Advanced Algebra Rational and Radical Relationships Rational and Radical Relationships Many people have an interest in pastimes such as diving, photography, racing, playing music, or just getting a tan. Or maybe you are considering a career in medicine, machinery, farming, banking, or weather. Welcome to the world of radical and rational relationships; not "far out" or "logical", but mathematical relationships. Radicals involve roots of a number and rationals are expressions written as a fraction. Pressure in diving, exposures in photography, average speed in racing, frequency in music and the sun's radiation can all be expressed as a radical or rational function. The careers listed are a few examples of jobs that use radical or rational functions in various aspects. Essential Questions What 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 Minute Rational expressions are fractions, so the rules are basically the same as for numerical fractions. The biggest difference will involve factoring parts of the expression. There are many 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 solution that doesn't work. An example would be a negative answer when calculating the length of a field. 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 Words Rational Expression an expression that can be written as a fraction Excluded 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 undefined Zeros the roots of a function, also called solutions or x intercepts. Intercepts points where a graph crosses an axis Domain the values for the x variable Range the values for the y variable Asymptotes vertical and horizontal lines where the function is undefined Extrema maximums and minimums of a graph End Behavior the rise or fall of the ends of the graph Radical Expression an expression that has a root Radicand the number or expression under the radical System of Equations n equations with n variables Point 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 Assignment Rational Functions Quiz Problem Challenge Discussion Rational graphs and Radical Equations Handout Assignment Radical Functions and Systems Quiz Rationals and Radicals Project Rational and Radical Relationships Test To view the standards from this unit, please download the handout from the sidebar. Rational Expressions and Operations Simplify Fractions The first thing to learn about algebraic fractions is simplifying them, which is to reduce them. Just like with number fractions, algebraic fractions are reduced by "canceling" factors common to both the numerator and the 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 of polynomials! 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 be considered. The denominator of a fraction can never be zero (try something divided by 0 in your calculator). Excluded values, also called restrictions, are numbers that would make the denominator zero, usually written " ". In functions, excluded values determine the domain. Here are some examples: Each of the excluded values would make the denominators zero. When reducing fractions, even if a factor in the denominator cancels, it must be considered for an excluded value because the original fraction would be undefined for that number. Excluded values can also be written as a domain. The form often used is "x such that x is not equal to 2 and 6"., which is read 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 to simplifying fractions. When you multiply fractions, you multiply the numerators and the denominators straight across, reducing (or canceling) factors that are common to the numerator and denominator. Again, NEVER cancel parts of polynomials! For practice, try "Practice 2" in the sidebar. To see more examples of multiply, go to Mult. Video in the sidebar. 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 and divide. the excluded values are x 5, 2, 3. Some problems have both multiply 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 see more 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 common denominator. If you are adding two fractions where the denominators are 2 and 3, you will find that the least common multiple is 6. This value will be the common denominator. For polynomials, the concept is the same. If you are adding two fractions where the denominators are x and multiple would be., the least common Once you have a common denominator, you add algebraic rational expressions just as you do basic fractions. Add the numerators and leave the denominators. Always make sure that your answers are in simplest form. Since addition and subtraction are a little more complicated, be sure to watch all of the examples in the 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 your answers in a separate document. Submit your completed assignment. Rational Equations In this learning task we will add, subtract, multiply, and divide rational expressions in order to solve rational equations. The first step is finding a common denominator for all of the fractions in the equation. Now, take that common denominator and multiply it by every term in the problem, including terms that are not fractions. This will eliminate 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 excluded values that would make the denominator zero. Remember, when you have a fraction, it will be undefined if the denominator equals zero. Here is an example. If one of the denominators was (x+2) and the solutions to the equation were x = 3 and 2, the 2 would be an extraneous solution. So you would cross it off and the only 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 denominators ONLY works for equations and inequalities, NEVER for operations with expressions like you were doing in the previous topic! This Video Showcase will show you a shortcut on certain types of equations, a more complicated example, 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 one third 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 the sidebar. Now do the Self Check. Solve the equations Rational Functions Quiz It is now time to complete the " Rational Functions " quiz. You will have a limited amount of time to complete your quiz; please plan accordingly. Graph Rational Functions
Let'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 rational functions. Though there are many variations of the graphs of rational functions, often the graphs will incorporate this basic shape. You will notice that these graphs are not continuous like the polynomial graphs were, there is a gap. Looking at 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 functions have excluded values or domain restrictions, which cause the gaps in the graph. As you go back through the 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 a dotted line. An asymptote is a vertical or horizontal line where the function is undefined. So the first 3 equations have a vertical asymptote at x = 0. The range is similar, with a gap that is a horizontal asymptote. In the 1 st, 4 th, and 5 th 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 be from 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 graphs have no extrema and the basic graph has no intercepts. Remember, to get the y intercept, put zero in for x (unless it is an excluded value). To get the x intercept, put 0 in for y and solve for x. From the graphs, you can 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 or decreasing 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 we are dealing with functions, there will never be x axis symmetry. (Functions must pass the vertical line test, so there can't be matching points across the x axis.) For y axis symmetry, an x value and its opposite must have 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 learn how. 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 in located in the sidebar. Here is an opportunity for you to challenge (or stump) your classmates. In this discussion, you will make up 2 problems for your classmates to simplify. First, create a multiplication or division problem with rational algebraic expressions. Be sure to include at least 3 expressions that require factoring to simplify the problem. Second, create an addition or subtraction problem with rational algebraic expressions, which requires finding a common denominator. (Do not just take a problem from the text, videos or resources for either part; come up with one of your own.) You will receive more points for more creative problems, but be sure you can find the correct answer. (Don't make your problems too hard...you want a classmate to give them a try.) Radical Expressions and Equations
Radical Expressions and Equations Radical 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. As you learned in Module 2, if you are taking the square root of a negative number, you get an imaginary number (i). Cube roots are the opposite of cubes;, so the cube root and cube cancel each other out. You can take the cube root of a negative number and your answer will be negative. NEVER use i with cube roots, or any other roots except square roots. To solve radical equations, you will use the opposite operation to get rid of the radical, just as you do with other operations. Remember, what you do to one side of an equation, you must do to the other side. Look at these examples. 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 your answers in a separate document. Submit your completed assignment. Graph Radical Functions Let'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 graphing square root functions, you cannot use imaginary numbers. Therefore, the domain of a square root function will always be limited to values that make the expression in the root non negative. (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 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.
Now do the Self Check. Systems of Equations In previous math courses, we learned that a system of equations is n equations with n variables. Here we will look at systems that involve rational and radical equations. The methods to use for these systems are graphing and substitution. To solve systems by graphing, we graph the equations to find the point of intersection, the point(s) where the graphs cross. Systems are consistent if they have one or more solutions and inconsistent if they have no solution. Check the answer by plugging it into both equations. To review this process with linear graphs, go to Graphing in the sidebar. To solve systems by substitution, first isolate one of the variables, then substitute the resulting expression into the second equation and solve. Substitute that answer back into either 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 limited amount of time to complete your quiz; please plan accordingly. Module Wrap Up Module Checklist
In this module you were responsible for completing the following assignments: Review Rational Expressions and Operations Handout Rational Functions Quiz Problem Challenge Discussion Rational graphs and Radical Equations Handout Radical Functions and Systems Quiz Rationals and Radicals Project Rational and Radical Relationships Test Now that you have completed the initial assessments for this module, review the lesson material with the practice activities and extra resources. Re watch videos and visit the extra resources in the sidebars as needed. Then, continue to the next page for your final assessment instructions. Standardized Test Preparation The following problems will allow you to apply what you have learned in this module to how you may see questions asked on a standardized test. Please follow the directions closely. Remember that you may 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 Assessments Rational and Radical Relationships Final Module Test It is now time to complete the " Rational and Radical Relationships " Test. Once you have completed all self checks, assignments, and the review items and feel confident in your understanding 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 Project Select " Rationals and Radicals Project " Handout from the sidebar. Record your answers in a
separate document. Submit your completed assignment. A rubric is available in the sidebar.