6-1. Roots and Radical Expressions. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary. 1. Circle the exponents in each expression.
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1 6- Roots and Radical Expressions Vocabulary Review. Circle the exponents in each expression. 7 t a 7 b x y 0. Underline the expression that is read five to the second power.. List the correct numbers or letters described by the vocabulary words for the expression x r s 7w in the correct space. Exponents:,,, Coefficients:,,, 7 Bases: x, r, s, w Vocabulary Builder root (noun) root Related Words: square root, cube root, nth root, power, radical, index, radicand Definition: The nth root of a given number is a specific number that when it is used as a factor n times, equals the given number. Using Symbols: 8, so!8 Use Your Vocabulary.. Draw a line from the number or symbol in Column A to each term in Column B that best describes a part of!8. Column A Column B radical 8 root index radical n a radicand root! index radicand Chapter 6
2 If a n b, with a and b real numbers and n a positive integer, then a is an nth root of b. If n is odd N there is one real nth root of b, denoted in radical form as!b. n The only nth root of 0 is 0. Key Concept The n th Root. Underline the correct words to complete the sentence. If n is even N and b is positive, there are two real nth roots of b. The positive root is the principal root (or principal nth root). Its symbol is!b. n The negative root is its opposite, or!b. n Since the index of!8 is even / odd and the radicand is negative / positive, there are two real fourth roots / no real fourth roots of 8. and b is negative, there are no real nth roots of b. Problem Finding All Real Roots Got It? What are the real fifth roots of 0,, and? 6. Cross out the question that will NOT help you find the roots. What number is the index? Is the radicand negative or positive? What number times equals? How many real roots of 0 are there? 7. Complete the equation to find the fifth root of The fifth root of 0 is Complete the equation to find the fifth root of. 0. The fifth root of is.. Complete the equation to find the fifth root of.. The fifth root of is. Problem Finding Roots Got It? What is each real-number root?!7!8 "(7)!9 Lesson 6-
3 . Circle the true equation. () 7 () 7 () 7 Underline the correct words to complete each sentence.. The index of!8 is even / odd and the radicand is positive / negative, so there is one real root / are no real roots.. The index of!9 is even / odd, and the radicand is positive / negative, so there is one real root / are no real roots. 6. What is the principal square root of the square of a number? Answers may vary. Sample: The principal square root of the square of a number is the absolute value of the original number. 7. Simplify each root.!7!8 "(7)!9 no real roots 7 no real roots Key Concept n th Roots of n th Powers For any real number a, "a n n a if n is odd e ua u if n is even. 8. The index of "() is even / odd, so "(). Problem Problem Simplifying Radical Expressions Got It? What is the simplified form of the radical expression "8x? 9. Complete the equation to simplify the radical expression. "8x " 9 Q x R 9 x Using a Radical Expression Got It? Some teachers adjust test scores when a test is difficult. One teacher s formula for adjusting scores is A 0!R, where A is the adjusted score and R is the raw score. What are the adjusted scores for raw scores of 0 and 00? 0. Circle the first thing you should do to solve this problem. Evaluate A 0!R for R 0 and R 00. Solve A 0!R for R. Chapter 6 6
4 Underline the correct number to complete each sentence.. The lowest possible raw score is 0 / 0 / / 00. The lowest possible adjusted score is 0 / 0 / / 00.. The highest possible raw score is 0 / 0 / / 00. The highest possible adjusted score is 0 / 0 / / 00.. So, the curved scores for raw scores of 0 and 00 are 0 and 00. Lesson Check Do you UNDERSTAND? Error Analysis A student said the only fourth root of 6 is. Describe and correct his error.. If a is a fourth root of 6, which statement is true? a? a? a? a 6 a 6 a 6 a 6. Look at the statement you circled in Exercise. Circle the values of a below that make the statement true There are / / / values of a that satisfy the statement you circled in Exercise. Therefore, the number of fourth roots of 6 is / / /. 7. Describe the error the student made. Answers may vary. Sample: The student may have forgotten that when n is even and b is positive, there are two nth roots of b, one positive, and one negative. 8. Complete the sentence to correct the error the student made. The fourth root of 6 is or. Math Success Check off the vocabulary words that you understand. nth root principal root radicand index Rate how well you can find nth roots. Need to review Now I get it! 7 Lesson 6-
5 6- Multiplying and Dividing Radical Expressions Vocabulary Review Write T for true or F for false. T F T T. All mathematical expressions can be written as an equivalent expression with a denominator of.. An expression can have a denominator equal to zero.. The expression above the fraction bar is the numerator.. Multiplying both the numerator and the denominator by the same nonzero number results in an equivalent fraction. Circle the numerator and underline the denominator in each expression r 6 7. r s c 6 Vocabulary Builder combine (verb) kum BYN Main Idea: Combine means to put things together or to get a total. Math Usage: To combine means to put together or add two like terms to get one term. Example: The like terms x and 7x can be combined to get x. Use Your Vocabulary 8. Circle the expression that shows the like terms in x x combined. x x 7x x x Property Combining Radical Expressions: Products If n!a and n!b are real numbers, then n!a? n!b n!ab. Chapter 6 8
6 Write T for true or F for false. F 9. Q!RQ!6R Q!R T 0. Q!6R Q!RQ!9R T. Q!RQ!8R Q!R Problem Multiplying Radical Expressions Got It? Can you simplify the product of the radical expression? Explain.!7?!7. Circle the indexes in!?!!7?!7.. Circle the indexes in!?!.. The indexes above are the same / different.. The indexes above are the same / different. 6. Can you simplify the product? Yes / No 7. Can you simplify the product? Yes / No If yes, write the product. If yes, write the product.!0 Problem Simplifying a Radical Expression Got It? What is the simplest form of "8x 7? QHint: Remember "b b.r 8. The index of "8x 7 is / 7 / Circle the perfect cube factors. "8x 7 "?? x? x? x 0. The simplest form of "8x 7 is x!x. Problem Simplifying a Product Got It? What is the simplest form of "x y? "xy?. Complete the steps to simplify "x y? "xy. "x y? "xy "(x y )? Q xy R " Q? RQx y? xy R " Q9??? 7 RQx6? y 7 R " Q?? 7RQx R? Q y R? y?? x? y?!7y x y!7y 9 Lesson 6-
7 Property Combining Radical Expressions: Quotients If n!a and n!b are real numbers and b 0, then n!a n!b n Î a b.. Multiple Choice Which expression is equivalent to Q!RQx!R x 7 Å!7 "x? 7 Åx x Problem Dividing Radical Expressions Got It? What is the simplest form of Q"0x6 R Q"x R?. Complete the expression. Q"0x 6 R Q"x R 0x 6 Use the Combining Radicals Property for Quotients. Å x " x Simplify under the radical sign. u x u Simplify the square root. Problem Rationalizing the Denominator!7x Got It? What is the simplest form of "y. The radicand in the denominator needs a and a y to make y a perfect cube.. You will need to multiply both the numerator and the denominator by the expression Å to rationalize the denominator. 6. Complete to show the rationalization of the denominator.!7x!7x "y? "y y Å Å? Å? Å 7 7 Å Å 7xy y y y xy xy y y Rationalize the denominator. Multiply. Find the cube root of the denominator. Simplify. Chapter 6 60
8 7. Explain why both the numerator and the denominator are multiplied by the expression used to rationalize the denominator. Answers may vary. Sample: When a ratio has the same number in the numerator as in the denominator, its value is. When you multiply an expression by, the value of the expression does not change. Lesson Check Do you know HOW? Divide and simplify "x0 "7x. 8. Circle the first step in simplifying the fraction. Underline the second step. Combine radical Divide out common Rationalize the Simplify each expressions. factors. denominator. root. 9. Now divide and simplify. "x 0? "7x "? 7? 7? x "7x "7x 7x 7x7 "x 7x x "x Lesson Check Math Success Need to review Do you UNDERSTAND? Vocabulary Write the simplest form of "x. 0. Complete to simplify. "x "? x? x x " x Check off the vocabulary words that you understand. simplest form of a radical Rate how well you can multiply and divide radical expressions rationalize the denominator Now I get it! 6 Lesson 6-
9 6- Binomial Radical Expressions Vocabulary Review Circle the like terms in each group.. y y y. b bc bc c. 8 a Vocabulary Builder binomial (adjective) by NOH mee ul Definition: A binomial expression is an expression made up of two terms. Related Words: monomial, binomial expression, trinomial Examples: monomial: a, x,, 7c,! binomial: a 7, x 0.9, ab, 7c, b! trinomial: a 7 x, x x 0.9, ab a, 7c c, b b! Use Your Vocabulary Write M if the expression is a monomial, B if the expression is a binomial, or T if the expression is a trinomial. T B. 7 00x y r M. 7t 6 6. s 0.9r T 7. 8a.b 7.8c Property Combining Radical Expressions: Sums and Differences Use the Distributive Property to add or subtract like radicals. a n!x b n!x (a b) n!x a n!x b n!x (a b) n!x Chapter 6 6
10 8. Underline the words that make each sentence true. To be like radicals, their indexes must be the same / different, and their radicands must be the same / different. To add or subtract two like radicals, you add or subtract their radicands / coefficients. Problem Adding and Subtracting Radical Expressions Got It? What is the simplified form of each expression? 7!! x!xy x!xy 9. Are the radicals in 7!! 0. Are the radicals in x!xy x!xy like radicals? like radicals? Yes / No Yes / No. Is 7!! simplified?. Is x!xy x!xy simplified? Yes / No Yes / No. Write the simplified form of each expression. 7!! x!xy x!xy 7!! 7x!xy Problem Using Radical Expressions Got It? In the stained-glass window design, the side of each small square is 6 in. Find the perimeter of the window to the nearest tenth of an inch.. The length of the window is made up of the diagonals / sides of three squares.. The width of the window is made up of the diagonals / sides of two squares. 6. Multiple Choice Which is the length of the diagonal of a square with side s? s!s s! s! 7. Write the length of the diagonal of a square with side 6 in. 8. Complete the following to find the length and the width of the window. Length of the window: 6! Width of the window: / Q 6! R 8! w Q 6! R! 6 Lesson 6-
11 9. Complete the steps to find the perimeter of the window. Perimeter / w Q 8! R Q! R Substitute for length and width. 6!! Simplify. 60! Add the coefficients of the like radicals. 8.9 Use a calculator to approximate to the nearest tenth. Problem Simplifying Before Adding or Subtracting Got It? What is the simplified form of the expression!0!!6? 0. Complete each factor tree to factor each radicand Complete the following by substituting the prime factorizations for 0,, and 6.!0!!6 Problem " Multiplying Binomial Radical Expressions Got It? What is the product A!BA!B?. Circle the expression that shows the FOIL method. ". Circle the simplified form of the expression!0!!6.!!! 6!!! A!BA!B A!BA!B??!!?!?!??!!?!. The product A!BA!B 6 6!. "? Chapter 6 6
12 Problem Multiplying Conjugates Got It? What is the product of the expression A6!B A6!B?. Use the FOIL method to find the product. A6!BA6!B 6? 6 6?!!? 6 Q!R?! 6 Problem 6 Rationalizing the Denominator!7 Got It? How can you write the expression with a rationalized!! denominator? 6. Circle the conjugate of the denominator.!! /!!!7 7. Use the conjugate of the denominator to write!! with a rational denominator.!7!!!7!!?!!!!!7a!!b!!!! A!B A!B!! Lesson Check Do you UNDERSTAND? Vocabulary Determine whether each of the following is a pair of like radicals. If so, combine them. x! and x!0!xy and 7!xy!y and!6y 8. Cross out the pairs that do NOT have the same index and the same radicand. x! and x!0!xy and 7!xy!y and!6y 9. The sum of the like radicals is 9!xy. Math Success Check off the vocabulary words that you understand. like radicals Need to review binomial radical expressions Rate how well you can add and subtract radical expressions. Now I get it! 6 Lesson 6-
13 6- Rational Exponents Vocabulary Review Examples may vary. Samples are given.. Use the terms in the box to complete the diagram. Then, give examples. Irrational Numbers Integers Natural Numbers Rational Numbers Whole Numbers Real Number Sets Irrational Numbers Rational Numbers Example: Example: Integers Example: Whole Numbers Example: 0 Natural Numbers Vocabulary Builder convert (verb) kun VURT Related Words: conversion (noun), convertible (adjective) Definition: To convert is to rewrite or change to another form. Use Your Vocabulary. Circle the expression that shows converted to a decimal. Example: Chapter 6 66
14 Complete each sentence with the correct form of the word convert.. NOUN Travelers to Europe calculate the 9 of dollars to euros.. ADJECTIVE A 9 sofa is a couch by day and a bed by night.. VERB To 9 fractions to decimals, divide the numerator by the denominator. conversion convertible convert Problem Simplifying Expressions With Rational Exponents Got It? What is the simplified form of the expression 6? 6. Circle the expression that shows 6 rewritten as a radical. (Hint: x n n!x)!6 7. The simplified form of 6 is 8.!6!6 Key Concept Rational Exponent If the nth root of a is a real number and m is an integer, then an!a n m and a n "a n m (!a n ) m. If m is negative, a Draw a line from each expression in Column A to its exponent form in Column B. Column A Column B!x x "x x Problem Converting Between Exponential and Radical Form Got It? What are the expressions w 8 and w 0. in radical form? 9. Is the exponent in w 8 in lowest terms? Yes / No 0. Identify the values of a, m, and n for w 8 using the rule a m n "a n m. a w m n. Use your values for a, m, and n to write. Convert w 0. to w raised to a power that is w 8 in radical form. a fraction in lowest terms. 8 "w w. Identify the values of a, m, and n for w 0. using the rule a m n n "a m. a w m n 8. Use your values for a, m, and n to write w 0. in radical form.!w 67 Lesson 6-
15 Problem Using Rational Exponents Got It? Kepler s third law of orbital motion states that you can approximate the period P (in Earth years) it takes a planet to complete one orbit of the sun using the function P d, where d is the distance (in astronomical units, AU) from the planet to the sun. Find the approximate length (in Earth years) of a Venusian year if Venus is 0.7 AU from the sun.. Complete the problem-solving model below. Know Need Plan the distance from Venus to the sun and the formula for the length of a planet year the length of a Venusian year in Earth years Use the formula to find the period. 6. Use the formula to find the length of a Venusian year. P d P (0.7) P N Key Concept Properties of Rational Exponents Let m and n represent rational numbers. Assume that no denominator equals 0. a m? a n a mn (a m ) n a mn (ab) m a m b m a m a m Complete each example below. Problem Combining Radicals With Like Radicands Got It? What is!(!) in simplest form?. Convert!(!) to exponential form. a m a n a mn Q a b Rm am b m 7. g? g g 7 g 8. (h ) h? h 9. (y)? y d 8 d d 8 d. Q R 6!(!)?. The bases of the factors are the same / different, so the Property you should use to 9 6 simplify the exponential form is (ab) m a m b m > a m? a n a mn.. In simplest form,!(!). Chapter 6 68
16 Problem Simplifying Numbers With Rational Exponents Got It? What is in simplest form? 6. Solve using two different methods. Complete each method. Method Method ( ) ()? Q" R () Q R 8 8 Lesson Check Do you UNDERSTAND? Error Analysis Explain why this simplification is incorrect. 7. The radical form of is!. The radical form of ( ) is!. The radical form of is!. 8. Explain why the simplification shown is incorrect. Math Success Need to review Now I get it! ( ) () () 0 Answers may vary. Sample: The student incorrectly simplified ( ). The answer should be 0!. Check off the vocabulary words that you understand. rational exponent exponential form radical form Rate how well you can simplify expressions with rational exponents. 69 Lesson 6-
17 6- Solving Square Root and Other Radical Equations Vocabulary Review. Circle the equation that is equivalent to!x.!x 0!x 6!x 8. Draw a line from each equation in Column A to its solution in Column B. Column A Column B x 0 y 6 0 z 0 w Circle the radicand in each expression..!x. 7!wz.!y 7 Vocabulary Builder reciprocal (noun) rih SIP ruh kul Related Term: multiplicative inverse Definition: Two numbers are reciprocals if their product is. Main Idea: To write the reciprocal of a fraction, switch the numerator and the denominator. Examples: and, 7 7 and, 6 and 6 Use Your Vocabulary Write T for true or F for false. T T 6. The reciprocal of is itself. 8. The reciprocal of a negative number is negative. F T Reciprocals a and b b a where a 0 and b 0 7. A decimal has no reciprocal. 9. The only real number without a reciprocal is 0. Chapter 6 70
18 Solving a square root equation may require that you square each side of the equation. This can introduce extraneous solutions. Problem Solving a Square Root Equation Got It? What is the solution of!x 0? 0. Circle the first step in solving the equation. Isolate the square root. Square each side.. Underline the correct word to complete each justification.!x Isolate the square root / variable. x x Take the square root / square of each side to remove the radical. Subtract to isolate the radical / variable term. x 6 Divide / Multiply by to solve for x. Problem Solving Other Radical Equations Got It? What are the solutions of (x ) 8?. Complete each step to find the solution. Isolate the radical term. (x ) 8 (x ) Raise each side of the equation to the reciprocal power. (x ) Solve for x. x 8 x 8 x or x Since the numerator of is even, (x ) x. 7 Lesson 6-
19 Problem Checking for Extraneous Solutions Got It? What is the solution of!x x? Check your results.. Use the justifications at the right to complete each step.!x x Write the original equation.!x x Isolate the radical. (!x ) (x ) Square each side of the equation. x x 6 x 9 Simplify. 0 x x 0 Combine like terms. 0 Qx RQx 0 R Factor. x or x 0 Use the Zero-Product Property.. Substitute each value into the original equation to check the solutions.!x x " Q R 0!x x " Q 0 R 0 0 " 0 " false 0 0. The solution is extraneous. 6. Multiple Choice What can cause an extraneous solution? raising each side of the equation to an odd power raising each side of the equation to an even power adding the same number to each side of an equation dividing each side of an equation by the same number 7. When should you check for extraneous solutions? Explain. You should check for an extraneous solution any time you raise both sides of an equation to an even power. Problem Solving an Equation With Two Radicals Got It? What is the solution of!x!x? Chapter 6 7
20 8. The equation has been solved below. Write the letter of the reason that justifies each step. Use the reasons in the box.!x!x!x!x x (!x ) x x 8!x 6 x 8!x x!x (x ) x x 6x 9 x x 0x 9 0 (x 9)(x ) x 9 or x 9. Only the solution x > x 9 satisfies the original equation. F A B E A C B E A D G A Addition Property of Equality B Square each side. C Division Property of Equality D Factor E Simplify. F Original equation G Zero Product Property Lesson Check Do you UNDERSTAND? Vocabulary Which value, or, is an extraneous solution of x 6!x? 0. The solution x satisfies / does not satisfy the original equation.. The solution x satisfies / does not satisfy the original equation.. The solution x / x is an extraneous solution of x 6!x. Math Success Check off the vocabulary words that you understand. radical equation Need to review square root equation Rate how well you can solve square root and other radical equations Now I get it! 7 Lesson 6-
21 6-6 Function Operations Vocabulary Review. In function notation, gx / g(x) / x(g) is read g of x.. Circle the equation that shows a function rule. x 7y.7 f (x) x 0. z(t). The function rule f (t).8t represents the cost of a number of tons of wheat t. The number of tons of wheat is the input / output. The output is the cost of the wheat / number of tons of wheat. Vocabulary Builder composite (adjective) kum PAHZ it Related Words: composite function, composite number Main Idea: Something that is composite is made up of more than one thing. Math Usage: A composite function uses the output of one function as the input of a second function. Use Your Vocabulary Domain of f Input of f, x Complete each sentence with the correct form of the word composite. composite composition compose. VERB The musician worked to 9 a new piece of music.. ADJECTIVE A 9 number has more than two factors. Range of f Output of f, f(x) Input of g, f(x) Domain of g Output of g, g(f(x)) Range of g compose composite 6. NOUN The poster was a 9 of photos and famous quotes. composition Chapter 6 7
22 Key Concept Function Operations Addition (f g)(x) f (x) g(x) Subtraction (f g)(x) f (x) g(x) Multiplication (f? g)(x) f (x)? g(x) Division Q f g R(x) f (x), g(x) 0 g(x) The domains of the sum, difference, product, and quotient functions consist of the x-values that are in the domains of both f and g. Also, the domain of the quotient function does not contain any x-value for which g(x) Let f (x) x and g(x) x 6. Complete the steps for finding f (x) g(x). f (x) g(x) (x ) (x 6) x x Problem Adding and Subtracting Functions Got It? Let f (x) x 8 and g(x) x. What are f g and f g? What are their domains? 8. Circle f g. 9. Circle f g. x x 8 x x x x 8 x x x x x x 0. Write the domain of f.. Write the domain of g. all real numbers all real numbers. The domain of both f g and f g is all real numbers. Problem Multiplying and Dividing Functions Got It? Let f (x) x x and g(x) x. What are f? g and f g and their domains?. Circle f? g. x x x. Circle f g. x x x Ax x BAx B Ax x BAx B x x x x x x. Simplify the product. 6. Simplify the quotient. 9x 0x x x 7. For what values of x does g(x) 0? 7 Lesson 6-6
23 8. Write the domain of f? g. 9. Write the domain of f g. all real numbers all real numbers except x Key Concept Composition of Functions The composition of function g with function f is written g + f and is defined as (g + f )(x) g( f (x)). The domain of g + f consists of the x-values in the domain of f for which f (x) is in the domain of g. (g + f )(x) g( f (x)). Evaluate f (x) first.. Then use f (x) as the input for g. Function composition is not commutative, since f (g(x)) does not always equal g( f (x)). If f (x) x and g(x) x, find each composition. 0. g + f g(x ) x. f + g f (x ) (x ) x 6 x 9 Problem Composing Functions Got It? Let f (x) x and g(x) x. What is ( f + g)()? Method. Find g().. Use g() as the input for f (x). ( f + g)() f(g()) f Q 9 R 9 Method. Simplify ( f + g)(x) f (g(x)).. Use the rule for f (g(x)) to find f (g()). f (g(x)) f Q R (f + g)() () Problem x x Using Composite Functions Got It? A store is offering a % discount on all items. Also, employees get a 0% employee discount. Write composite functions. Model taking the % discount and then the 0% discount. Model taking the 0% discount and then the % discount. 9 Chapter 6 76
24 6. Let x be the price of an item. Write functions to model each discount. C(x) cost using the storewide discount x 0. x 0.8 x D(x) cost using the employee discount x 0. x 0.8 x 7. Draw a line from each composition to its rule. (D + C)(x) 0.8(0.8x) (C + D)(x) 0.8(0.8x) 8. Simplify each function. (D + C)(x) 0.68x (C + D)(x) 0.68x 9. If you were an employee, which discount would you take first? Why? Either discount can be taken first, since in both cases the result is that you pay 68% of the original price. Lesson Check Do you UNDERSTAND? Open-Ended Find two functions f and g such that f (g(x)) x for all real numbers x. 0. If g(x) x, write a function f (x). If g(x) x, write a function f (x) to give f (g(x)) x. to give f (g(x)) x. f(x) x f(x) x. Write a function g(x). Then, find f (x) such that f (g(x)) x. Math Success Check off the vocabulary words that you understand. composite function Need to review Answers will vary. The functions should be inverses of one another function operations Rate how well you can find the composition of two functions. Now I get it! 77 Lesson 6-6
25 6-7 Inverse Relations and Functions Vocabulary Review. Underline the correct term to complete each sentence. The domain of a relation is the set of inputs, also called the x- / y- coordinates of the ordered pair. The range of a relation is the set of outputs, also called the x- / y- coordinates of the ordered pair.. Use the relation {(, ), (6, 7), (, 0), (8, ), (, 7)}. Write the domain and range of the relation. Domain Range {,, 6, 8, } {,, 7, 0} Vocabulary Builder inverse (noun) in VURS Related Words: opposite, reverse Math Usage: The inverse of a function is found by reversing the order of the elements in the ordered pairs. Example: The inverse of the function {(, ), (, ), (, 6)} is {(, ), (, ), (6, )}. Use Your Vocabulary. Complete the diagram below. Use relation r {(0, ), (, ), (, ), (8, )}. Relation r Domain Range 0,,, 8, Inverse of Relation r Domain Range, 0,,, 8 Chapter 6 78
26 Problem Finding the Inverse of a Relation Got It? What are the graphs of t and its inverse?. Complete the table of values for the inverse of relation t. Relation t x y 0 Inverse of Relation t x y 0. Plot the points from the Relation t table and from the Inverse of Relation t table. Relation t y O Problem Finding an Equation for the Inverse Got It? What is the inverse of y x 8? 6. Switch the x and y values in the function. function inverse x y x 8 y 8 x Inverse of Relation t y O x 79 Lesson 6-7
27 7. Solve the inverse equation for y. x y 8 x 8 y x 8 y Problem Graphing a Relation and Its Inverse Got It? What are the graphs of y x 8 and its inverse? 8. Complete the table for y x Complete the table for the inverse of y x 8. x 6 0 x 0 8 y 0 8 y Plot and draw a line through the points from the y x 8 table.. On the same grid, plot and draw a line through the points from the inverse of y x 8 table.. Draw a dashed line to show the line that reflects the equation y x 8 to its inverse. Problem Finding the Inverse of a Formula Got It? The function d v 9.6 relates the distance d, in meters, that an object has fallen to its velocity v, in meters per second. Find the inverse of this function. What is the velocity of the cliff diver in meters per second as he enters the water?. Solve the function for v. d v d v m!9.6d v y x 8. Let d meters. Write the value of the velocity v, to the nearest hundredth meter per second, of the diver as he enters the water. 8 8 O 8 8 y meters y x y x 8 x.69 Chapter 6 80
28 Key Concept Composition of Inverse Functions If f and f are inverse functions, then Q f + f R(x) x and Q f + f R(x) x for x in the domains of f and f, respectively. Problem 6 Composing Inverse Functions Got It? Let g(x) x. What is g (x)?. Complete each step. x y Write the inverse function. xqy R Multiply. (y ) x Divide. y x Subtract. 6. So, g (x) x. Lesson Check Do you UNDERSTAND? Reasoning A function consists of the pairs (, ), (x, ), and (, 6). What values, if any, may x not assume? 7. Each x-value in the domain of a function corresponds to exactly one y-value / many y-values in the range. 8. What values, if any, may x not assume? Answers may vary. Sample: x cannot equal or. Math Success Check off the vocabulary words that you understand. inverse relation inverse function one-to-one function Rate how well you can find the inverse of a relation or function. Need to review Now I get it! 8 Lesson 6-7
29 6-8 Graphing Radical Functions Vocabulary Review Write T if the figures show a translation or N if they do NOT show a translation.. T. N. N Vocabulary Builder vertical (adjective) VUR tih kul Related Word: horizontal Main Idea: A vertical line is straight up and down. A vertical shift moves something up or down. Math Usage: A vertical translation moves the graph of a relation parallel to the y-axis. Use Your Vocabulary Underline the correct word to complete each sentence.. A helicopter takes off vertically / horizontally.. A package on a flat conveyor belt moves vertically / horizontally. 6. Stepping side-to-side is a vertical / horizontal movement. vertical horizontal Chapter 6 8
30 Key Concepts Families of Radical Functions Square Root Radical Parent function y!x y!x n Reflection in x-axis y!x y!x n Stretch (a. ), shrink (0, a, ) by factor a y a!x y a!x n Translation: horizontal by h, vertical by k y!x h k y!x n h k Problem Translating a Square Root Function Vertically Got It? What are the graphs of y!x and y!x? 7. The graph of y!x is a horizontal / vertical translation of y!x up / down / left / right units. 8. What does the translation y!x look like? Answers may vary. Sample: It is a vertical translation of y!x. It is shifted down units. 9. Draw the graph of the function y!x. Then, use that graph to draw the graphs of y!x and y!x on the same grid. y O Problem Translating a Square Root Function Horizontally Got It? What are the graphs of y!x and y!x? 0. The graph of y!x is a horizontal / vertical translation of y!x up / down / left / right units.. What does the translation y!x look like? It is a horizontal translation of y!x. It is shifted units left. x 8 Lesson 6-8
31 . Draw the graph of the function y!x. Then, use that graph to draw the graphs of y!x and y!x on the same grid. y O x Problem Graphing a Square Root Function Got It? What is the graph of y!x?. Complete the table.. Multiplying the y-coordinates of x x 6 y!x by shrinks / stretches the graph.. Explain how the graph of y!x relates to the graph of y!x. x 6 Answers may vary. Sample: It is a translation of y!x that is units left and units down, and stretched by a factor of. Problem Graphing a Cube Root Function Got It? What is the graph of y!x? 6. Write y!x in standard form. y!x Underline the correct numbers or words to complete each sentence. 7. The shift is / units left / right and / units up / down. 8. There is a shrink / stretch by a factor of / and the result is / is not reflected. 9. Circle the graph of y!x. x y x y 6 y x x y Chapter 6 8
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