Graphing square root functions. What would be the base graph for the square root function? What is the table of values?

Transcription

1 Unit 3 (Chapter 2) Radical Functions (Square Root Functions Sketch graphs of radical functions b appling translations, stretches and reflections to the graph of Analze transformations to identif the of radical functions. The graph of is related to the graph of = f () and the domain and range for each function are compared. finding approimate solutions to radical equations. (This was done algebraicall in Math 2200) Graphing square root functions What would be the base graph for the square root function? What is the table of values? Page 1

2 A) Sketch the following using B) State the mapping rule and the transformational form of the function C) Determine the domain and range 1. Vertical translation of -3, Horizontal translation of -2 (, ) (0, 0) (1, 1) (4, 2) (9, 3) 2. Vertical stretch 2, Horizontal translation of 1 left (, ) (0, 0) (1, 1) (4, 2) (9, 3) Page 2

3 ( 2) (, ) (0, 0) (1, 1) (4, 2) (9, 3) Page 3

4 4. Eplain how to transform the graph of to obtain Sketch the graph of each function. Then, identif the domain and range of each function Solution Begin b identifing the parameters and the effect each has on the base function. Parameter a =, resulting in a b a factor of. Since a is negative, the graph is reflected in the. Parameter b =, resulting in a b a factor of. Parameter h =, so the graph is translated b units. Parameter k =, so the graph is translated b units. Appl the transformations to sketch the graph of transformed function. (, ) The domain of the base function is { 0, R} and its range is { 0, R}. The domain and range of the transformed function are domain: range: Page 4

5 5. Eplain how to transform the graph of to obtain Sketch the graph of each function. Then, identif the domain and range of each function Solution Begin b identifing the parameters and the effect each has on the base function. Parameter a =, resulting in a b a factor of. Since a is negative, the graph is reflected in the. Parameter b =, resulting in a b a factor of. Parameter h =, so the graph is translated b units. Parameter k =, so the graph is translated b units. Appl the transformations to sketch the graph of transformed function. (, ) The domain of the base function is { 0, R} and its range is { 0, R}. The domain and range of the transformed function are domain: range: Page 5

6 6. Using the four points on the graph of the base function, complete the table to determine the resulting coordinates on the graph of domain: range: Page 6

7 Now that we have worked through a number of eamples, can ou see a pattern that would allow us to determine the domain and range of a radical epression such without creating an accurate graph. This can be done b eamining the The the domain and range of radical functions. We have a reflection in the. Horizontal Translation is. Thus the domain is Vertical translation is Thus the range is Eamples 1. Determine the domain and range of A) o Horizontal Translation is. Thus the domain is o We have a reflection in the. o Vertical translation is Thus the range is Page 7

8 B) o We have a reflection in the o Horizontal Translation is. Thus the domain is o We have a reflection in the. o Vertical translation is Thus the range is 2. Write the equation of a radical function with each domain and range. (Don t use stretch factors) A) Domain: 7 B) Domain: (,2] Range: [-8, ) Range: 2 Page , 3, 4, 5. 11, 13 Page 8

9 Section 2.2 Sketch the graph of the function, given the equation or graph of the function = f(), and eplain the strategies used. Compare the domain and range of the function, to the domain and range of the function = f(), and eplain wh the domains and ranges ma differ. Working Eample 1: Compare Graphs of a Linear Function and the Square Root of the Function a) Given f () = 4 3, graph the functions = f () and b) Compare the graphs. f ( ) Solution a) Determine the -value in the second column of the table. Then, complete the third column b taking the square root of the second column. Use the table of values to sketch the graphs of = f () and f ( ). (Hint: You could graph = f () on our graphing calculator and then use the table function to complete the second column of the table.) Page 9

10 b) From our table of values, determine the points of intersection: (, 0); (, 1 ) How is the -intercept of the graph of = 4-3 related to the graph of the function 4 3? o Wh are these points of intersection referred to as invariant points? o These points are the because when f() = 0, or when f() = 1, For which values of is the graph of 4 3 above the graph of =4-3? o o How are these values related to the invariant points? o Wh is the graph of 4 3 above = 4-3 between these points? For which values of is the graph of 4 3 below the graph of = 4-3? o To see a similar question, refer to Eample 1 on pages in Pre- Calculus 12. Page 10

11 Eample 2. Compare Graphs of a Quadratic Function and the Square Root of the Function Graph 2 1 Consider the graph of = 2 1 One method to produce the graph of 2 1is to first generate for = 2-1 from the given graph. Then, to graph take the. 2 1 f( ) f( ) For = f() state: Domain: Range: Page 11

12 Wh is the graph undefined from ( 1, 1)? Are there an invariant points? If so, what are the? f ( ) Where is the graph of above = f ()? Below = f ()? These results, along with other ke points, can then be used to help create the graph of when = f() is given. Page 12

13 Practice: 1. Given the graph of =f() sketch the graph of = f() A) Page 13

14 B) 2. What are all of the invariant points for the graphs of f() = and = f()? Page 14

15 f ( ) Another wa to sketch the graph of is to use equation = f() to generate. f ( ) From this, ou could then graph. The graph and table of values of = f() can also be generated with the use of. The graphs of = f() are limited to. Eample 1. Graph f ( ) A) f() = Page 15

16 B) f() = 4 2 C) f() = ( 1) 2 2 Page 16

17 In General: Tet Page 86-7: # 1, 2, 3, 8 Page 89 # 16, 17 Page 17

18 Page 18

19 Finding Domain and Range of Square Root Functions Consider: To find the domain simpl solve the inequalit: The range consists of the of all of the values in the of for which is defined. Eamples: Find the domain and range of the following: A) = B) = 2 5 Page 19

20 2 C) 2 8 D) = Page 87 5a) b) 6, 10, 11 Page 20

21 2.3 Solving Radical Equations Graphicall In Mathematics 2200, ou solved radical equations This ear we will solve radical equations Solving Radical Equations Algebraicall o ensuring that it does not contain o solutions that do not satisf the equation or when substituted in the equation. Eamples : Solve A) Page 21

22 B) C) D) Page 22

23 Solving Radical Equations Graphicall Method 1: Method 2: Graph the and find the of the function. Graph on the same grid, and find the Eample: Eample: Graph Graph: and Page 23

24 Graph and Eamples : Solve A) Page 24

25 B) C) Page 25

26 D) E) Page 26