problem type 2
Solving a rational equation that simplifies to a linear equation: Problem type 3
Solving an equation with radicals: Problem type 1
Solving an equation with radicals: Problem type 3
Solving an equation with radicals: Problem type 4
Finding the roots of a quadratic equation with leading coefficient 1
Finding the roots of a quadratic equation with leading coefficient greater than 1
Completing the square
Solving a quadratic equation using the quadratic formula
Finding the discriminant of a quadratic equation
Solving equations written in factored form
Solving a quadratic equation needing simplification
Solving a word problem using a linear equation: Problem type 3

Section 1.6 Modeling with Equations


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 4
Solving a word problem using a quadratic equation with rational roots
Solving a word problem using a quadratic equation with irrational roots

Section 1.7 Inequalities


Solving a linear inequality: Problem type 2
Solving a linear inequality: Problem type 3
Solving a linear inequality: Problem type 4
Solving an inequality involving absolute value
Solving a quadratic inequality

Section 1.8 Coordinate Geometry


Graphing a parabola: Problem type 1
Midpoint of a line segment in the plane
Distance between two points in the plane

Section 1.10 Lines


Graphing a line given its equation in slope-intercept form
Graphing a vertical or horizontal line
Graphing a line given its equation in standard form
Writing the equation of a line given the slope and a point on the line
Graphing a line through a given point with a given slope
Finding the slope of a line given its equation
Determining the slope of a line given its graph
Writing the equation of the line through two given points
Writing the equations of vertical and horizontal lines through a given point
Slopes of parallel and perpendicular lines: Problem type 1
Slopes of parallel and perpendicular lines: Problem type 2
Writing equations and drawing graphs to fit a narrative

Section 9.9 Systems of Inequalities (This topic shows up earlier in ALEKS than in the book)


Graphing a linear inequality in the plane: Problem type 1
Graphing a linear inequality in the plane: Problem type 2

Chapter 2: Functions

Section 2.1 What is a Function?


Introduction to functions: Notation and graphs

Section 2.2 Graphs of Functions


Domain and range: Problem type 1
Domain and range: Problem type 2
Vertical line test

Section 2.3 Increasing and Decreasing Functions; Average Rate of Change


NOTE: The ideas in this section will come up in some of the first questions in Chapter 3

Section 2.4 Transformations of Functions


Vertical translation of the graph of a function
Vertical and horizontal translations of the graph of a function

Section 2.5 Quadratic Functions; Maxima and Minima


Graphing a parabola: Problem type 2
Graphing a parabola: Problem type 3

Section 2.6 Modeling with Functions


Solving a word problem by finding a local extremum of a polynomial function

Section 2.7 Combining Functions


Sum, difference, and product of two functions
Quotient of two functions
Composition of two functions: Domain and range
Composition of two functions: Basic

Section 2.8 One-to-One Functions and Their Inverses


Horizontal line test
Inverse functions: Basic
Inverse functions: Advanced

Chapter 3: Polynomial and Rational Functions

Section 3.1 Polynomial Functions and Their Graphs


Simplifying a ratio of polynomials: Problem type 1 (NOTE: This is really a problem from Section 1.4)
Inferring properties of a polynomial function from its graph

Solving a word problem involving a polynomial of degree 3 (NOTE: Section 2.7 may be helpful for this, too).

Section 3.2 Dividing Polynomials


Polynomial long division: Linear divisor
Polynomial long division: Quadratic divisor
Synthetic division
Finding a polynomial of a given degree with given zeros

Section 3.3 Real Zeros of Polynomials


Using a given zero to write a polynomial as a product of linear factors

Section 3.4 Complex Numbers


Adding and subtracting complex numbers
Multiplying complex numbers
Dividing complex numbers
Solving a quadratic equation with imaginary roots

Section 3.5 Complex Zeros and the Fundamental Theorem of Algebra


N zeros theorem and conjugate zeros theorem
Using the conjugate zeros theorem to find all zeros of a polynomial

Section 3.6 Graphing Rational Functions


Sketching the graph of a rational function: Problem type 1
Choosing the form of a rational function given its graph

Chapter 4: Exponential and Logarithmic Functions

Section 4.1 Exponential Functions


Sketching the graph of an exponential function: Basic
Sketching the graph of an exponential function: Advanced

Section 4.2 Logarithmic Functions


Exponential and logarithmic equations
Evaluating a logarithmic expression
Sketching the graph of a logarithmic function
Translating the graph of a logarithmic or exponential function (NOTE: This is really a problem from Section 2.4)

Section 4.3 Laws of Logarithms


Basic properties of logarithms
Change of base for logarithms

Section 4.4 Exponential and Logarithmic Equations


Solving a logarithmic equation: Problem type 1
Solving a logarithmic equation: Problem type 2
Solving an exponential equation: Problem type 1
Solving an exponential equation: Problem type 2
Solving a word problem using an exponential equation: Problem type 1
Solving a word problem using an exponential equation: Problem type 2

Section 4.5 Modeling with Exponential and Logarithmic Functions


Solving a word problem using an exponential equation: Problem type 3

Chapter 5: Trigonometric Functions of Real Numbers

Section 5.1 The Unit Circle


NOTE:To understand the following sections, a strong understanding of the Unit Circle is necessary.

Section 5.2 Trigonometric Functions of Real Numbers


Common Angles and Trig Functions (This is also in Sec 6.3)
Finding values of trigonometric functions given information about an angle: Problem type 1 (This is also in Sec 6.3)
Finding values of trigonometric functions given information about an angle: Problem type 2 (This is also in Sec 6.3)
Finding values of trigonometric functions given information about an angle: Problem type 3 (This is also in Sec 6.3)

Section 5.3 Trigonometric Graphs


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

Section 5.4 More Trigonometric Graphs


Sketching the graph of a secant or cosecant function
Sketching the graph of a tangent or cotangent function: Problem type 1
Sketching the graph of a tangent or cotangent function: Problem type 2

Chapter 6: Trigonometric Functions of Angles

Section 6.1 Angle Measure


Converting between degree and radian measure
Coterminal angles
Arc length and central angle measure
Sketching an angle in standard position

Section 6.2 Trigonometry of Right Triangles


Special right triangles
Sine, cosine, and tangent ratios
Using a trigonometric ratio to find a side length in a right triangle
Using a trigonometric ratio to find an angle measure in a right triangle
Finding trigonometric ratios given a right triangle

Section 6.3 Trigonometric Functions of Angles


Common Angles and Trig Functions (This is also in Sec 5.2)
Finding values of trigonometric functions given information about an angle: Problem type 1 (This is also in Sec 5.2)
Finding values of trigonometric functions given information about an angle: Problem type 2 (This is also in Sec 5.2)
Finding values of trigonometric functions given information about an angle: Problem type 3 (This is also in Sec 5.2)

Section 6.4 The Law of Sines


Solving a triangle with the law of sines: Problem type 1
Solving a triangle with the law of sines: Problem type 2

Section 6.5 The Law of Cosines


Solving a triangle with the law of cosines

Chapter 7: Analytic Trigonometry

Section 7.1 Trigonometric Identities


Cofunction identities
Verifying a trigonometric identity
Proving a trigonometric identity: Problem type 1
Proving trigonometric identities: Problem type 2

Proving trigonometric identities: Problem type 3

Section 7.2 Addition and Subtraction Formulas


Sum and difference identities
Proving trigonometric identities: Problem type 4

Section 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas


Product-to-sum and sum-to-product identities
Double-angle identities

Section 7.4 Inverse Trigonometric 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 3

Section 7.5 Trigonometric Equations


Solving a basic trigonometric equation involving sine or cosine
Solving a basic trigonometric equation involving tangent, cotangent, secant, or cosecant
Solving a trigonometric equation involving a squared function
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
Solving a trigonometric inequality (This is not covered in detail in the book)

Chapter 8: Polar Coordinates and Vectors

Section 8.1 Polar Coordinates


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

Section 8.2 Graphs of Polar Equations


NOTE: The graphs of polar equations can be very interesting, not to mention beautiful.

Section 8.3 Polar Form of Complex Numbers; DeMoivre’s Theorem


Writing a complex number in trigonometric form
De Moivre's theorem
Finding the nth roots of a number


Last updated September 5, 2007 by William P. Clement.
© William P. Clement 2007