Math 120 Final Review: Coordinate Planes, Functions, Quadratic Functions, and Trigonometry, Study notes of Pre-Calculus

Download Math 120 Final Review: Coordinate Planes, Functions, Quadratic Functions, and Trigonometry and more Study notes Pre-Calculus in PDF only on Docsity! Final Review - Math 120 • Chapter 1, 2, 3, 4 - The Coordinate Plane, Lines and Circles – You should understand the idea of imposing a coordinate system and the use of the distance formula. – You should understand the various forms of the equations of a line, and be able to determine a line given either two points on the line, or the slope and a point on a line. – You should understand the equation of a circle, and be able to find the intersections of lines with lines, and lines with circles. You should be able to do this in the context of a model. See, e.g., problems 3.7, 3.9, 4.8, 4.14 – You should be able to create a linear model for the relationship between two quantities (e.g., population varying with time). If you have two such models, you should be able to answer various questions about the two quantities modeled (e.g., when are they equal? when is one twice as large as the other?) See problems e.g 4.5, 4.9. • Chapters 5, 6 - Functions and Graphs – You should know a function is. You should know what the domain, range and graph of a function are, and, if asked be able to find the domain and range of a particular function. You should know what the vertical line test is all about. – You should be able to graph linear functions, and multipart functions whose parts are linear. – You should understand what a multipart function is. You should be able to describe rela- tionships between two quantities with a multipart function, i.e., you should be able to model with them. See problems like 6.3, 6.4 and 6.6. • Chapter 7 - Quadratic Functions – You should know that quadratic functions are those of the form f(x) = ax2 + bx + c and that these can always be put into vertex form f(x) = a(x− h)2 + k. You should be able to find the vertex of a quadratic function. – You should be able to create quadratic models given three generic points, or the vertex and one other point. See problems like 7.1, 7.3, 7.12. – You should be able to find the maximum or minimum value of a quantity determined by a quadratic function by considering the vertex. I like problems 7.9, 7.10, and 7.11 a lot. • Chapter 8 - Composition – You should know what it means to compose two functions. You should understand what is meant by f(g(x)). You should know that f(g(x)) and g(f(x)) are generally different func- tions. You should be able to write simplified rules for compositions f(g(x)) and g(f(x)) given rules for f(x) and g(x). I like problems 8.2 and 8.4. • Chapter 9 - Three Construction Tools – You should understand horizontal and vertical shifting, and horizontal and vertical scaling (aka dilating) – You should understand how to derive the graph of g(x) = af(bx + c) + d from the graph of f(x) (see, e.g., problem 9.2) – I especially like problem 9.2, 9.3, and 9.7 • Chapter 10 - Arithmetic – This is a very short chapter. An important topic in this chapter is step functions, which are a nice example of multipart functions. – You should understand how to graph functions built up from the unit step function (see problem 10.8) – You should be able to combine multipart functions and come up with the rule for the new function. – I really like problem 10.4 and 10.8. • Chapter 11 - Inverse Functions – Another very short chapter. – You should understand what an inverse function is, what conditions a function must satisfy in order to have an inverse (do all functions have inverses? can you tell if a function has an inverse by looking at its graph?), and how to find the inverse of a given function – You should understand what a one-to-one function is, and what is special about the graph of a one-to-one function – I like problem 11.6, 11.8 and 11.9. • Chapter 12 - Rational Functions – A very important chapter. We spent two days in lecture on this instead of the usual one. – You should be able to find the asymptotes (horizontal and vertical) of a rational function, and be able to sketch the graph of a rational function like those in problem 12.1 – You should be able to model with linear-to-linear rational functions. This comes down to finding a rational function of the form f(x) = ax + b x + c whose graph 1. passes through three given points or 2. has a given asymptote and passes through two given points or 3. has two given asymptotes and passes throuh one given point You will need to translate the language of the modeling problem. Pay particularly close attention to the words “linear-to-linear”. Note that a linear-to-linear function is not a linear function. – I like the problems from the chapter 12 supplement, and problems 12.1, 12.8 and 12.9. • Chapter 13 - Measuring an Angle – You should understand how to convert between degrees and radians – You should understand and be able to use the relationships between radii, angle, arc length and area – I like problems 13.3 and 13.8. • Chapter 14 - Measuring Circular Motion – You should understand the various measures of angular speed (aka angular velocity), like rpm, radians per second, or degrees per hour – You should understand the relationship between radius, angular speed and linear speed