# Precalculus Math119

## Week One

### The Arithmetic of Exponents

• Skim Blitzer’s Precalculus 6th Edition, Chapter P, Sections 1–3
• Be able to clean up all kinds of expressions involving exponents
• Practice with these exercises, these exercises, and their solutions
• Practice with these exercises, these exercises, and their solutions
• Practice with these exercises, these exercises, and their solutions
• Key search phrases: rules of exponents, fractional exponents, negative exponents

### The Arithmetic of Polynomials

• Skim Blitzer’s Precalculus 6th Edition, Chapter P, Sections 4–6
• Know what a polynomial and rational expression are
• Be able to multiply, manipulate, and factor polynomials and rational expressions
• Practice with these exercises and their solutions
• Practice with these exercises, these exercises, and their solutions
• Practice with these exercises, these exercises, these exercises, and their solutions
• Key search phrases: roots of polynomials, degree of a polynomial, coefficients of a polynomial, factoring quadratics, multiplying polynomials, rational expressions

## Week Two

### Functions

• Watch What is a Function?
• Skim Blitzer’s Precalculus 6th Edition, Chapter 1, Section 2
• Understand what is (and is not) a function in general
• Understand the notation of real valued functions
• Practice with these exercises and their solutions
• Key search phrases: function, function evaluation, function domain/range, function input/output, piecewise-defined function

### Composite Functions and Function Inverses

• Skim Blitzer’s Precalculus 6th Edition, Chapter 1, Sections 7–8
• Understand what a composite of two functions is
• Be able to write a formula for the sum, product, or composite of two real functions
• Practice with these exercises/exercises/exercises and their solutions/solutions
• Be able to write down the formula for the inverse of an invertible real-valued function
• Practice with these exercises and their solutions
• Know that if a function is not one-to-one, it doesn't have an inverse
• Know that you can restrict the domain of a non-invertible function to create a new function that is invertible
• Key search phrases: composing functions, inverse function

### The Graph of a Function

Late Start registration deadline is February 17, or February 24 with instructor permission.

## Week Three

### Transformations of Function Graphs

• Skim Blitzer’s Precalculus 6th Edition, Chapter 1, Section 6
• Be able to graph any real-valued function as a transformation of some “parent graph”
• Practice with these exercises/exercises/exercises and their solutions/solutions
• Key search phrases: transformations of functions, parent graphs

### Modelling and Miscellany

• Skim Blitzer’s Precalculus 6th Edition, Chapter 1, Sections 9–10
• Given two points, be able to calculate the distance between them and their midpoint
• Practice with these exercises/exercises and their solutions
• Be familiar with the equation of a circle in Cartesian coordinates
• Be able to work with basic linear and quadratic models
• Practice with these exercises/exercises/exercises and their solutions
• Key search phrases: linear modelling, quadratic modelling, distance formula, midpoint formula

### Direct Variation and Inverse Variation

• Skim Blitzer’s Precalculus 6th Edition, Chapter 2, Section 8
• Understand the phrases “directly proportional” and “inversely proportional”
• Practice with these exercises/exercises/exercises and their solutions
• Key search phrases: directly proportional, inversely proportional

## Week Four

### Complex Numbers

• Skim Blitzer’s Precalculus 6th Edition, Chapter 2, Section 2
• Be able to factor and graph any quadratic polynomial function
• Practice with these exercises/exercises and their solutions
• Be able to write any quadratic as a square quadratic plus a constant
• Practice with these exercises
• Stare at this GIF until you understand why they call this “completing the square”
• Practice analysing situations that are modelled by quadratic equations
• Know that there are also equations for the solution to a general cubic
• Know that you can build a solar death ray from a parabolic mirror
• See this, or watch MythBusters Season 4 Episode 7
• Key search phrases: quadratics, quadratic formula, vertex form of a quadratic, completing the square, parabola, parabolic focus

## Week Five

### Graphing Polynomials

• Skim Blitzer’s Precalculus 6th Edition, Chapter 2, Section 3
• Be able to graph any polynomial presented in factored form
• Practice with these exercises/exercises/exercises and their solutions/solutions
• Understand the implications of the Intermediate Value Theorem
• Key search phrases: polynomial degree, multiplicity (of a root), graphing polynomials, Intermediate Value Theorem

### Polynomial Division

• Skim Blitzer’s Precalculus 6th Edition, Chapter 2, Sections 4–5
• Be able to perform polynomial division (either long division or synthetic division)
• Practice with these exercises/exercises/exercises and their solutions
• Gain fluency with vocabulary of polynomial division
• Understand that $$r$$ is a root of a polynomial $$f$$ if and only if $$(x-r)$$ divides $$f(x)$$
• Practice with these exercises and their solutions
• Be passively aware of the Rational Roots Theorem and Descarte’s Rule of Signs
• Key search phrases: polynomial long division, polynomial synthetic division, Remainder Theorem, dividend/divisor/quotient, Rational Roots Theorem, Descarte’s Rule of Signs

### Graphing Rational Functions

• Skim Blitzer’s Precalculus 6th Edition, Chapter 2, Sections 6–7
• Be able to sketch a graph of any “nicely” presented rational function
• Practice with these exercises/exercises/exercises and their solutions/solutions
• Key search phrases: graphing rational functions, asymptote, zeros and poles of rational functions, multiplicity (of a pole), end-behavior of a function

## Week Six

### Exponential Functions

• Skim Blitzer’s Precalculus 6th Edition, Chapter 3, Section 1
• Understand what an exponential function is and how to manipulate exponential expressions algebraically
• Know how to graph elementary transformations of an exponential function
• Practice with these exercises/exercises and their solutions/solutions
• Read An Intuitive Guide To Exponential Functions and $$e$$ by Kalid Azad
• Key search phrases: exponential function, exponential constant e, rules of exponents

### Modelling with Exponential Functions and Logarithms

• Skim Blitzer’s Precalculus 6th Edition, Chapter 3, Section 5
• Understand some applications of exponential functions and situations they model
• Practice with these exercises/exercises/exercises and their solutions
• Know how to model situations using exponential and logarithmic functions
• Practice with these exercises/exercises/exercises/exercises and their solutions
• Be familiar with logistic growth and Newton’s law of cooling
• Practice with these exercises/exercises and their solutions
• Key search phrases: modelling populations, modelling radioactive decay, exponential growth and decay, compound interest, logistic growth, Newton’s law of cooling

## Week Seven

### Compass Directions, Latitude/Longitude, and Global Positioning

• Read the Wikipedia articles on Cardinal Directions and Points of the Compass
• Read the Wikipedia article on the Earth’s Geographic Coordinate System
• Read the Wikipedia article on the many Map Projections of the Earth
• Have a working understanding and knowledge of the vocabulary of the fundamentals of navigation and global positioning
• Key search phrases: heading, bearing, latitude and longitude, cardinal directions, intercardinal directions, half-winds, quarter-winds, map projections

### Linear vs Rotational Speed

• Read the Chapter Measuring Circular Motion
• Model situations involving wheels connected by belts or by common axels
• Practice with these exercises
• Key search phrases: linear and rotational speed

## Week Eight

Exercise-driven review of first-half of the term

Midterm Break

## Week Ten

### Converting Between Polar & Rectangular Coordinates

• Skim Blitzer’s Precalculus 6th Edition, Chapter 4, Sections 1–2
• Understand what the sine, cosine, and arctangent functions are
• Be able to convert between rectangular and polar coordinates
• Be familiar with some of the “nice” input-outputs pairs of sine/cosine/arctangent
• Practice with these exercises and their solutions
• Key search phrases: polar coordinates, sine, cosine, arctangent, unit circle

### Trigonometric Functions

• Skim Blitzer’s Precalculus 6th Edition, Chapter 4, Sections 2–4
• Be able to derive or recall common coordinates of points on the unit circle
• Know how tangent, secant, cosecant, and cotangent relate to sine and cosine
• Understand the symmetry and geometry of the trigonometric functions
• Understand how trig functions can be used to describe general right triangles
• Practice with these exercises/exercises and their solutions
• Practice with these exercises/exercises/exercises and their solutions
• Practice with these exercises/exercises/exercises and their solutions
• Practice with these exercises/exercises/exercises and their solutions
• Key search phrases: unit circle, tangent, secant, cosecant, cotangent, right triangle, reference angle

## Week Eleven

### Inverse Trigonometric Functions

• Skim Blitzer’s Precalculus 6th Edition, Chapter 4, Section 7
• Understand what the inverse trig functions are and know their domains
• Practice with these exercises/exercises/exercises and their solutions
• Key search phrases: inverse trigonometric functions

### The Graphs of Trigonometric Functions and their Inverses

• Skim Blitzer’s Precalculus 6th Edition, Chapter 4, Sections 5–6
• Know the key features of the graphs of the trigonometric functions
• Be able to graph transformations of the sine and cosine functions
• Practice with these exercises/exercises/exercises and their solutions/solutions
• Be able to graph transformations of the (co)secant and (co)tangent functions
• Practice with these exercises/exercises/exercises and their solutions/solutions
• Know what the graphs of the inverse trig functions must look like
• Key search phrases: amplitude, period, graph of sine and cosine

## Week Twelve

Exercise-driven review of trigonometry

## Week Thirteen

### Pythagorean Identities and Symmetry

• Skim Blitzer’s Precalculus 6th Edition, Chapter 5, Section 1
• Know the Pythagorean identities, and even/odd properties of trig functions
• Be able to fluently use identities to rewrite trigonometric expressions
• Practice with these exercises/exercises/exercises
• Key search phrases: Pythagorean identities, even odd trigonometry

### Sum-/Difference-of-Angles & Double-/Half-Angle Formulas

• Skim Blitzer’s Precalculus 6th Edition, Chapter 5, Sections 2–3
• Be able to apply identities to calculate trigonometric outputs exactly
• Practice with these exercises/exercises, and their solutions
• Practice with these exercises/exercises
• Be able to fluently use identities to rewrite trigonometric expressions
• Practice with these exercises/exercises/exercises, and their solutions
• Practice with these exercises/exercises/exercises
• Key search phrases: sum-of-angles formula, difference-of-angles formula, double-angle formula, half-angle formula, power-reduction formula

### Sum-to-Product & Product-to-Sum Formulas

• Skim Blitzer’s Precalculus 6th Edition, Chapter 5, Sections 4–5
• Be able to fluently use identities to rewrite trigonometric expressions
• Practice with these exercises
• Be able to fluently use identities to solve equations involving trigonometric functions
• Practice with these exercises/exercises/exercises, and their solutions
• Practice with these exercises/exercises/exercises, and their solutions
• Practice with these exercises/exercises, and their solutions
• Key search phrases: sum-to-product formulas, product-to-sum formulas, solving trigonometric equations

## Week Fourteen

### Graphing in Polar Coordinates

• Skim Blitzer’s Precalculus 6th Edition, Chapter 6, Section 4
• Be aware of how to graph a function in polar coordinates
• Practice with these exercises/exercises, and their solutions
• Key search phrases: polar graphing

### Complex Exponentials & Hyperbolic Trigonometry (Optional)

• Skim Blitzer’s Precalculus 6th Edition, Chapter 6, Section 5
• Key search phrases: Euler’s formula, DeMoivre’s formula, hyperbolic trig functions

## Week Fifteen

### Vectors

• Skim Blitzer’s Precalculus 6th Edition, Chapter 6, Sections 6–7
• Be fluent in the notation and basic calculations of vectors
• Practice with these exercises/exercises, and their solutions
• Have a hazy idea of what a dot product of vectors is
• Practice with these exercises, and their solutions
• Key search phrases: vectors, dot product

### Systems of Equations

• Skim Blitzer’s Precalculus 6th Edition, Chapter 7, Sections 1–2
• Be able to solve a system of system of equations
• Practice with these exercises, and their solutions
• Practice with these exercises, and their solutions
• Understand the geometric interpretation of the solution to a system of equations
• Key search phrases: solving systems of equations

### Matrices

• Skim Blitzer’s Precalculus 6th Edition, Chapter 8, Section 1
• Know that you can solve systems of linear equations quickly using matrix arithmetic
• Practice with these exercises, and their solutions
• Key search phrases: matrices, linear algebra

## Finals Week

Final Exam Wednesday May 18, 8am–10am