Preface Here are my online notes for my Algebra course that I teach here at Lamar University, although Ihave to admit that it’s been years since I last taught this course. At this point in my career Imostly teach Calculus and Differential Equations.Despite the fact that these are my “class notes” they should be accessible to anyone wanting tolearn Algebra or needing a refresher in for algebra. I’ve tried to make the notes as self containedas possible and do not reference any book. However, they do assume that you’ve has someexposure to the basics of algebra at some point prior to this. While there is some review ofexponents, factoring and graphing it is assumed that not a lot of review will be needed to remindyou how these topics work.Here are a couple of warnings to my students who may be here to get a copy of what happened ona day that you missed.1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learnalgebra I have included some material that I do not usually have time to cover in classand because this changes from semester to semester it is not noted here. You will need tofind one of your fellow class mates to see if there is something in these notes that wasn’tcovered in class.2. Because I want these notes to provide some more examples for you to read through, Idon’t always work the same problems in class as those given in the notes. Likewise, evenif I do work some of the problems in here I may work fewer problems in class than arepresented here.3. Sometimes questions in class will lead down paths that are not covered here. I try toanticipate as many of the questions as possible in writing these up, but the reality is that Ican’t anticipate all the questions. Sometimes a very good question gets asked in classthat leads to insights that I’ve not included here. You should always talk to someone whowas in class on the day you missed and compare these notes to their notes and see whatthe differences are.4. This is somewhat related to the previous three items, but is important enough to merit itsown item. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!!Using these notes as a substitute for class is liable to get you in trouble. As already notednot everything in these notes is covered in class and often material or insights not in thesenotes is covered in class.
Hyperbolas The next graph that we need to look at is the hyperbola. There are two basic forms of ahyperbola. Here are examples of each.Hyperbolas consist of two vaguely parabola shaped pieces that open either up and down or rightand left. Also, just like parabolas each of the pieces has a vertex. Note that they aren’t reallyparabolas, they just resemble parabolas.There are also two lines on each graph. These lines are called asymptotes and as the graphs showas we make x large (in both the positive and negative sense) the graph of the hyperbola gets closerand closer to the asymptotes. The asymptotes are not officially part of the graph of the hyperbola.However, they are usually included so that we can make sure and get the sketch correct. Thepoint where the two asymptotes cross is called the center of the hyperbola.There are two standard forms of the hyperbola, one for each type shown above. Here is a tablegiving each form as well as the information we can get from each one.
Form( ) ( )2 22 21x h y ka b− −− =( ) ( )2 22 21y k x hb a− −− =Center ( ),h k ( ),h kOpens Opens left and right Opens up and downVertices ( ),h a k+ and ( ),h a k− ( ),h k b+ and ( ),h k b−Slope of Asymptotesba±ba±Equations of Asymptotes ( )by k xah= ± − ( )by k xah= ± −Note that the difference between the two forms is which term has the minus sign. If the y termhas the minus sign then the hyperbola will open left and right. If the x term has the minus signthen the hyperbola will open up and down.We got the equations of the asymptotes by using the point-slope form of the line and the fact thatwe know that the asymptotes will go through the center of the hyperbola.Let’s take a look at a couple of these.Example 1 Sketch the graph of each of the following hyperbolas.(a)( ) ( )2 23 1125 49x y− +− = [Solution](b) ( )2229yx− + =1 [Solution]Solution(a) Now, notice that the y term has the minus sign and so we know that we’re in the first columnof the table above and that the hyperbola will be opening left and right.The first thing that we should get is the center since pretty much everything else is built aroundthat. The center in this case is and as always watch the signs! Once we have the centerwe can get the vertices. These are(3, 1−(8,))1− and ( )2, 1− − .Next we should get the slopes of the asymptotes. These are always the square root of the numberunder the y term divided by the square root of the number under the x term and there will alwaysbe a positive and a negative slope. The slopes are then75± .Now that we’ve got the center and the slopes of the asymptotes we can get the equations for theasymptotes. They are,
( ) ( )7 71 3 and 15 5y x y x= − + − = − − −3We can now start the sketching. We start by sketching the asymptotes and the vertices. Oncethese are done we know what the basic shape should look like so we sketch it in making sure thatas x gets large we move in closer and closer to the asymptotes.Here is the sketch for this hyperbola.[Return to Problems](b) In this case the hyperbola will open up and down since the x term has the minus sign. Now,the center of this hyperbola is ( . Remember that since there is a y2term by itself we had tohave . At this point we also know that the vertices are)2,0−0k = ( )2,3− and ( )2, 3− − .In order to see the slopes of the asymptotes let’s rewrite the equation a little.( )22219 1xy +− =So, the slopes of the asymptotes are331± = ± . The equations of the asymptotes are then,( ) ( )0 3 2 3 6 and 0 3 2 3 6y x x y x x= + + = + = − + = − −Here is the sketch of this hyperbola.