[Vertical Shifting Of Sinusoidal Graphs Algebra 2 With Trigonometry Homework
Betty Neyhart
Make sure you understand Examples 1, 2, 3, 5, and 6b). It is notnecessary to know the names of the properties. You should spendonly a little time studying open and closed interval notation.Interval notation is important but you will have many opportunitiesto practice it throughout the semester. Make sure you review orderof operations or PEMDAS and review adding and subtracting fractionsby finding a LCD. Surprisingly it is easier to multiply and dividefractions than it is to add and subtract fractions, since you donot need a common denominator to multiply or divide fractions.
Read through this section to make sure you understand all theLaws of Exponents and can do the included Examples 1-5. You mayskip the Examples 6-8 using calculators and scientific notation. Wewill not be using calculators or scientific notation in thiscourse.
vertical shifting of sinusoidal graphs algebra 2 with trigonometry homework
Download ⚙ https://t.co/W6pbiEmDC4
Read through this section to make sure you understand how to useexponential notation and radical notation as in Examples 1, 3-5.Skip "Rationalizing the Denominator". You may need to spend alittle extra time studying rational exponents since it may be newto some students.
Exponent notation is often preferable to radical notationbecause in exponential notation you can use the laws of exponents.Many exercises challenge you to rewrite radical notation inexponential notation and then apply one of the laws ofexponents.
In general the webassign exercises are a bit misleading in thatafter doing the exercises you may think that most roots (rationalexponents) 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 notsimplify further. They are perfectly good numbers that are alreadysimplified and may arise as a solution to a problem in math,science, or engineering. You must become comfortable working withthese numbers.
Adding and subtracting polynomial, combining like terms tosimplify, and expanding polynomials using multiplication. Often insection P.5, you will be asked to expand by distributing or usingFOIL and then combining like terms to simplify. You do not oftenwant to expand a factored expression unless you have a good reasonto do so. One of the goals of middle school math is to factorintegers. One of the goals of high school math is to factorpolynomials. You need not memorize the Special Product Formulas byname. You should however be able to expand and simplify any of theproducts yourself.
One of the goals of middle school math is to factor integers.One of the goals of high school math is to factor polynomials. Inboth 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 easierto understand than the whole organism. Physicists factor atoms intoprotons, neutrons, and electrons.... Factoring can be hard work.Therefore you do not often want to expand a factored object unlessyou have a good reason to do so. Make sure you understand theExamples in this section but you may skip the Examples on"factoring the sum and differences of cubes", "factoring withfractional exponents", and "factoring with more than onevariable".
A rational expression is a fraction of polynomials. Luckilythere is not much new here. All you do is mix together the basicalgebraic techniques for fractions from middle school (and fromsection 1.1) with the factoring and reducing techniques forpolynomials from section 1.3. Just like in middle school you needto be able to reduce, add (find the LCD), subtract (find the LCD),multiply, and divide rational expressions. One of the keys to manyexam problems is to factor and reduce as much as possible beforeadding, subtracting, multiplying and dividing. You should workthrough the Examples in this section. There is a nice table on"Avoiding Common Errors" at the end of this section. You shouldread this table of common errors multiple times and try not to makeany of these errors.
One technique often used to solve linear equations is to isolatethe variable on one side (it does not matter what side). In orderto isolate the variable correctly you must recognize when theequation is linear. In section 1.4 you will study quadraticequations when isolating the variable is not usually a good idea.For many quadratic equations it is better to set the equation tozero. Power equations can often be solved by taking the appropriateroot of both sides. You must be careful when taking roots becauseeven roots require a different technique than taking odd roots.
You should memorize the distance and midpoint formulas. Youshould understand that the distance formula is derived from thePythagorean Theorem. Do a couple problems from section 1.1practicing the midpoint and distance formulas. Do enough examplesso that you are comfortable.
Then move to section 1.2. Most of the important material is insection 1.2. You should spend a lot (lots and lots) of timegraphing equations by making tables and plotting points. Later inthe course you will learn faster ways to trace some graphs. Thetrouble with these faster graphing methods is that they do notalways work. Sketching by making a table always works! You shouldlearn it well. Much later, after you learn some calculus, you willsketch by using a mixture of these faster techniques, some calculustechniques, as well as making a table of points and plottingpoints. 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 basicgraphs with labeling some points on the graphs.
Work through all the Examples in this section. Make sure youunderstand the standard form of the circle and how to convert anyequation of a circle to the standard form using completing thesquare. The standard form of a circle tells you its geometry: itscenter and its radius. It is remarkable that algebra used incompleting the square is related to the center and radius of acircle.
This is truly a review section. Every student has studied linesin high school and middle school. Nevertheless, you should spendsome time doing many problems in this section. You need tounderstand lines well. One of the key ideas of calculus is toapproximate curves by lines. It makes sense to do this becauselines are easy and curves can be very complicated. However you willnever be able to approximate complicated curves by lines if you donot first understand lines well.
In calculus the point-slope form is often used since in calculusyou often end up with one point (not necessarily the y-intercept)of a line and its slope. You should become comfortable usingpoint-slope form now. Students sometimes are uncomfortable usingpoint-slope form because each line has an infinite number ofpoint-slope forms. This is because there are an infinite number ofpoints on a line and each point will give a different point-slopeform. This is a good thing. Be confident with your point-slope formof a line even if it looks different from other students'point-slope form. Any correct point-slope form is acceptable for ananswer. Don't forget: Although each line has an infinite number ofpoint-slope forms, each line has only one slope--except forvertical lines whose slope is undefined. If you find the wrongslope you're equation is sure to be incorrect.
This is a review of equation solving techniques from highschool. You should already know the quadratic formula, the zeroproduct property, and cross multiplying for solving rationalequations. You may not yet be familiar with completing the squareand solving equations involving a radical. When learning andreviewing the equation solving techniques in this section do notget lost in the techniques. The adding, subtracting, multiplying,dividing, reducing, and power techniques learned in previoussections are all helpful here but remember: you are searching for anumber that you can plug in for x so that the equation is true. Ifyou can guess a solution go ahead. You don't need any of the fancytechniques. Just make sure that you know that you can check anysolution by substituting the solution for x in the originalequation.
Do not get carried away in substituting your solution for x tocheck your answer. Many times students have gotten the correctsolution to an equation and then made an error while checking thesolution and convinced themselves that the correct answer wasincorrect. Whether or not you decide to check your solutions toequations is a personal matter. You need to practice solving manyequations 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 tofirst set one side of the equation equal to zero and then try for acouple 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 nosolutions. Complex solutions are not accepted in this course.Sometimes webassign tells you that you must use such and suchtechnique (factoring, quadratic formula, or completing the square)to solve a particular problem. This is good for homework andclasswork problems. On the quizzes and exams you may use anytechnique you choose. You can even try to guess the correctsolution---though it is not recommended to guess every solution onthe test.
Most students do not use the completing the square method onquizzes and exams. Most students prefer to use the quadraticformula or trial and error factoring on exams. It is important knowthat any method will work in theory. Completing the square is agood method for math majors who want to know why the quadraticformula works without just memorizing it. Before an exam everyoneshould memorize the quadratic formula. A good exercise to do is tosolve x^2 - 4x - 5 = 0 using all three methods: (1) trial and errorby saying "two numbers that multiply to -5 and add to -4, (2) thequadratic formula, and (3) completing the square. You will get thesame answers using each method.
795a8134c1