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