Graphing Radical Functions

Transcription

1 17 LESSON Graphing Radical Functions Basic Graphs of Radical Functions UNDERSTAND The parent radical function, 5, is shown The function takes the principal, or positive, square root of. So, the -values will never be negative. The range is $ 0. The square root of a negative number is not a real number, so the -values are also nonnegative. The domain is $ 0. The graph resembles one arm of a parabola, flipped on its side. UNDERSTAND Recall that a graph can be reflected over the line 5 b swapping and in the equation and solving for. Tr this with the parent quadratic function, 5. Its graph is shown on the right Swap and. 5 5 Solve for. Take the square root of both sides. The result is two separate equations: 5 and 5. Both graphs are shown on the right. Together, the graphs make up the reflection of 5 over the line Duplicating this page is prohibited b law. 01 Triumph Learning, LLC 1 Unit 1: Polnomial, Rational, and Radical Relationships

2 Connect Graph 5, and compare it to the graph of 5. 1 Create a table of values Remember that ou can take the cube root of a negative number. Find coordinate pairs for negative values of as well. Graph 5 using the - and -values from the table Add the graph of 5 on the same coordinate plane, and compare. Duplicating this page is prohibited b law. 01 Triumph Learning, LLC DISCUSS Find and compare the domain and range for the portion of each graph that is shown on the coordinate plane. What do ou notice? The dashed line shows 5. 0 The graphs are reflections of each other over the line 5. Lesson 17: Graphing Radical Functions 15

3 Transforming Basic Radical Equations UNDERSTAND Adding a constant to the radical epression in a radical equation moves the equation s graph up or down. The graph of 5 1 k is a vertical translation of 5 b k units. If k is positive, it is a translation up. If k is negative, it is a translation down. On the other hand, adding or subtracting a constant to the radicand moves the equation s graph left or right. The graph of 5 h is a horizontal translation of 5 b h units. If h is negative, it is a translation to the left. If h is positive, it is a translation to the right. The coordinate plane on the right shows 5 1, a translation of the parent function three units up. It also shows 5 1, a translation of the parent function three units left. 0 UNDERSTAND Negating the radical epression in a radical equation reflects the equation s graph across the -ais. But negating the radicand in the equation reflects the graph across the -ais. The coordinate plane on the right shows 5, a reflection of the parent graph across the -ais. It also shows 5, a reflection of the parent graph across the -ais. 0 UNDERSTAND Multipling the radical epression in a radical equation b a constant stretches or shrinks the graph verticall. Conversel, multipling the radicand in a radical equation b a constant stretches or shrinks the graph horizontall. The coordinate plane on the right shows 5, a vertical stretch of the parent graph. It also shows 5, a horizontal shrink of the parent graph Duplicating this page is prohibited b law. 01 Triumph Learning, LLC 1 Unit 1: Polnomial, Rational, and Radical Relationships

4 Connect Graph the function f () Determine the parent function and how this function differs. This is a radical function. The parent function is 5. The radical epression has a coefficient of, and 1 is subtracted from the radicand. Think of the coefficient,, as two separate transformations: a multiplication b followed b a multiplication b 1. Alwas start with stretches or shrinks. Start with the parent graph. Multipling the radical epression b stretches it verticall b. To stretch b, multipl each -value of the parent graph b. Multipling the radical epression b 1 reflects it over the -ais. Reflect the stretched graph over the -ais. Duplicating this page is prohibited b law. 01 Triumph Learning, LLC Perform the final transformation. Since 1 is subtracted from the radicand, it results in a translation of 1 unit right. Translate the last graph from the previous step to the right 1 unit CHECK 0 Stretch b 10 Reflect over -ais Identif some ordered pairs on the final graph, and check to make sure the satisf the function. Lesson 17: Graphing Radical Functions 17

5 EXAMPLE A Graph the function g() Determine the parent function and how this function differs. This is a radical function. The parent function is 5. The radicand is multiplied b 1, then is added to the radical epression. Perform the first transformation. Start with the parent graph. Multipling the radicand b 1 reflects it over the -ais. Perform the second transformation. Adding to the radical epression translates the graph units up. Translate the reflected graph up b units. Reflect over -ais 0 0 TRY Use the graph to identif the zeros of the function. Duplicating this page is prohibited b law. 01 Triumph Learning, LLC 1 Unit 1: Polnomial, Rational, and Radical Relationships

6 EXAMPLE B Solve the equation algebraicall and graphicall: 5 1 Solve algebraicall. Square both sides. ( ) 5 ( ) ( )( 1) 5 and 5 1 Check both solutions. Test: 5 5 Test: The solution 5 1 is etraneous. The onl solution is 5. Solve graphicall. Graph 5 and 5 on the same coordinate plane, and find the point(s) of intersection Duplicating this page is prohibited b law. 01 Triumph Learning, LLC Analze the results. The graphs intersect at the point (, ). Since the original equation is in one variable, the -value of the point of intersection is the solution. This is the same solution found algebraicall. The solution to the equation is 5. DISCUSS Add the graph of 5 onto the coordinate plane above. How does the etraneous solution occur? Lesson 17: Graphing Radical Functions 19

7 Practice For each graph, determine the zeros and the domain of the function zero(s): zero(s): domain: domain: Graph each function on the coordinate plane.. f () 5. g() REMEMBER Start with the parent function. HINT The constant is subtracted in the radicand. Duplicating this page is prohibited b law. 01 Triumph Learning, LLC 150 Unit 1: Polnomial, Rational, and Radical Relationships

8 Graph each function on the coordinate plane. 5. p() 5. q() Choose the best answer. 7. The graph of a function is shown.. The graph of a function is shown. 0 0 Duplicating this page is prohibited b law. 01 Triumph Learning, LLC Which of the following is not true? A. The parent function is 5. B. The radicand has been multiplied b 1. C. The radical epression has been multiplied b 1. D. The radical epression has been increased b. Which of the following is true? A. The epression under the radical has been increased b. B. The epression under the radical has been decreased b. C. The rational epression has been increased b. D. The rational epression has been decreased b. Lesson 17: Graphing Radical Functions 151

9 Solve. 9. For the function f () 5 1, identif the parent function. Then, describe the transformation made to the parent graph in order to graph f (). Graph f () Solve the equation algebraicall and graphicall: 5. Solve algebraicall. Show our work. Graph 5 and 5 on the coordinate plane. Name the points of intersection: Identif the solution(s) to the equation: 0 Duplicating this page is prohibited b law. 01 Triumph Learning, LLC 15 Unit 1: Polnomial, Rational, and Radical Relationships

10 11. COMPARE The function s() is shown on the coordinate plane. 0 Compare the function s() with the function t() 5. Discuss both the equations and graphs of the functions. Compare the domain and range of the functions. 1. EXAMINE A cube has a volume of cubic units. Write a function that describes the side length of the cube in units. f () 5 Identif the domain of the function. Think carefull about the contet. Eplain. Duplicating this page is prohibited b law. 01 Triumph Learning, LLC 1. CRITIQUE Malik wants to graph the radical equation 5 1 b performing transformations on the graph of the parent radical function, 5. He notices the variable is negative in the radicand, so he reflects the parent graph over the -ais. Then he translates the reflected parent graph units to the left since is added to the radicand. Critique Malik s approach. Lesson 17: Graphing Radical Functions 15