Rational and Radical Relationships - Classroom Blogplanemath.weebly.com/.../3/5/...rational_and_radical_relationships.pdf · Rationals and Radicals Project Rational and Radical Relationships

6/19/2015 CCGPS Advanced Algebra

http://cms.gavirtualschool.org/Shared/Math/CCGPS_AdvancedAlgebra/RationalandRadicalRelationships/AdvAlgebra_RationalandRadicalRelationships_SHA… 1/12

CCGPS Advanced AlgebraRational and Radical Relationships

Rational and Radical RelationshipsMany people have an interestin pastimes such as diving,photography, racing, playingmusic, or just getting a tan. Or maybe you areconsidering a career inmedicine, machinery,farming, banking, or weather.Welcome to the world ofradical and rationalrelationships; not "far out" or"logical", but mathematicalrelationships. Radicalsinvolve roots of a number andrationals are expressionswritten as a fraction. Pressurein diving, exposures inphotography, average speedin racing, frequency in musicand the sun's radiation can allbe expressed as a radical orrational function. The careerslisted are a few examples ofjobs that use radical orrational functions in variousaspects.

Essential QuestionsWhat are the rules for operations on rational expressions?How are equations used in problem solving?In solving rational and radical equations, what are extraneous solutions?What do the key features of a graph mean in the context of the problem?

Module MinuteRational expressions are fractions, so the rules are basically the same as for numericalfractions. The biggest difference will involve factoring parts of the expression. There aremany situations in life that are modeled with radical and rational equations, as you will see. An interesting part of solving these equations is an extraneous solution, which is a solutionthat doesn't work. An example would be a negative answer when calculating the length of afield. The key features in graphing these functions provide information like maximums,minimums, positive values, and extraneous solutions. All of these give us valuable

information in the context of the problem, helping us understand the situation we're exploring.

Key WordsRational Expression an expression that can be written as a fractionExcluded Values values that make the expression undefined (0 in the denominator)Like Terms terms having the exact same variable(s) and exponent(s).



Extraneous Solutions solutions that make the expression undefinedZeros the roots of a function, also called solutions or xintercepts.Intercepts points where a graph crosses an axisDomain the values for the x variableRange the values for the y variableAsymptotes vertical and horizontal lines where the function is undefinedExtrema maximums and minimums of a graphEnd Behavior the rise or fall of the ends of the graphRadical Expression an expression that has a rootRadicand the number or expression under the radicalSystem of Equations n equations with n variablesPoint of Intersection the point(s) where the graphs cross.

A handout of these key words and definitions is also available in the sidebar.

What To Expect

Rational Expressions and Operations Handout AssignmentRational Functions QuizProblem Challenge DiscussionRational graphs and Radical Equations Handout AssignmentRadical Functions and Systems QuizRationals and Radicals ProjectRational and Radical Relationships Test

To view the standards from this unit, please download the handout from the sidebar.

Rational Expressions and OperationsSimplify FractionsThe first thing to learn about algebraic fractions is simplifying them, which is to reduce them. Just like withnumber fractions, algebraic fractions are reduced by "canceling" factors common to both the numerator andthe denominator.

Let's look at some monomial fractions first.

In order to reduce fractions with polynomials, you must factor the polynomials first. NEVER cancel parts ofpolynomials!

Watch this video for more monomials and reducing polynomials.

If you want to see more examples, go to Simplify in the sidebar. To practice go to Practice 1 in the sidebar.



In dealing with fractions, excluded values must beconsidered. The denominator of a fraction can never be zero(try something divided by 0 in your calculator). Excludedvalues, also called restrictions, are numbers that would makethe denominator zero, usually written " ". In functions,excluded values determine the domain.

Here are some examples:

Each of the excluded values would make the denominatorszero. When reducing fractions, even if a factor in thedenominator cancels, it must be considered for an excludedvalue because the original fraction would be undefined forthat number.

Excluded values can also be written as a domain. The form often used is , which is read "xsuch that x is not equal to 2 and 6".

Now try these Self Check problems.

Multiply and Divide

In learning operations with rational polynomials, we will start with multiplying because the process is similar tosimplifying fractions. When you multiply fractions, you multiply the numerators and the denominators straightacross, reducing (or canceling) factors that are common to the numerator and denominator. Again, NEVERcancel parts of polynomials!

For practice, try "Practice 2" in the sidebar. To see more examples of multiply, go to Mult. Video in thesidebar.

Do these self check problems for more practice.

Dividing is almost the same as multiplying. First invert the fraction after the divide sign, then multiply.

See the steps in this video.

When finding excluded values for division, you must include what makes both the top and bottom of the



inverted fraction equal to zero, since both quantities are in the denominator at some point. For example, in

the problem the excluded values are x ≠ 5, 2, 3. Some problems have both multiply anddivide.

Watch this video for an example.

For more practice, go again to "Practice 2" in the sidebar and scroll down to the divide problems. To seemore examples of divide, go to Div Video in the sidebar.

Now do the Self Check.

Add and Subtract

The next operations to explore are addition and subtraction. To add fractions, they must have a commondenominator. If you are adding two fractions where the denominators are 2 and 3, you will find that the leastcommon multiple is 6. This value will be the common denominator. For polynomials, the concept is thesame. If you are adding two fractions where the denominators are x and , the least common multiplewould be .

Once you have a common denominator, you add algebraic rational expressions just as you do basicfractions. Add the numerators and leave the denominators. Always make sure that your answers are insimplest form.

Since addition and subtraction are a little more complicated, be sure to watch all of the examples inthe Video Showcase.

To see more examples and to practice, go to Add/Sub 1, Add/Sub 2, and Practice 3 in the sidebar.

Do the Self Check.

Rational Expressions and Operations Assignment

Select the " Rational Expressions and Operations " Handout from the sidebar. Record youranswers in a separate document. Submit your completed assignment.

Rational EquationsIn this learning task we will add, subtract, multiply, and divide rational expressions in order to solve rationalequations.

The first step is finding a common denominator for all of the fractions in the equation. Now, take that commondenominator and multiply it by every term in the problem, including terms that are not fractions. This willeliminate all of the denominators and you will be left with a basic linear or quadratic equation to solve.

Watch the video below to learn more about eliminating denominators.



Sometimes to find the common denominator, you have to factor the denominators.

Once you have solved the equation, you must check for extraneous solutions. These are the excludedvalues that would make the denominator zero. Remember, when you have a fraction, it will be undefined ifthe denominator equals zero. Here is an example. If one of the denominators was (x+2) and the solutions tothe equation were x = 3 and 2, the 2 would be an extraneous solution. So you would cross it off and theonly solution would be x = 3.

Watch this video to see some examples.

When solving rational equations, ALWAYS check to see if the solutions are extraneous!

Caution: This process of multiplying by the common denominator to get rid of the denominatorsONLY works for equations and inequalities, NEVER for operations with expressions like youwere doing in the previous topic!

This Video Showcase will show you a shortcut on certain types of equations, a more complicatedexample, and an example of how this is applied.

Here is another application example.

Jolynn went shopping for her family for Christmas. She spent half her money on her mom and dad, and onethird of her money on her sister. If she had $14 left over, how much did she start with?

Let x = starting amount, so would be half and would be a third.

The equation is

Mult. by the common denominator

Solve

Jolynn had $84 to start with.

For more examples, go to Equations in the sidebar. To practice go to Practice 4 and Word problems in thesidebar.


Solve the equations

Rational Functions Quiz

It is now time to complete the " Rational Functions " quiz. You will have a limited amount of time tocomplete your quiz; please plan accordingly.

Graph Rational FunctionsLet's look at the graphs of rational functions. Just as with polynomial functions, these graphs have a basic



shape. The most basic rational function is .

As you can see, the transformations are the same whether the graphs are polynomial functions or rationalfunctions. Though there are many variations of the graphs of rational functions, often the graphs willincorporate this basic shape.

You will notice that these graphs are not continuous like the polynomial graphs were, there is a gap. Lookingat some key characteristics will help to understand the differences.

Recall that the domain is the x values and the range is the y values. As you just learned, rational functionshave excluded values or domain restrictions, which cause the gaps in the graph. As you go back throughthe album, the domain of each graph is all real numbers except what makes the denominator zero. Compare this to the vertical gaps.

The domain for the basic equation is stated , or more simply, .

At this excluded value is a key feature called a vertical asymptote, usually drawn by hand on the graph as adotted line. An asymptote is a vertical or horizontal line where the function is undefined. So the first 3equations have a vertical asymptote at x = 0.

The range is similar, with a gap that is a horizontal asymptote. In the 1st, 4th, and 5th graphs the range is

, or more simply, and there is a horizontal asymptote at y = 0.

Horizontal asymptotes are easiest to find by looking at the graph and the shifts.

Determine range by using horizontal asymptotes and extrema. For example, the range of a parabola will befrom the minimum y value to infinity, or negative infinity to the maximum y value.

To understand asymptotes, watch this video.

Other key features of graphs you have learned are intercepts, extrema, and end behavior. These graphshave no extrema and the basic graph has no intercepts. Remember, to get the yintercept, put zero in for x(unless it is an excluded value). To get the xintercept, put 0 in for y and solve for x. From the graphs, youcan see the end behavior of each of these graphs is rising on the left and falling on the right.

There are 3 new key features that we will learn here. The first 2 are where a graph is increasing ordecreasing and where the graph is positive or negative.

Watch the video to see how to find these intervals.

The third feature is symmetry. Graphs have symmetry on the x axis, the y axis and the origin. Since weare dealing with functions, there will never be x axis symmetry. (Functions must pass the vertical line test, sothere can't be matching points across the x axis.) For y axis symmetry, an x value and its opposite musthave the same y value. For origin symmetry, an x value and its opposite must have the opposite y values. Stated more formally:



Our basic rational graph has origin symmetry and the other graphs in the album have no symmetry.

These 3 new key features apply to all types of graphs, not just rational functions.

Using the key features, we can graph rational functions by hand. Watch the video below to learnhow.

For more on the key features, see Rationals and Intercepts in the sidebar.

Now try the Self Check.

Problem Challenge Discussion

It is now time to complete the " Problem Challenge " discussion. A rubric for your discussion inlocated in the sidebar .

Here is an opportunity for you to challenge (or stump) your classmates. In this discussion, you willmake up 2 problems for your classmates to simplify. First, create a multiplication or division problem withrational algebraic expressions. Be sure to include at least 3 expressions that require factoring to simplify theproblem.

Second, create an addition or subtraction problem with rational algebraic expressions, which requires findinga common denominator. (Do not just take a problem from the text, videos or resources for either part; comeup with one of your own.)

You will receive more points for more creative problems, but be sure you can find the correct answer, sinceyou will be checking your classmates' work. (Don't make your problems too hard...you want a classmate to



give them a try.)

Lastly, reply to the students who answered your problems, telling them whether or not they were correct.

Radical Expressions and EquationsRadical Expressions are expressions involving a root. Roots and powers are opposite operations, just like

add and subtract or multiply and divide. We will be looking at square roots ( ), or cube roots ( ) here. Let's start with a little background.

Square roots are the opposite of squares; ,so the square root and square cancel each other out. Asyou learned in Module 2, if you are taking the square root ofa negative number, you get an imaginary number (i).

Cube roots are the opposite of cubes; , sothe cube root and cube cancel each other out. You can takethe cube root of a negative number and your answer will benegative.

NEVER use i with cube roots, or any other roots exceptsquare roots.

To solve radical equations, you will use the oppositeoperation to get rid of the radical, just as you do with otheroperations. Remember, what you do to one side of anequation, you must do to the other side. Look at theseexamples.

Example 1:

Example 2:

Watch these videos to see how to solve radical equations with one radical.



Sometimes equations have more than one radical. This is how to solve those.

ALWAYS check radical equations for extraneous solutions!

For more examples, go to Radical 1 and Radical 2 in the sidebar.

Here is an application of radical functions.

Now try the Self Check.

Rational Graphs and Radical Equations Assignment

Select the " Rational Graphs and Radical Equations " Handout from the sidebar. Record youranswers in a separate document. Submit your completed assignment.

Graph Radical FunctionsLet's look at the graphs of radical functions. Just as with polynomial and rational functions, these graphs

have a basic shape. The most basic radical function is .

As you can see, the transformations are the same no matter what type of function you are graphing. The

function could have been used in the album.

In your calculator, graph the equations in the album using cube root instead of square root, to see that the

transformations are the same. (You can use an exponent of (1/3) to graph a cube root .) In graphingsquare root functions, you cannot use imaginary numbers. Therefore, the domain of a square root functionwill always be limited to values that make the expression in the root nonnegative. (You can have zero.) Numbers that make the expression in the root negative would be excluded values.

Most of the other key features will also apply to these graphs.

Graph these 2 equations in your calculator, then "rollover" the equation to see the key features.

Example 1

Example 2


http://cms.gavirtualschool.org/Shared/Math/CCGPS_AdvancedAlgebra/RationalandRadicalRelationships/AdvAlgebra_RationalandRadicalRelationships_SH… 10/12

Let's look at how to graph radical expressions in the videos below.

For more information and practice, see Graph 1 and Graph 2 in the sidebar.


Systems of EquationsIn Module 2, we learned that a system ofequations is n equations with n variables.Here we will look at systems that involverational and radical equations. The methodsto use for these systems are graphing andsubstitution.

To solve systems by graphing, we graph theequations to find the point of intersection,the point(s) where the graphs cross. Systemsare consistent if they have one or moresolutions and inconsistent if they have nosolution. Check the answer by plugging itinto both equations.

To review this process with linear graphs, goto Graphing in the sidebar.

To solve systems by substitution, first isolateone of the variables, then substitute theresulting expression into the second equationand solve. Substitute that answer back intoeither of the original equations to find the second variable.

To review this process with linear graphs, go to Substitution in the sidebar.

Watch the Video Showcase to see examples of graphing, substitution and an application.

Do the Self Check problems.

Radical Functions and Systems Quiz

It is now time to complete the " Radical Functions and Systems " quiz. You will have a limitedamount of time to complete your quiz; please plan accordingly.



Module Wrap UpModule ChecklistIn this module you were responsible forcompleting the following assignments:

Rational Expressions andOperations HandoutRational Functions QuizProblem Challenge DiscussionRational graphs and RadicalEquations HandoutRadical Functions and SystemsQuizRationals and Radicals ProjectRational and Radical RelationshipsTest

ReviewNow that you have completed the initialassessments for this module, review thelesson material with the practice activities andextra resources. Rewatch videos and visitthe extra resources in the sidebars asneeded. Then, continue to the next page for your final assessment instructions.

Standardized Test PreparationThe following problems will allow you to apply what you have learned in this module to how you maysee questions asked on a standardized test. Please follow the directions closely. Remember that youmay have to use prior knowledge from previous units in order to answer the question correctly. If you

have any questions or concerns, please contact your instructor.

Final AssessmentsRational and Radical Relationships Final Module Test

It is now time to complete the " Rational and Radical Relationships " Test . Once you havecompleted all self checks, assignments, and the review items and feel confident in yourunderstanding of this material, you may begin. You will have a limited amount of time to complete



your test and once you begin, you will not be allowed to restart your test. Please plan accordingly.

Rationals and Radicals ProjectSelect " Rationals and Radicals Project " Handout from the sidebar. Record your answers in aseparate document. Submit your completed assignment. A rubric is available in the sidebar.

Documents

Rational and Radical Relationships - Classroom Blogplanemath.weebly.com/.../3/5/...rational_and_radical_relationships.pdf · Rationals and Radicals Project Rational and Radical Relationships