Math 190 Topics
P.2 Real Numbers:
This section is review. Make sure you understand Examples 1, 2, 3, and 5. 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 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 Exercises for practice: 29-38, 47-66
Order of Operations three videos, one overview, and one practice problem sets.
Rational and Irrational Numbers four videos, one overview, and two practice problem sets.
P.3 and P.4 Exponents and Radicals:
Most of this section should be review of high school math. Read through this section to make sure you understand all the Laws of Exponents and can do the included Examples. You may skip the Example about using a calculator. We will not be using calculators in this course. You may also skip "Rationalizing the Denominator". You may need to spend a little extra time studying rational exponents? We will not cover scientific notation.
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 Exercises for practice P3: 1-7, 9-34
Textbook Exercises for practice P4: 9-26, 49-62
Exponents Properties Review two videos, three practice problem sets, and one quiz.
Exponents Properties Intro three videos, one overview, and four practice problem sets.
Properties of Exponents (rational exponents) three videos and three practice problem sets.
Properties of exponents (rational exponents) two videos, two practice problem sets, and one quiz.
Evalulating exponents and radicals five videos and one practice problem set.
P.5 Algebraic Expressions and P.6 Factoring:
A review of factoring and expanding polynomials. Make sure you understand the difference. 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", "factoring by grouping", and "factoring with more than one variable".
Textbook Exercises for practice P5: 9-78
Textbook Exercises for practice P6: 7-36, 61-80
Adding and Subtracting Polynomials two videos, one overview, and two practice problem sets.
Multiplying Binomials four videos, two overviews, and three practice problem sets.
Special Products Binomials two videos, one overview, and one practice problem set and one quiz.
Factoring polynomials by taking common factors five videos, two overviews, and one practice problem set.
Factoring quadratics: Difference of Squares four videos, one overview, and one practice problem set.
Factoring quadratics: Perfect squares six videos, two overviews, and two practice problem sets.
Factoring quadratics 1 three videos, two overviews, and one practice problem set.
Factoring quadratics 2 two videos, two overviews, and one practice problem set.
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 but skip the last two on Rationalizing the Denominator and Numerator. You need not know how to rationalize the denominator or numerator (skip EX 9 and EX 10). 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 Exercises for practice: 1-64
Rational Expressions to lowest terms one video, two overviews, and two practice problem sets.
Multiplying and Dividing Rational Expressions three videos and two practice problem sets.
Adding and Subtracting Rational Expressions two overviews, seven videos, and two practice problem sets and Quiz 2.
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 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 Exercises for practice P8: 1-78, 87-97
Equations with variables on both sides four videos and two practice problem sets.
Equations with parenthesis one video, one overview, two practice problem sets.
1.1 Coordinate Geometry and 1.2 Graphs
This should be a review of graphing from high school. You should memorize the distance and midpoint formulas. You should understand that the distance formula is derived from the Pythagorean Theorem. You should spend a lot 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 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 useful.
You should work through EX1, EX2, EX3, EX5, EX8, EX9 in section 1.2.
You may skip the part of the subsection about symmetry and EX11, EX12, EX13 in section 1.2.
Textbook Exercises 1.1: 21-30
Textbook Exercises 1.2: 9-40, 47-56, 67-82
Distance, Midpoints, Lines two videos, five overviews, and two practice problem sets.
Graphs of Circles Intro three videos and two practice problem sets.
Standard Equation of a Circle three videos and three practice problem sets and one quiz.
Intercepts from an Equation one video
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 become 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".
Textbook Exercises for practice: 9-52, 57-78
Forms of Linear Equations 17 videos, eight overviews, 11 practice problem sets, three quizzes, and one test.
Equations of Parallel and Perpendicular Lines eight videos and two practice problem sets.
1.4 Quadratic Equations and 1.6 Other 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.
The methods in section 1.6 are of secondary importance in this course. 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"Fourth-Degree of Quadratic Type" and "Fractional Powers". Have fun working on the exercises. Some are easy; some are challenging.
Textbook Exercises for practice 1.4: 5-46
Textbook Exercises for practice 1.6: 5-8, 25-30, 37-44
Solving Quadratic Equations by Factoring five videos, two overviews, and three practice problem sets.
Solving Quadratics by taking square roots four videos, two overviews, and four practice problem sets.
The quadratic formula four videos, three overviews, and two practice problem sets.
Completing the square Intro four videos and two practice problem sets.
More on Completing the Square five videos, three overviews, and two practice problem sets.
Strategizing to solve quadratic equations one video, one practice problem set.
Rational Equations, Square Root Equations, Extraneous Roots four videos and three practice problem sets.
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 Exercises: 5-24, 33-54
Testing Inequalities one video.
Testing Inequalities one practice problem set.
Two step inequalities one video, one practice problem set. Skip all word problem videos and problem sets.
Multistep Inequalities three videos and one practice problem set.
Quadratic Inequalities three videos.
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 Exercises: 17-65
Evaluating Functions five videos and three practice problem sets.
Inputs and Outputs of a Function three videos and two practice problem sets and one quiz.
Introduction Domain and Range of a function four videos and one practice problem set.
Determining the Domain of a function five videos and three practice problem sets.
Recognizing functions five videos and two practice problem sets.
Maximum and Minimum Points of Functions two videos and two practice problem sets.
Functions Domain, Range, Intervals three videos and two practice problem sets. Do not do or watch anything related to domains in word problems.
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 Exercises for practice: 2-28, 33-46, 49-56
Piecewise Functions five videos and three practice problem sets.
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.
One confusing thing though 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 Exercises for practice: 7-16, 31-34, 43-46
Graph Info Max/Min two videos and two practice problem sets.
Graph Info Increasing/Decreasing two videos and two practice problem sets.
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.
Textbook Exercises for practice: 1-32
Average Rate of Change three videos and one practice problem set.
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 Exercises: 1-74
Shifting Functions three videos and one practice problem set.
Reflecting Functions two videos and one practice problem set.
Symmetry of Functions five videos, two overviews, and two practice problem sets and one quiz.
Scaling Functions four videos and two practice problem sets.
Putting it all together one video and one practice problem set.
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.
Textbook Exercises for practice: 27-72
Composing Functions six videos, two overviews, and three practice problem sets.
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 Exercises for practice: 1-74
Invertible Functions two videos, one overview, and two practice problem sets.
Inverse Functions in Graphs and Tables two videos and one practice problem set.
Verifying Inverse functions by composition four videos, one overview, two practice problem sets.
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.
Additional Textbook Exercises for practice: 1-44, 47, 48, 51, 52
Quadratics: Solving and Graphing in Factored Form two videos and two practice problem sets. Skip word problems.
Quadratics: Vertex Form three videos and 2 practice problem sets.
Quadratics: Standard Form two videos and one practice problem set. Skip word problems.
Comparing Quadratics two videos and one practice problem set.
Quadratics: Transformations three videos and two practice problem sets.
3.2 Polynomial Functions and Their Graphs
The graphs of a degree one polynomial is a line. The graph of a degree two polynomial (a quadratic function) is a parabola. You learned to graph lines in section 1.10 and parabolas in section 3.1. In this section you will learn to graph some polynomials of higher degree. The method we will learn is to factor the polynomial is to factor the polynomial to find the x-intercepts and then to choose test values to make a sign table. After you have the sign table and x-intercepts you can make a rough sketch of the graph. You should know that this method rarely works because higher degree polynomials are mostly impossible to factor. Yet it does work on many final exam questions and you should learn the method. Mastering this technique is a good way to appreciate the intimate relationship between algebra (factoring) and geometry (graphing intercepts). Finally you should understand the end behavior of a higher degree polynomial and how it relates to its degree being even or odd.
You should understand Examples 4, 5, 6, and 8. You may skip the other EXAMPLES in the section. You may skip the "Intermediate Value Theorem for Polynomials" and subsection "Local Maxima and Local Minima of Polynomials".
Additional Textbook Exercises for practice: 1-36, 51-54
Zeros of Polynomials six videos and two practice problem sets.
Positive and Negative Intervals of Polynomials three videos, two overviews, and two practice problem sets.
End Behavior of polynomials one video, one overview, and one practice problem set.
Putting it all together two overviews and one test.
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 Exercises: 2,3,5-20,25-44
Graphs of exponential functions three videos and one practice problem set.
Solving exponential equations using properties of exponentials two videos and two practice problem sets.
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 want to graph or make a table of a log function, first sketch or make a table of the corresponding exp 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).
When graphing a log function you should know that each log function has one vertical asymptote and no horizontal asymptote. You should find the vertical asymptote and trace it (with a different color pen) before tracing the log graph. In order to find the vertical asymptote of a log graph use the transformation method of section 2.6 and the fact that all basic log functions have the y-axis as the vertical asymptote.
You may skip Examples 7, 8, 11 in this section.
Textbook Exercises: 1-44, 49-66
Introduction to Logs five videos, one overview, and three practice problem sets.
Graphs of log functions three videos and one practice problem set.
The constant e and the natural log three videos
4.4 Laws of Logs
I know that I said last section that, in order to save brain space, you should not learn anything about log functions but instead convert all things log into their corresponding exponential things. I lied. You should memorize the laws of logs. You should know that the laws of logs are nothing more than the laws of exponents converted into logs (see the proof in our textbook). However in this case you should not convert the laws of exponents yourself on an exam. Memorize the laws of logs instead. It is safer that way.
This is a quick section. Do not spend much time on it. You will see the power of the laws of logs in the next section when you will solve exponential and log equations. ONLY go over Examples 1, 2, 3. Skip the others.
Textbook Exercises: 9-16, 49-51
Properties of Logs seven videos, two overviews, one practice problem set, and one quiz.
The change of base formula three videos, two overviews, and two practice problem sets.
4.5 Exp and Log Equations
In this section we add two new tools to our equation solving toolkit. We can now 1. take log (using the same base) of both sides of an equation and 2. take exp (using the same base) of both sides of an equation. This is no different from middle school when you learned how to solve equations by adding the same number to both sides of an equation or by dividing both sides of an equation by the same number. The key is that you MUST do the same thing to both sides of an equation.
This all seems too easy---its just middle school all over again? This is not true. There are two tricky details you must practice: 1. Every time you learn a new equation solving technique you must mix the technique in with all the other techniques in the correct order (PEMDAS...) and 2. There is a domain issues with log (you may end up with false roots). In order to become comfortable with doing things in the correct order you should solve many equations in this section. In order to not have false roots, you should test your solutions to log equations to make sure they are in the domain.
You should skip Examples 4, 5, 6, 10 (graphical part), 11, 12, 13, 14. You should skip subsection "Compound Interest". You do not need to know how to solve geometrically or how to use a calculator to approximate an exact solution.
Textbook Exercises: 1-18, 49-51, 55-66
Solving Exponent Equations with Logs two videos, one overview, and one practice problem set.
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. You should study Examples 1, 2, 3, 4. You should skip subsections "Area" and "Circular Motion" and Examples 5, 6, 7.
Additional Textbook Exercises: 1-62
Arc Length (from degrees) two videos, two overviews, and one practice problem set.
Intro to Radians five videos, one overview, and one practice problem set.
Arc Length (from radians) two videos, two overviews, and one practice problem set.
Sectors one video, one practice problem set, one quiz.
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).
All included Examples should be understood without a calculator.
Addition Textbook Exercises: 15-33, 53-59.
Intro to the Trig Ratios two videos, one overview, and one practice problem set.
Solving for a side in a right triangle using trig ratios one video, one overview, one practice problems set
Modeling with right triangles 1 video, 2 overviews, and 1 practice problem set.
5.3 Trig Functions and Angles
Very similar to section 6.2 and should be studied simultaneously with that section.
In this section you will learn how to leverage the table of "special" trig angles you found using basic geometry and then memorized in section 6.2. This method allows you to determine many many more trig values then the trig values for the ''special'' angles. It is best to follow our textbook's method in this case:
Find the reference angle associated to the given angle.
Hope that the reference angle matches with one of the "special angles". If the reference angle does indeed match with one of the ''special'' angles then the answer will be the almost the same ratio as the matching ''special'' angle.
Determine if you will use + or - the answer from 2. by finding the quadrant in which the original angle lies. You need not memorize the formula for the area of a triangle.
Skip "Areas of Triangles" and Example 8.
Textbook Exercises: 1-54
Special Trig Values to Memorize two videos and one practice problem set.
Trig values of special angles one practice problem set.