Radical Functions

7 Key excerpts on "Radical Functions"

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  eBook - PDF

  Intermediate Algebra

  Connecting Concepts through Applications

  • Mark Clark, Cynthia Anfinson(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  The name radical is given to the symbol ! , and the expression inside of the radical is called the radicand. The word radical is also used to describe any expression or function that uses a radical with variables in the radicand. In order to represent a root other than a square root, we use an index in the nook of the radical symbol. Whenever the index is greater than 2, the radical is considered as a higher root. Radical Functions LEARNING OBJECTIVES Recognize the relationship between radicals and rational exponents. Use radicals that model applications. Find the domain and range of a radical function in a context. Calculate square roots and higher roots. Simplify radical expressions. 8.1 DEFINITION Radical Expression !x ! n x A square root or nth root is called a radical expression. In these examples, x is the radicand and n is the index. Square roots have an index of 2, but the 2 is not written in the nook of the radical. ! 5 2x Rational Exponents x 1 n 5 ! n x Raising a base to a rational exponent with a denominator of n is the same as taking the nth root of the base. 8 1 3 5 ! 3 8 5 2 If x is negative, n must be odd. If x is positive, n can be any whole number greater than or equal to 2. Relationships between Radicals and Rational Exponents A rational exponent is another way of writing a radical such as a square root or cube root. !25 5 25 1 2 5 5 ! 3 27 5 27 1 3 5 3 The rational exponent 1 2 represents a square root, the rational exponent 1 3 represents a cube root, and so on. index radicand Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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  Intermediate Algebra

  • Mark D. Turner, Charles P. McKeague(Authors)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Chapter Outline 7.1 Roots and Radical Functions 7.2 Rational Exponents 7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radical Expressions 7.5 Multiplication and Division of Radical Expressions 7.6 Equations Involving Radicals 7.7 Complex Numbers 7 iStockphoto.com © trait2lumiere 489 E cology and conservation are topics that interest most college students. If our rivers and oceans are to be preserved for future generations, we need to work to eliminate pollution from our waters. If a river is flowing at 1 meter per second and a pollutant is entering the river at a constant rate, the shape of the pollution plume can often be modeled by the simple equation y = √ — x The following table and graph were produced from the equation. Distance from Width of Source (meters) Plume (meters) x y 0 0 1 1 4 2 9 3 16 4 Width of a Pollutant Plume 4 0 8 12 16 20 4 0 8 12 16 20 x y Distance from source (m) Width of plume (m) To visualize how the graph models the pollutant plume, imagine that the river is flowing from left to right, parallel to the x -axis, with the x -axis as one of its banks. The pollutant is entering the river from the bank at (0, 0). By modeling pollution with mathematics, we can use our knowledge of mathematics to help control and eliminate pollution. Roots and Rational Exponents 490 Success Skills If you have made it this far, then you have the study skills necessary to be successful in this course. Success skills are more general in nature and will help you with all your classes and ensure your success in college as well. Let's start with a question: Question: What quality is most important for success in any college course? Answer: Independence. You want to become an independent learner. We all know people like this. They are generally happy. They don't worry about getting the right instructor, or whether or not things work out every time.

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  Elementary and Intermediate Algebra

  • Mark D. Turner, Charles P. McKeague(Authors)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Chapter Outline 10.1 Roots and Radical Functions 10.2 Rational Exponents 10.3 Simplified Form for Radicals 10.4 Addition and Subtraction of Radical Expressions 10.5 Multiplication and Division of Radical Expressions 10.6 Equations Involving Radicals 10.7 Complex Numbers 10 iStockphoto.com © trait2lumiere 699 E cology and conservation are topics that interest most college students. If our rivers and oceans are to be preserved for future generations, we need to work to eliminate pollution from our waters. If a river is flowing at 1 meter per second and a pollutant is entering the river at a constant rate, the shape of the pollution plume can often be modeled by the simple equation y = √ — x The following table and graph were produced from the equation. Distance from Width of Source (meters) Plume (meters) x y 0 0 1 1 4 2 9 3 16 4 Width of a Pollutant Plume 4 0 8 12 16 20 4 0 8 12 16 20 x y Distance from source (m) Width of plume (m) To visualize how the graph models the pollutant plume, imagine that the river is flowing from left to right, parallel to the x -axis, with the x -axis as one of its banks. The pollutant is entering the river from the bank at (0, 0). By modeling pollution with mathematics, we can use our knowledge of mathematics to help control and eliminate pollution. Roots and Rational Exponents 700 Success Skills Think about the most successful people you have met or heard about. What are the qualities they tend to have in common? One of these qualities usually involves making a resolute commitment. If you are not firmly committed to something, then you will tend to give less than your full effort. Consider this quote from Faust by Johann Wolfgang Von Goethe: Until one is committed, there is hesitancy, the chance to draw back, always ineffectiveness.

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  Beginning Algebra

  Connecting Concepts through Applications

  • Mark Clark, Cynthia Anfinson(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  There are also excluded values for radical expressions, namely, those values that result in negative numbers under the square root. Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. C H A P T E R 8 R a d i c a l E x p r e s s i o n s a n d E q u a t i o n s 684 Evaluating Radical Equations Square roots often show up in applications and equations. A radical equation is an equation in which the variable is in the radicand (under the radical). DEFINITION Radical Equation A radical equation is an equation in which a variable expression is under the radical. For instance, ! x 2 6 5 2 is a radical equation. We now learn how to evaluate radical equations. Solving radical equations is covered in Section 8.4. Example 5 Using a formula to model the height of a falling object A science class tests how long it takes a ball to fall to the ground when dropped from various heights. The class determines that the time, T , in seconds that it takes the ball to fall to the ground after being dropped from a height of h feet is given by the equation T 5 0.243 ! h a. What units does h represent? What units does T represent? b. How long will it take the ball to fall to the ground when the ball is dropped from 15 feet? Round the answer to the hundredths place if necessary. c. How long will it take the ball to fall to the ground when the ball is dropped from 50 feet? Round the answer to the hundredths place if necessary. SOLUTION a. T , in seconds, is the time it takes the ball to fall to the ground. The variable h , in feet, is the initial height the ball is dropped from.

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  Elementary Algebra

  • Alan Tussy, R. Gustafson(Authors)
  • 2012(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  In fact, it is its own square root, because . A radical symbol represents the positive or principal square root of a positive number. When reading this symbol, we usually drop the word positive (or principal ) and simply say square root. Since 3 is the positive square root of 9, we can write represents the positive number whose square is 9. Read as “the square root of 9 is 3.” The symbol is used to represent the negative square root of a positive number. It is the opposite of the principal square root. Since is the negative square root of 144, we can write Read as “the negative square root of 144 is ” or “the opposite of the square root of 144 is .” The notation represents the negative number whose square is 144. If the number under the radical symbol is 0, we have The number or variable expression within (under) a radical symbol is called the radicand, and the radical symbol and radicand together are called a radical. Radical symbol Radicand Radical An algebraic expression containing a radical is called a radical expression. Some examples of radical expressions are To evaluate (find the value of) square roots, you need to quickly recognize each of the following natural-number perfect squares shown in red: 400 20 2 256 16 2 144 12 2 64 8 2 16 4 2 361 19 2 225 15 2 121 11 2 49 7 2 9 3 2 324 18 2 196 14 2 100 10 2 36 6 2 4 2 2 289 17 2 169 13 2 81 9 2 25 5 2 1 1 2 2 100 , 2 2 3, 2 x 2 , and B a 1 49 ⎫ ⎬ ⎭ 2 16 2 0 0. 2 144 12 12 2 144 12 12 1 2 9 2 9 3 1 0 2 0 12 3 8.1 An Introduction to Square Roots 611 The Language of Algebra The word radical comes from the Latin word radix, meaning root. The radical symbol has evolved over the years from in the 1300s, to in the 1500s, to the familiar in the 1600s. 1 If is a positive real number, 1. represents the positive or principal square root of . It is the positive number we square to get . 2. represents the negative square root of . It is the opposite of the principal square root of : .

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  eBook - ePub

  Making Sense of Mathematics for Teaching High School

  Understanding How to Use Functions

  • Edward C. Nolan, Juli K. Dixon, Farhsid Safi, Erhan Selcuk Haciomeroglu(Authors)
  • 2016(Publication Date)
  • Solution Tree Press

    (Publisher)

  2 simultaneously (such as circles, ellipses, and other conic sections).

  The Mathematics

  A focus on function types must include connections to parent functions. When students make sense of parent functions and transformations of those functions, students are able to make inferences regarding the behaviors of functions and link characteristics of different types of functions together. Knowing the general behavior of a function helps in determining if the application of that function to model a real-world context is a reasonable choice. High school students must be able to determine the impact of parameters on function behavior, make connections through representations, connect types of functions using symmetry, and explore rational functions.

  Determining the Impact of Parameters on Function Behavior

  Functions are often presented, explored, and analyzed as families. Families of functions involve grouping functions by specific parameters to provide a deeper understanding of how variations in these parameters impact the behavior of the graph of a function. For instance, in early high school coursework, students examined y = mx + b and discussed the role of parameters m and b on the behavior of the line. The learning progression extends this analysis to various function types including quadratic (see figure 4.4 ), exponential, and logarithmic functions.

  Figure 4.4: Transformations of quadratic functions task.

  Students explore the impact of changing different parameters of the quadratic family of functions presented in vertex form f(x) = a(x − h)2 + k. The impact of changing the numerical value of the parameter a affects how narrow or wide the graph will appear. In fact, the parameter a provides a sense of how quickly the quadratic function increases or decreases, identifying the rate of change of the quadratic function. Additionally, understanding the role of h and k as they relate to the vertex of the parabola will link to the study of axis of symmetry and locus of points, and extend to the study of other conic sections. In the case of f(x) = (x − 3)2 − 2, the parent graph of y = x2

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  Introductory Algebra

  Concepts and Graphs 2E

  • Charles P. McKeague(Author)
  • 2020(Publication Date)
  • XYZ Textbooks

    (Publisher)

  476 CHAPTER 8 Roots and Radicals Getting Ready for the Next Section Simplify. 73. √ — 4x 3 y 2 74. √ — 9x 2 y 3 75. 6 __ 2 √ — 16 76. 8 __ 4 √ — 9 77. √ — 2 _ √ — 4 78. √ — 6 _ √ — 9 79. 3 √ — 18 _ 3 √ — 9 80. 3 √ — 12 _ 3 √ — 8 Multiply. 81. √ — 2 _ √ — 3 ⋅ √ — 3 _ √ — 3 82. √ — y _ √ — 2 ⋅ √ — 2 _ √ — 2 83. 3 √ — 3 ⋅ 3 √ — 9 84. 3 √ — 4 ⋅ 3 √ — 2 477 Simplified Form of Radicals Radical expressions that are in simplified form are generally the easiest form to work with. A radical expression is in simplified form if it has three special characteristics. A radical expression is in simplified form if 1. There are no perfect squares that are factors of the quantity under the square root sign, no perfect cubes that are factors of the quantity under the cube root sign, and so on. We want as little as possible under the radical sign. 2. There are no fractions under the radical sign. 3. There are no radicals in the denominator. DEFINITION: SIMPLIFIED FORM A radical expression that has these three characteristics is said to be in sim- plified form. As we will see, simplified form is not always the least complicated expression. In many cases, the simplified expression looks more complicated than the original expression. The important thing about simplified form for radicals is that simplified expressions are easier to work with. A Properties of Radicals The tools we will use to put radical expressions into simplified form are the properties of radicals. We list the properties again for clarity. If a and b represent any two nonnegative real numbers, then it is always true that 1. √ — a √ — b = √ — a ⋅ b 2. √ — a _ √ — b = √ __ a __ b b ≠ 0 3. √ — a √ — a = ( √ — a ) 2 = a This property comes directly from the definition of radicals PROPERTY: PROPERTIES OF RADICALS The following examples illustrate how we put a radical expression into simplified form using the three properties of radicals. Although the properties are stated for square roots only, they hold for all roots.

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