The City College of New YorkCCNY
Department of Mathematics
Division of Science

Math 190 Topic Summaries

P.2 Real Numbers.

Make sure you understand Examples 1, 2, 3, 5, and 6b). It is not necessary to know the names of the properties. You should spend only a little time studying open and closed interval notation. Interval notation is important but you will have many opportunities to practice it throughout the semester. Make sure you review order of operations or PEMDAS and review adding and subtracting fractions by finding a LCD. Surprisingly it is easier to multiply and divide fractions than it is to add and subtract fractions, since you do not need a common denominator to multiply or divide fractions.

Textbook Examples: 1-3, 5-8

Textbook Exercises: 29-38, 47-61, 63, 67, 68.

Adding Integers

Subtracting Integers

Multiply and Divide Integers

Properties of Reals

Ex1: Simplifying Fraction

Ex2: Simplifying Fraction

Multiplying Fractions

Dividing Fractions

Ex 1: Simplify a Complex Fraction

Interval Notation

More Interval Notation

Evaluating Basic Absolute Values

Evaluate Absolute Value Expressions: |-a-b|,|-a|-|b|

Simplify Absolute Value Expressions

P.3 Integer Exponents:

Read through this section to make sure you understand all the Laws of Exponents and can do the included Examples 1-5. You may skip the Examples 6-8 using calculators and scientific notation. We will not be using calculators or scientific notation in this course.

Textbook Examples: 1-5

Textbook Exercises: 1-7, 9-32

Properties of Exponents

Negative Exponents

The Zero Exponent

Ex: Simplify Exponential Expressions - Product and Power Rule

Ex: Simplify Exponential Expressions - Quotient Rule

P.4 Exponents and Radicals:

Read through this section to make sure you understand how to use exponential notation and radical notation as in Examples 1, 3-5. Skip "Rationalizing the Denominator". You may need to spend a little extra time studying rational exponents since it may be new to some students.

Exponent notation is often preferable to radical notation because in exponential notation you can use the laws of exponents. Many exercises challenge you to rewrite radical notation in exponential notation and then apply one of the laws of exponents.

In general the webassign exercises are a bit misleading in that after doing the exercises you may think that most roots (rational exponents) simplify like the cube root of -125 simplifies to -5. This is not so. Most roots (rational exponents) do not simplify. For instance the cube root of -124 or the cube root of -126 do not simplify further. They are perfectly good numbers that are already simplified and may arise as a solution to a problem in math, science, or engineering. You must become comfortable working with these numbers.

Textbook Examples: All 1-6

Textbook Exercises: 7-58, 63, 64

Introduction to Radicals

Adding and Subtracting Radicals

Introduction to Multiplying Radicals

Multiplying Radicals Containing Variables

Rational Exponents

Ex: Evaluate an Expression with Rational Exponents Using Radicals

Ex: Simplify an Expression with Rational Exponents and Write in Radical Form

P.5 Algebraic Expressions.

Adding and subtracting polynomial, combining like terms to simplify, and expanding polynomials using multiplication. Often in section P.5, you will be asked to expand by distributing or using FOIL and then combining like terms to simplify. You do not often want to expand a factored expression unless you have a good reason to do so. One of the goals of middle school math is to factor integers. One of the goals of high school math is to factor polynomials. You need not memorize the Special Product Formulas by name. You should however be able to expand and simplify any of the products yourself.

Textbook Examples: ALL 1-5

Textbook Exercises: 1-78

Introduction to Variables and Variable Expressions

Introduction to Polynomials

Adding and Subtracting Polynomials

Multiplying Polynomials Review

Special Polynomial Products

P.6 Factoring.

One of the goals of middle school math is to factor integers. One of the goals of high school math is to factor polynomials. In both cases you factor a complicated thing into simpler parts. Scientists do the same thing but they do not call it factoring. Biologist factor organisms into cells because the cells are easier to understand than the whole organism. Physicists factor atoms into protons, neutrons, and electrons.... Factoring can be hard work. Therefore you do not often want to expand a factored object unless you have a good reason to do so. Make sure you understand the Examples in this section but you may skip the Examples on "factoring the sum and differences of cubes", "factoring with fractional exponents", and "factoring with more than one variable".

There is no need to memorize all of the Special Factoring Formulas. You only need to memorize the first three: difference of squares and perfect square. You can skip difference and sum of cubes.

Textbook Examples: 1-7, 9, 11

Textbook Exercises: 1-36, 47-52, 61-78, 85-88.

Factoring Out The Greatest Common Factor

Factor Trinomials When A equals 1

Factoring Trinomials: Trial and Error and Grouping

Factor By Grouping

Factoring a Perfect Square Trinomial

P.7 Rational Expressions.

A rational expression is a fraction of polynomials. Luckily there is not much new here. All you do is mix together the basic algebraic techniques for fractions from middle school (and from section 1.1) with the factoring and reducing techniques for polynomials from section 1.3. Just like in middle school you need to be able to reduce, add (find the LCD), subtract (find the LCD), multiply, and divide rational expressions. One of the keys to many exam problems is to factor and reduce as much as possible before adding, subtracting, multiplying and dividing. You should work through the Examples in this section. There is a nice table on "Avoiding Common Errors" at the end of this section. You should read this table of common errors multiple times and try not to make any of these errors.

Textbook Examples: ALL 1-10

Textbook Exercises: 1-64, 87, 88, 93, 94.

Reducing Rational Expressions to Lowest Terms Video

Multiplying and Dividing Rational Expressions Video

Adding and Subtracting Rational Expressions

Simplify a Complex Fraction

P.8 Equations.

One technique often used to solve linear equations is to isolate the variable on one side (it does not matter what side). In order to isolate the variable correctly you must recognize when the equation is linear. In section 1.4 you will study quadratic equations when isolating the variable is not usually a good idea. For many quadratic equations it is better to set the equation to zero. Power equations can often be solved by taking the appropriate root of both sides. You must be careful when taking roots because even roots require a different technique than taking odd roots.

Textbook Examples: ALL 1-8.

Textbook Exercises: 1-78, 87-97

Understanding Expressions and Equations

The difference between an Expressions and an Equations

Introduction to Algebraic Equations

Solving One Step Equations: The Basic

Solve One Step Equations with Fractions

Solving Two Step Equations: The Basics

Solve an Equation with Variable on Both Sides

Solving One Step Literal Equations

Solving Literal Equations - Part 1

Solving Literal Equations - Part 2

Example A: Solving a Formula for a Variable

Example B: Solving a Formula for a Variable

1.1 Coordinate Geometry and 1.2 Graphs of Equations in Two Variables

You should memorize the distance and midpoint formulas. You should understand that the distance formula is derived from the Pythagorean Theorem. Do a couple problems from section 1.1 practicing the midpoint and distance formulas. Do enough examples so that you are comfortable.

Then move to section 1.2. Most of the important material is in section 1.2. You should spend a lot (lots and lots) of time graphing equations by making tables and plotting points. Later in the course you will learn faster ways to trace some graphs. The trouble with these faster graphing methods is that they do not always work. Sketching by making a table always works! You should learn it well. Much later, after you learn some calculus, you will sketch by using a mixture of these faster techniques, some calculus techniques, as well as making a table of points and plotting points. You will always be making tables. Making tables is boring, but it is essential to understanding. To make it through calculus, you must be able to quickly make a rough sketch of many basic graphs with labeling some points on the graphs.

Textbook Examples: ALL in section 1.1 and Examples 1-5 in section 1.2.

You may skip the part of section 1.2 about symmetry.

Textbook Exercises 1.1: 9-12, 21-30, 35-37

Textbook Exercises 1.2: 1-40.

Introduction to the Cartesian Plane - Part 1

Introduction to the Cartesian Plane - Part 2

Plotting Points on the Coordinate Plane

Determine the Ordered Pairs for Points Plotted on the Coordinate Plane

Plot Points Given as Ordered Pairs on the Coordinate Plane

The Distance Formula

Example: Determine the Distance Between Two Points

Ex: Midpoint of a Segment

Ex: Find the Endpoint of a Segment Given the Midpoint and One Endpoint

1.3 Circles

Work through all the Examples in this section. Make sure you understand the standard form of the circle and how to convert any equation of a circle to the standard form using completing the square. The standard form of a circle tells you its geometry: its center and its radius. It is remarkable that algebra used in completing the square is related to the center and radius of a circle.

Textbook Examples: ALL

Textbook Exercises: 1-42

Conic Sections: The Circle

Ex: Write the Standard Form of a Circle From a Graph

Ex: Graph a Circle in Standard Form

Ex: Find Standard Equation of a Circle Given Center and Point on the Circle

Ex: Find Standard Equation of a Circle Given the Endpoints of a Diameter

1.4 Lines

This is truly a review section. Every student has studied lines in high school and middle school. Nevertheless, you should spend some time doing many problems in this section. You need to understand lines well. One of the key ideas of calculus is to approximate curves by lines. It makes sense to do this because lines are easy and curves can be very complicated. However you will never be able to approximate complicated curves by lines if you do not first understand lines well.

In calculus the point-slope form is often used since in calculus you often end up with one point (not necessarily the y-intercept) of a line and its slope. You should become comfortable using point-slope form now. Students sometimes are uncomfortable using point-slope form because each line has an infinite number of point-slope forms. This is because there are an infinite number of points on a line and each point will give a different point-slope form. This is a good thing. Be confident with your point-slope form of a line even if it looks different from other students' point-slope form. Any correct point-slope form is acceptable for an answer. Don't forget: Although each line has an infinite number of point-slope forms, each line has only one slope--except for vertical lines whose slope is undefined. If you find the wrong slope you're equation is sure to be incorrect.

You may skip EX 10 "Graphing a Family of Lines". You should work through the other exercises.

Textbook Examples: 1-9, 11

Textbook Exercises: 1-52, 57-78

Slope and y-intercept form equation Video

Intro to Point-Slope Form Video

Point-Slope and Slope-Intercept Equations Video

Point-Slope Form Exercises

Graphing a Linear Equation Video

Graph From Linear Standard Form Exercises

Writing Linear Equations in All Forms Video

Parallel Lines From Equation Video

Writing Equations of Perpendicular Lines

1.5 Solving Quadratic Equations

This is a review of equation solving techniques from high school. You should already know the quadratic formula, the zero product property, and cross multiplying for solving rational equations. You may not yet be familiar with completing the square and solving equations involving a radical. When learning and reviewing the equation solving techniques in this section do not get lost in the techniques. The adding, subtracting, multiplying, dividing, reducing, and power techniques learned in previous sections are all helpful here but remember: you are searching for a number that you can plug in for x so that the equation is true. If you can guess a solution go ahead. You don't need any of the fancy techniques. Just make sure that you know that you can check any solution by substituting the solution for x in the original equation.

Do not get carried away in substituting your solution for x to check your answer. Many times students have gotten the correct solution to an equation and then made an error while checking the solution and convinced themselves that the correct answer was incorrect. Whether or not you decide to check your solutions to equations is a personal matter. You need to practice solving many equations to decide when it is a good idea to check your answer. Sometimes I do check sometimes I don't.

When solving a quadratic equation it is usually a good idea to first set one side of the equation equal to zero and then try for a couple of minutes to factor the quadratic by using trial and error. If your trial and error method does not work after a few minutes, use the quadratic formula. The quadratic formula will always work. The quadratic formula will even tell you when there are no solutions. Complex solutions are not accepted in this course. Sometimes webassign tells you that you must use such and such technique (factoring, quadratic formula, or completing the square) to solve a particular problem. This is good for homework and classwork problems. On the quizzes and exams you may use any technique you choose. You can even try to guess the correct solution---though it is not recommended to guess every solution on the test.

Most students do not use the completing the square method on quizzes and exams. Most students prefer to use the quadratic formula or trial and error factoring on exams. It is important know that any method will work in theory. Completing the square is a good method for math majors who want to know why the quadratic formula works without just memorizing it. Before an exam everyone should memorize the quadratic formula. A good exercise to do is to solve x^2 - 4x - 5 = 0 using all three methods: (1) trial and error by saying "two numbers that multiply to -5 and add to -4, (2) the quadratic formula, and (3) completing the square. You will get the same answers using each method.

Textbook Examples: 1-4

Textbook Exercises: 1-46, 57-62.

Solving Quadratics by Factoring Video

Quadratics by Factoring Exercises

Solving Quadratics by Taking Square Roots Video

Solve Quadratics by Taking Square Roots Exercises

Worked Example: Quadratic Formula Video

Quadratic Formula Exercises

Using the Quadratic Formula: Number of Solutions Video

Worked Example: Rewriting and Solving Equations by Completing the Square Video

1.7 Other Equations

The methods in section 1.6 are of secondary importance in this course. Do not spend much time on them. Focus your efforts on understanding section 1.5 first. You will use the techniques from section 1.6 again in math 195 next semester so don't worry if you cannot master all the techniques in 1.6. However after you are confident solving the equations from 1.5 go ahead and try to solve the exercises in 1.6. The techniques in 1.6 are not so difficult: (1) factoring with powers higher than 2, (3) cross multiplying to solve rational equations, and (3) squaring both sides to solve an equation with a root. You may skip "Equations of Quadratic Type". Have fun working out many of the exercises. Some are easy; some are challenging.

Textbook Examples: 1-4

Textbook Exercises: 5-9, 25-30, 33-40

Rational Equations Intro Video

Rational Equations Intro Exercises

Equations with Rational Expressions Video

Square-Root Equations Intro Video

Square-Root Equations Intro Exercises

Square-Root Equations: One Solution Video

1.8 Solving Inequalities

Here you will solve inequalities. You can use many of the same techniques to solve inequalities that you used to solve equations. One technique that you absolutely cannot use is you cannot multiply or divide both sides of an equality by a negative number. If you do multiply both sides of an inequality by a negative number you must remember to flip the inequality symbol. You can however add and subtract both sides by the same number without changing anything. Most inequalities have an infinite number of solutions so you cannot write them all in order. You must use interval notation or graph notation to write your answers. This is why you learned interval notation in section P.2.

For linear inequalities you want to isolate x, just as you did for linear equations. Make sure the flip the inequality sign every time you multiply or divide both sides of an inequality by a negative number while isolating x.

For nonlinear inequalities you do not want to isolate x. Instead, like solving nonlinear equations, you want to set one side of the inequality to zero. Then factor the other side or reduce it to a simplified and factored fraction. Then plug in test points to check which interval determined by your factorization is part of the solution. In this method plugging in test points is not an optional part of checking your solution. Plugging in test points is a crucial part of the technique. You must do it.

In order to complete the problems in this section successfully you must be able to recognize the difference between linear and nonlinear inequalities. In fact one of the first things you should do when studying an exam question is to ask yourself, "Is this linear?"

You may skip the parts of this section on Absolute Value Inequalities and Modeling with Inequalities. You need ONLY understand EX 1--EX 4.

Textbook Examples: 1-4

Textbook Exercises: 5-24, 35-48

Testing Solutions to Inequalities Video

Two-Step inequalities Video

Two-Step Inequalities Exercises

Inequalities with Variables on Both Sides Video

Quadratic Inequalities Video

2.1 Functions

In middle school you used numbers to describe the world around us. You used numbers to measure distances (D) and weigh (W) things. In calculus the weight of some substance may change with time giving a function W(t). In middle school you always needed new numbers (first fractions and then irrationals) to describe the world because the world is a complicated place. Using functions to describe the world is no different. We always need new functions because the world is a complicated place. One of the main goals of precalculus and college algebra is to become comfortable with as many functions as possible before starting a calculus course. Calculus studies functions in a more sophisticated way. If you are not comfortable with the basic function techniques introduced in college algebra and precalculus you will struggle in calculus.

So where does one begin to study functions when there are so many of them? One answer is that we will divide functions into families: the polynomials (Chapter 3), the rational functions (Chapter 3), the algebraic functions (Chapter 2), the exponential functions (Chapter 4), and the trigonometric functions (Chapter 5). We will study the formulas of functions from each family in this course. By the end of the course you should be able to recognize the family from the formula.

Unfortunately these families are still not enough. Most phenomena studied with functions in science and engineering are more complicated and do not fit into any of the families. What do you do in that case? Calculus has a clever answer to this problem. In calculus you will learn how to approximate the complicated functions you will study in advanced science and engineering with the simple families we study here in college algebra and then in precalculus. It is for this reason that you must be comfortable with as many families of functions as possible.

One great way to expand the amount of functions that you are comfortable working with is to learn inverse functions (section 2.8). In some sense learning inverse functions doubles the amount of functions you know. Thats all there is to it. Learn about inverse functions and then you "double" the functions you know to describe the world we live in. Most inverse functions are just called "inverse functions". Yet some inverse functions are so special that they get their own name: inverse exponential functions are called "logarithmic functions". Inverse trig functions are called "inverse trig functions" or "arc..." functions. It can get confusing.

And I am getting too far ahead. Please start reading section 2.1. Do not fuss too much about the definition of a function. You already know this definition, more or less. Instead start doing the webassign problems on domains, piecewise functions, evaluating functions, and making tables. This is most important. I will never ask you to state the definition of a function on a final exam.

Textbook Examples: Read each one.

Textbook Exercises: 1-8, 17-65

Evaluating Functions Video

Evaluate Functions Exercises

Equations vs. Functions Video

Determining Whether Values are in Domain of Function Video

Identifying Values in Domain Exercises

Examples Finding the Domain of Functions Video

Determining the Domain of Functions Exercises

What is the Range of a Function

2.2 Graphs of Functions

Every college algebra student must know how to graph functions. A very common final exam question asks a student to sketch a graph of function given by a formula. There are probably five or more questions like this on each final exam? You need to know how to do it. The most important and most basic way to graph a function is to first make a table of values like EX 1 "Graphing Functions by Plotting Points". You should know this example well. Unfortunately humans are not so good at making tables; computers are better. Humans make simple errors and are slow. Later (in section 2.6) we will learn a faster way to graph functions for humans.

Yet the faster way is not necessarily a better way. The faster way relies on recognizing the formula from one of our families of functions. Although one can often recognize the family and shape of the basic problems on precalculus exams, it is rarely possible to do this when graphing in the real world. In those cases we must make a table and plot points. Though it is slow and error prone, making a table and then plotting points always works. Eventually when you get to a science or engineering lab you will have a computer sketch graphs for you by plotting points. This may ultimately be the best method? Nevertheless it is a useless method until you have sketched many graphs by plotting points yourself, without a computer. If you do not put in the work and sketch many graphs on your own, you will never know what the computer is doing when it makes a sketch for you.

You should skip the formal definition of a graph with set builder notation on page 159. Instead work through EX 1 on page 160. You should skip the subsection "Graphing Functions with a Graphing Calculator" and EX 2 and Ex 3. You should spend some time to understand EX 4 and EX 5 on p. 162. These are important graphs that you should understand. You should skip EX 6 on the Greatest Integer Function and EX 7 on the Cost Function.

The best definition of a function in precalculus is this: Functions have graphs that pass the vertical line test. You can better understand the vertical line test by doing EX 8 and EX 9. Finally you should memorize the shapes (except fo the Greatest Integer Function) on p. 166. In order to graph many precalculus functions fast (without making a big table) you must memorize basic shapes.

Textbook Examples: 1, 2, 4, 5, 9. You should memorize the formulas and their shapes in the box on p. 208.

Textbook Exercises: 2-32, 37-46, 49-56

Worked Example: Graphing Piecewise Functions Video

2.3 Getting Information From the Graph

This is an easy section. There is a lot of important information you can get from studying the shape of the graph of a function, like the domain and range of the graph, where the function is increasing and decreasing, the maximum and minimum values of a function. You should be able to figure out most of these ideas on your own. They follow from common sense. The hard part is figuring out how to write your answers, mostly using interval notation. We are often interested in the y-values (are they maximums, minimums, increasing, or decreasing). Strangely we answer these questions about the y-values using x-values. It takes some getting used to. You need to practice many problems in order to really get it. Luckily in calculus you will study graphs in the same way.

One particularly confusing aspect of graphs is the interval of increase and decrease. The interval of increase and decrease is related to when the y-values of the graph of the function are increasing or decreasing. However the intervals of increase and decrease are on the x-axis. Get it straight. It is easy to get it straight when looking at the graph but you can get nervous on the exam so practice a lot before the exam. There is also some confusion about the maximum and minimum value. The maximum and minimum value of a function it its maximum or minimum y-value. Yet often we say the maximum occurs when x = 2/7, when we really mean the maximum y-value occurs when x = 2/7. This way of talking about maximum and minimum values happens all the time in calculus. It is a little odd but you'll get used to it.

In calculus you will learn how to find these same important values without looking at the graph of the function. This is harder. Calculus is harder because you are forced to answer questions about the shape of a graph without the graph itself. You will have to wait until you take calculus before you understand how to find when a function is increasing and decreasing without looking at its graph. You may skip the subsection "Comparing Function Values: Solving Equations and Inequalities Graphically" and EX 3 and EX 4. You may also skip EX 9.

Textbook Examples: 1, 2

Textbook Exercises for practice: 1-5, 7-22, 37-40, 51-54

Worked Example: Domain and Range from Graph Video

Domain and Range from Graph Exercises

Intro to Minimum and Maximum Points Video

Worked Example: Absolute and Relative Extrema Video

Absolute Maxima and Minima Exercises

Worked Example: Domain and Range From Graph Video

Domain and Range from Graph Exercises

2.4 Average Rate of Change

The two fundamental ideas in calculus, the derivative and the integral, are respectively related to the average rate of change and the total change. We will not study the derivative and integral in this class but you should know how to compute (1) the average rate of change and (2) the total change of a function. If you already know these concepts, learning calculus will be easier.

The problems in this section often appear on exams because the algebra techniques used to simplify net change and the average rate of change are used all the time in calculus. You should practice these problems repeatedly throughout the semester.

Textbook Examples: 1-4

Textbook Exercises: 1-32

Introduction to Average Rate of Change Video

Worked Example: Average Rate of Change from Graph Video

Worked Example: Average Rate of Change from Table Video

Average Rate of Change: Graphs and Tables Exercises

2.6 Transformation of Functions

Here finally we meet a faster way to sketch some functions. Rather than making a table, plotting points, and then connecting the points, we will use geometric transformations to graph functions whenever possibles. When first learning the transformation method to sketch graphs it seems wildly improbable that the method is useful. Yet useful it is. Many basic graphs that show up on precalculus exams can be sketched much faster using the transformation method than by making a table. Indeed if you continue to sketch graphs by making tables, you will most likely run out of time while working on an exam.

The transformation method to sketch graphs boils down to this: Memorize some basic graph shapes (like the shapes on p.166) and then learn how to shift, stretch, and reflect these graphs into new graphs. This is a clever way to quickly expand your ability to sketch basic shapes. All you need to do is memorize a few basic shapes and the learn how to transform them. Since humans have limited brain capacity and limited computational abilities but have strong geometrical intuition , this is a useful method for humans. It only requires brain space to memorize a few shapes and then learn how to transform these shapes. This method is a little like the way you learned to multiply numbers in 4th grade. You first memorized how to multiply numbers up to twelve and then learned the multiplication algorithm to multiply any two numbers.

The trouble with the transformation method to sketch graphs, unlike the multiplication algorithm, is that the transformation method often does not work, though it does often work on college algebra final exams. The graphs that show up in science and engineering problems are usually much more complicated. In these cases we have a computer make a table and plot points. Unlike humans, computers can make tables quickly and then flawlessly plot the points. At some point after this class you will learn how to program a computer to sketch graphs for you. Yet even after you learn to program a computer sketch for you, you will still use the transformation method. The transformation method is helpful theoretically and also to visualize basic waves and vibrations that arise when you study differential equations.

When you are asked to graph a function on an exam, first try to decide if its graph is a transformed image of a basic shape you previously memorized, like y = x^2 or y = sqrt(x), or y = |x|. If you can recognize the formula as a shape you've previously memorized, use the transformation method to graph the function. It will be the fastest and best method. If after two minutes you cannot recognize the formula as a basic shape you've previously memorized, start making a table. Making a table is long and error-prone but it will always work. If you have to make a table for each exam question you will probably run out of time. Hopefully you only have to use the table method once or twice on an exam. Warning: even if you use the transformation method to sketch a graph you should still make a very small table of points. The transformation method forces you to pick the important points of the graph when you make your table.

Textbook Examples: 1-6

Textbook Exercises: 1-74

Shifting Functions Examples Video

Graphing Shifted Functions Video

Scaling Functions Introduction Video

Scaling Functions Vertically: Examples Video

Scale Functions Vertically Exercises

Shift Functions Exercises

2.7 Composing Functions

You should be able to go through this section quickly. Do not spend much time on "Sums, Differences, Products, and Quotients" of functions. Instead spend some time doing some problems related to composition of functions (like Examples 3 and 4), which is review material from high school math. You may skip Examples 5 "A Composition of Three Functions", 6 "Recognizing Composition of Functions", and 7 "An Application".

Do enough problems until you are confident that you remember how to compose function. Like section 2.4 the algebraic techniques used to simplify composed functions often appear on exams in calculus class.

Textbook Examples: 3, 4, 5, 6

Textbook Exercises: 27-72

Intro to Composing Functions Video

Evaluating Composite Functions Video

Evaluate Composite Functions Exercises

Evaluating Composite Functions: Using Tables Video

Evaluating Composite Functions: Using Graphs Video

Evaluate Composite Functions: Graphs and Tables Exercises

2.8 Inverse Functions

This is an important section. You will have to spend a lot of time on it. Learning about inverse functions is a great way to expand you library of functions. After this section you will "almost" double the number of functions that you know. For each function f(x) you know you will get an inverse function, more or less. Since one of the goals of college algebra is to learn as many functions as possible, learning about inverse functions is helpful. In fact, even if you do not want to double your library of functions, you still need to learn about inverse functions. Inverse functions arise naturally in applications, even students who do not know much math will ask about inverse functions without knowing it.

For example, the simple interest equation is I(t) = Prt when I(t) is the interest you receive for investing a principal amount P with rate r for time t. So if you invest $1000 at a 5% rate for 3 years, the interest you receive is I(3) = (1000)(.05)(3) = $150. Great you make $150 after three years. This is how we have though of functions. Simply plug in the domain value t=3 and out comes the range value I(3) = $150. But what if this is not the way you want to think of the relationship between I and t. What if you want I to be the domain and t to be the range? Suppose you do not ask "how much interest will I make after 3 years?" but ask instead "after how much time will I make $95 in interest?" Both questions are totally natural, even to someone who knows nothing about inverse functions. But the second question is an inverse question to the function I(t). In the second question I = $95 is the domain and t is the range. Inverse questions arise naturally in science and engineering all the time. You must learn about inverses.

n this course we have four ways to think about functions: 1. Verbally, 2. as a Formula, 3. as a Table, and 4. as a Graph. You should be able to think of inverse functions in these four ways too. I just explained how to think of inverse functions verbally in the last paragraph. It is usually impossible to find a formula for an inverse function but there is a method you will learn in this section that sometimes works---and it shows up on final exams. Finding a table for an inverse is easy: Just switch the x (domain) column of your table with the y (range) column. That is all there is to it. The graph of an inverse function is the mirror reflection of the function over the line y = x.

There is one slight problem with all this. We know that the graph of a function passes the vertical line test. This is our definition of a function. It sometimes happens that after mirror reflecting a graph about the line y = x the new shape does not pass the vertical line test. Then the function does not have an inverse. All functions must pass the vertical line test. So we want to know when an inverse will pass the vertical line test and become an inverse function. The answer is that only functions that are "one to one" have an inverse function. A "one to one" function is a function that passes the horizontal line test. This is almost the end of the inverse story. It may happen that the function y = f(x) you are studying does not pass the horizontal line test (it is not "one to one") so it does not have an inverse but you still need the inverse. In that case you must restrict the domain of y = f(x) so that it passes the horizontal line test. Then it will have an inverse. Restricting the domain is a messy part of the subject but you must learn it.

Textbook Examples: 1-10

Textbook Exercises: 1-74

Finding Inverses of Rational Functions Video

Finding Inverses of Rational Functions Exercises

Restrict Domains of Functions to Make Them Invertible Exercises

Reading Inverse Values from a Graph Video

Reading Inverse Values from a Table Video

Inverse Functions: Graphs and Tables Exercises

Using Specific Values to Test for Inverses Video

Verifying Inverses by Composition Video

Verify Inverse Functions by Composition: Not Inverse Video

Verify Inverse Functions Exercises

3.1 Quadratic Functions

All students in this course should be able to recognize a quadratic function y = f(x) and how to graph a quadratic function. The graph of a quadratic is a parabola. Some parabolas point up; some point down. Some parabolas have two x-intercepts; some have one x-intercept, some have zero x-intercepts---just like some quadratic equations have two solutions, some have one solution, and some have none. Finding x-intercepts of quadratic functions and solving quadratic equations are the same thing. You can find the quadratics x-intercepts by solving the quadratic equation f(x) = 0 by factoring or using the quadratic formula. One way to sketch the graph of a quadratic function is to use completing the square to put the quadratic into standard form. If you hate completing the square you can just memorize the formula for the x-value of the vertex, x = -b/(2a), and then make a small table by choosing one x-value greater than and one x-value less than the x-value at the vertex. Most students memorize x = -b/2a rather than use completing the square on exams.

After you know where the vertex is, it is easy to determine the maximum or minimum of the quadratic function as well as when the function is increasing and decreasing.

You may skip the subsection on "Modeling with Quadratic Functions" and Examples 5 and 6.

Textbook Examples: All

Textbook Exercises: 1-44, 51-54

Vertex Form Introduction Video

Graphing Quadratics: Vertex Form Video

Graph Quadratics in Vertex Form Exercises

Finding the Vertex of a Parabola in Standard Form Video

Graphing Quadratics: Standard Form Video

Graphing Quadratics in Standard Form Exercises

Comparing Maximum Points of Quadratic Functions Video

4.1 Exponential Functions

We have now studied polynomial, algebraic, and rational families of functions. In Chapter 4 you will learn about a new and important family of functions: the exponential functions and their inverse functions, the log functions. You absolutely need to understand this family well. These functions arise frequently as solutions to basic science and engineering problems. Since you do not know how to graph a basic exponential function, you should make a table to graph your first couple exponential functions. Making a table of a basic exponential graph, y = a^x, will illustrate the concept of a horizontal asymptote.

A table is the foundation of any graph. Once you graph one exponential function like f(x) = 2^x and then g(x) = (1/3)^x, you do not want to continue to make tables for every exponential function that you need to graph. You want to transform the basic exponential graphs as you transformed basic graphs in section 2.6 so you do not have to make many tables. You will be able to graph faster this way.

The most important fact about the graph of a basic exponential function is that it has the x-axis as a horizontal asymptote. The x-axis is not part of the graph of y = 2^x. It serves as a backbone of the graph. All exponential graphs will have a horizontal asymptote. You cannot graph an exponential functions without finding its horizontal asymptote. The graph of each basic exponential function y = a^x passes through the point (0,1) and has the x-axis as a horizontal asymptote. Do not forget this. It then becomes easy to sketch the graph of transformed exponential functions like y = 2^x + 5. The first thing you do is move the horizontal asymptote, the x-axis, up 5 units. So the horizontal asymptote of y = 2^x + 5 is then the horizontal line y = 5.

You may skip the subsection on "Compound Interest" as well as Examples 5, 6, and 7. You will not be tested on this material.

Textbook Examples: 2-4

Textbook Exercises: 1-3,11-40

Transforming Exponential Graphs Video

Transforming Exponential Graphs (example 2) Video

Graphing Exponential Functions Video

Graphs of Exponential Functions Exercises

4.3 Log Functions

In this section you will utilize the full power of the inverse method you studied in section 2.8. Until this section you have not gained much by learning the inverse method. Each inverse function that you found so far has been a function that we already knew, an known algebraic function or a rational function. One of the goals of precalculus is to learn as many functions as possible. The inverse method is most powerful when it gives you new functions, functions that you did not already know. We then expand our library of functions. The inverse method explained in section 2.8 only works to your advantage if you memorize little to nothing about the inverse function. All the information you get from the inverse function should derive from section 2.8. The brain space you will save by not memorizing specific facts about the inverse function can be used to learn something else, maybe more math...

Ask yourself this: Why do we memorize a multiplication table in grade school but do not memorize a division table? The answer is that division is the inverse of multiplication. It is thus unnecessary to memorize a division table. Any division problem like 35/7 and converted to a multiplication problem 7x = 35. Do not waste your brain cells memorizing a division table.

Exponential functions are one-to-one (they pass the horizontal line test) so each exponential function has an inverse. The inverse exponential function is so important that it is given a special name, the log function. If you need to compute the value of a log function, first convert to exponential form (this is how you first learned to divide---you converted division into multiplication).

We will not graph log functions in this class.

You may skip Examples 4, 5, 6, 7, 8, 9, 10, 11 in this section.

Textbook Examples: 1-3. Skip graphing log functions.

Textbook Exercises: 9-14, 17-22, 25-31, 35, 36.

Intro to Logs Video

Evaluate Logarithms Exercises

Evaluating Logarithms (Advanced) Video

Evaluating Logarithms (Advanced) Exercises

Relationship between Exponentials and Logarithms Video

CHAPTER 5: Trig Functions: A Right Triangle Approach

In this chapter you begin your study of your final family of functions, the trig functions that arise in studying periodic (repetitive) behavior. Of course this is all routine by now (we do the same things every time we learn a new family of functions): 1. make a table to graph a couple (sin, cos) basic trig graphs, 2. then use transformations from section 2.6 to realize many more trig graphs, 3. finally study inverse trig functions. This would be fast if only it was easy to make a table for basic trig functions. Unfortunately it is not. In order to compute a table of trig values, you need to do geometry (luckily only basic high school geometry of triangles and circles is needed). So far in this class when you've made a function table you've substituted some number for x in f(x) and then made some computations. In trig tables you substitute some number for the Greek letter "theta" (our text uses "theta" for the independent variable in chapter 6) and then draw a triangle. It takes some getting used to.

And you should know that we will only make a table this semester. We are running out of time. You will graph trig functions and study their inverse functions next semester in math 195.

5.1 Angle Measure

In this section you will learn how to measure an angle using two measurements: 1. degrees and 2. radians. Even though you probably have already studied degree measure in high school, it turns out that radian measure is more useful. All your trigonometric formulas in calculus will be easier if you measure angles in radians. If you insist on measuring angles with degrees in calculus, your formulas trig formulas will all have ugly constants in them. All calculus students measure in radians rather than degrees. It is easier that way. Your future self studying calculus will appreciate it if you now learn to use radians. One way to begin to appreciate the power of measuring in radians is to look at the formulas in the blue boxes for the "Length of a Circular Arc" and "Area of a Circular Sector". These formulas are easy to memorize if you measure in radians (you must memorize them). They are not so easy to memorize using degrees.

You should spend some time learning about the "Standard Position" of an angle and the complications that arise with "Coterminal Angles". You should understand how to convert from degrees to radians and from radians to degrees.

Textbook Examples: 1-5

Textbook Exercises: 1-70

Intro to Radians Video

Radians and Degrees Video

Degrees to Radians Video

Radians to Degrees Video

Radian Angles and Quadrants Video

Arc Length from Subtended Angle Video

Subtended Angle from Arc Length Video

Arc Length Exercises

Arc Length from Subtended Angle: Radians Video

Radians and Arc Length Exercises


5.2 Trig of Right Triangles

You should memorize ONLY the trig ratios for sin t = opp/hyp, cos t = adj/hyp, and tan t = opp/adj. You can get the other three trig ratios taking the reciprocals of these three.

You should practice the basic high school geometry of triangles used in getting the "Special Ratios" from the "Special Triangles" at home. You should not mess around with these triangles on an exam. Just memorize the table of special values of sine and cosine (you can get the other four trig "special ratios" from these two). Some students like to write this table on the back of their exam once the get it so they do not forget anything.

One more thing: It is rare that we can find numbers we already know as sin(t) and cos(t). The special ratios are really special. Most theta-values that we plug in give new irrational numbers that we do not already know. For instance if someone asks you to compute the sine and cosine determined by theta = 2. The best answer is the exact answer the point cos(2) and sin(2. You could plug these into your calculator but this will only give an approximate answer. The best answer is the exact answer cos(2) and sin(2).

Textbook Examples: 1, 2, 4, 5, 6 (do not use a calculator to approximate final answer in this class).

Textbook Exercises: 1-12, 17-46, 55-61.

Use a Trig Equation to Determine the Length of a Side of a Right Triangle Video

Find the Reach of a Ladder Video

Triangle Similarity and the Trigonometric Ratios Video

Trigonometric Ratios in Right Triangles Video

Trigonometric Ratios in Right Triangles Exercises

Solving for a Side in Right Triangles with Trigonometry Video

Solve for a Side in Right Triangles Exercises

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