2. Domain: The set of all abscissas (x s) of the ordered pairs (abscissa is the first element of an ordered pair)

Transcription

1 . Relations and Functions. Relation: A set of ordered pairs E:,4,,5,,, 8,4. The set of all abscissas s of the ordered pairs abscissa is the first element of an ordered pair. Range: The set of all ordinates s of the ordered pairs ordinate is the second element of an ordered pair 4. Function: A relation in which each element in the domain is paired with eactl one element in the range. 5. Given,,,8,,, 5, state the domain and range. Is this relation is a function? 6. If is a negative integer greater than -4, state the relation representing the equation 5 Then state the domain and range. 7. Given is an integer and 9, state the ordered pairs from 86. Is this a function? 8. Spec one number in the table ou could change so that the relation would NOT represent a function ,000,000 9,700,000 8,000,000 6,700,000 9,700, Vertical Line Test: If a vertical line passes through a graph more than once, the graph is not the graph of a function. 0. Determine the graph of each relation is the set of a function a. b. c. Yes/ No Yes/ No Yes/ No

2 . Function Notation: f, which is read f of. Interpreted as the value of the function f at. f indicated that for each element in the domain that replaces, the functions assigns one and onl one replacement for. Coordinates are, or, f.. Find f f. Find f f 4. Factor each of the following epression. a. 7 8 b. n n 4 c. m 7m 0 d. 4m m e. w 8w 5. To find domain for a function, it is easiest to start b finding values that are not in the domain. Once ou have found the values that are not in the domain then ou can eclude those values when stating the domain. 6. State the domain in interval notation. a. f b. f c. f 5 d. f e. f. g.

3 7. General Rules for finding a. Linear function b. Quadratic Functions c. Square Root Functions d. Rationals fractions with variables in the denominator 7. Greatest Integer Function also known as the birthda function A tpe of step function. The smbol means the integer Not Greater than. ROUND DOWN!!!!! 8. Fill in the table and graph for Evaluate each for and = b. = c. = d. e. = f. = g. = h. Homework: Function Wksht & Domain Wksht

4 . Function Worksheet Da Homework I. State the domain and range each relation. Then state whether the relation is a function. Write es or no..,,,0,,0,,. 0,,,6,,,,.,4,,8,,4 4.,,,54,, 6,,8 II. Given that is an integer, state the relation representing each of the following b listing a set of ordered pairs. Then state whether the relation is a function. Write es or no. 5. 5and and 7. 4 and 0 8. and 0 III. The smbol means the greatest integer not greater than. If f, find each value. 9. f0 0. f0.. f-.9. f VI. Given f 4 5, find each value.. f 4. f f f5d V. Find the domain of each function. Put each answer in interval notation. 7. f 8. f 9. f f. 6 5 f. 5 7 f

5 Dom Wksht: Find the domain of the following graphs and equations. Write our answer in interval notation Function? Function? Function? Function? Function? Function? Function? Function? Function? 0. f 8. f. 4 h. 4 9 g

6 . Families of Graphs:. Reflection Flips a figure over a line called the ais of smmetr.. Linear Translation Relocates a graph on the coordinate plane but does not change the shape or size.. Geometric Transformation: Sometimes called dilations. 4. Summar of Translations and Transformations If the equation of is change to: Then the graph of is: f Reflected in the -ais f Reflected in the -ais cf, c Stretched verticall b a factor of c cf, 0 c Shrunk verticall b a factor of c f h Translated h units horizontall f k k Translated k units verticall f c, c Shrunk horizontall b a factor of c f c, 0 c Stretch horizontall b a factor of c 5. When listing translations/transformation, ORDER MATTERS!!!!! 6. List the transformations, in order, that have occurred when compared to the parent graph. a. g b. f c. h d. n e. m f. s 5 g. t h. h 8 6 i. a j. g 6 k. l. f h

7 7. Use the graph of the function f to sketch the graph of the given function g. State the domain and range. f a. g f b. g f c. g f d. g f e. g f f. g f g. g f h. g f i. g f

8 AFM Unit Da notes Parent Graphs a. b. c. d. or e. f. 8. Graph each function. Then describe how each graph is related to the parent graph. Then state the domain and range. a. f b. g c. h Etra Eamples: On Board Homework: Transformation Worksheet

9 . Graphing Transformations Da Worksheet Homework I. Graph each function as a transformation of its parent graph.. f. f. p 4. p 5. h. 6. h 7. h 8. h 9. g 4 0. g. g. g 6

10 II. Write the transformations, in order, for each function when compared to their parent graph.. f 4 4. f f f 7. f 5 III. Evaluate each. 8. [6.] 9. [0.7] 0. [-0.5]. [6] +. [-.] 5. [.]. 4. [7.9] 7 5. [-8.]

11 . Da Notes 9. Graph each transformation of a. b. c. d. e. f. g. h. + i.

12 0. Use the graph of the function f to sketch the graph of the given function g. Then state the domain and range. f b. g f b. g f c. g f d. g f e. g f f. g f Write our own equation for g and graph below: Homework: Graphing Practice Da

13 . Transformation Graphing Practice Da Graph and list the transformations. Da Homework. f

14

15 .4 Rational Functions Da 4. Rational Function g The equation of two polnomials, where f 0 Parent Rational Function is. f Definition of a Vertical and Horizontal Asmptotes a. The line = a is a vertical asmptote of the graph of f f or f as a, either right or from the left. b. The line is a horizontal asmptote of the graph of f f b as or.. Holes: Occur when there is a common FACTOR in the numerator and denominator. To find the eact location of the hole, set the common factor equal to zero, then take that value and substitute it back into the SIMPLIFIED rational equation. E: h therefore the hole is located at, f which is,. 6. Vertical Asmptotes: these occur at the -value that makes the denominator equal to zero. E: E: 5 g has a vertical asmptote at =. f has two vertical asmptotes one at = 4 and one at = Horizontal Asmptotes: A rational function has at most one horizontal asmptote. Three conditions: Given f = p q = b m a n m n b a.. a n 0 m m... b 0. n > m then f has no horizontal asmptote it ma have a slant asmptote 4 E. p n a. n = m then f has a horizontal asmptote at = m b leading coefficients 4 E: g horizontal asmptote at =

16 . n < m then = 0 is the horizontal asmptote E: f horizontal asmptote at = 0. 6 Slant Asmptote: The are eactl how the sound. *Neither vertical or horizontal!! These occur onl the degree of the numerator is eactl one more than the degree of the denominator. E. Need to do snthetic or long division to find it. 4 p E: w 4. Determine the vertical and horizontal asmptotes of the function, include slant, one. Also list an holes, an. Then state the domain. 5 a. f b. f c. f intercept: -intercept: -intercept: -intercept: -intercept: -intercept: d. g e g f. 6 f intercept: -intercept: -intercept: -intercept: -intercept: -intercept: g. p h. g i. 9 g -intercept: -intercept: -intercept: -intercept: -intercept: -intercept: Homework Rational Worksheet

17 Rational Worksheet Da 4 Homework I. List all the horizontal, vertical and slant asmptotes, holes, and state the domain.. f. 4 h. 5 g 4 -intercept: -intercept: -intercept: -intercept: -intercept: -intercept: 4. f 5. p 6. p 4 -intercept: -intercept: -intercept: -intercept: -intercept: -intercept: 7. 4 g 8. f h -intercept: -intercept: -intercept: -intercept: -intercept: -intercept: 0. f g. 6 f -intercept: -intercept: -intercept: -intercept: -intercept: -intercept:

18 .4 Rational Notes Da 5 5. Use the parent graph f to graph each equation. Describe transformations that have taken place. Then list the domain and range. 4 a. g b. f c. k d. f 6. Write an equation of a function that has a vertical asmptote at and a hole at 7. When graphing a rational function that is not a standard transformation of then there are some steps to take to make it easier to graph, without using a graphing calculator. a. Factor and simpl possible. Look for an Holes! b. Ident an Vertical, Horizontal, Slant asmptotes. c. Find the -intercept, one d. Find the -intercepts, an. e. Start to sketch the function with the information ou have! 8. Graph each function. a. f

19 b. f 5 7 c. f Homework: Da Rational Function Worksheet

20 Rational Functions Da 5 Worksheet. Describe the transformations that have taken place when compared to a the parent graph of f then graph each one. a. f b. g c. p 4 5. Create a function of the form = f that satisfies each set of conditions. a. Vertical asmptotes at hole at b. Vertical asmptotes at and hole at c. Holes at and, resembles Graph each function. 5. g 4. g

21 5. g 6. f 7. g 8. 5 f 9. g 4 0. f -intercept: -intercept: -intercept: -intercept: -intercept: -intercept: -intercept: -intercept: -intercept: -intercept: -intercept: -intercept:

22 Piecewise Functions: Da 6 Notes. Piecewise Functions- A functions that is defined b two or more equations over a specied domain.. Graph each of the following, given the domain restrictions. List the domain and range in interval notation. a., b., 0 c., d.,. Evaluate each piecewise function at the given values of the independent variable. a. 5 0 g g- = g0 = g = b. f f0 = f-6 = f- = c. 9 h h5 = h0 = h = 6

23 4. Sketch the graph of each function. Then list the domain and range in interval notation. a. h b. 0 0 g c. h d g e. f Homework Piecewise Worksheet

24 Piecewise Homework Worksheet. f. 0 0 h. g h 5. f 6. 4 p 7. int int even an is odd is an g 8. 4 f f

25 0. ; 4 ; ; f. ; ; f. ; ; f II. Evaluate.. 5 f a. f-. = b. f.8 = c. f-.5 = 4. 4 ; 5; ; g a. g- = b. g = c. g =

26 Piecewise Application Notes. An amusement park charges $00 for groups of 0 or less people. For groups of more than 0 the charge the $00 fee plus an additional $4 per person. The park does not allow groups larger than 40. a. A group of 5 would pa: b. A group of 0 would pa: Let = c. Write a function C = f, that represents the cost as a function of the number of people going to the amusement park. Graph the function: Label aes. f. A tai in Los Angeles costs $.00 for the first half mile and then $0.50 for each additional half mile. The tais round up to the net half-mile. So someone on a. mile ride would be charged for.5 mile: $ = $4.00 a. Finish the table Length of Ride Cost b. Graph the function: 0 < < ½ $5.00 < < $.50 Cost $4.00 < < < < $4.50 $.00 Trip Length miles

27 . A long distance telephone charges 99 cents for an call up to 0 minutes in length and 7 cents for each additional minute. a. Use bracket notation to write a formula for the cost, C, of a call as a function of its length time, t, in minutes. b. How much does it cost to talk for 0 minutes? 5 minutes? 4. A compan charges $00 a month to organize a compan s paroll for up to 0 emploees and an additional $00 a month for each 0 emploees over 0. Find a function, P = f, that gives the paroll amount for 00 emploees in one month. 5. An econom car costs $95 per week. Etra das cost $4 per da until the rate eceeds the weekl rate, in which case the weekl rate applies. Find the cost C of renting an econom car as a piecewise-defined function of the number of das used, where 7 4 Homework Piecewise Application Worksheet

28 Piecewise Application Worksheet. A museum charges $40 for a group of 0 or fewer people. A group of more than 0 people must, in addition to the $40, pa $ per person for the number of people above 0. For eample, a group of pas $44 and a group of 5 pas $50. The maimum group size is 50. a. Find a function, C = f, that represents the cost as a function of the number of people going to the museum. b. How much would the museum charge for a group of 8? c. Group with 5 people?. The charge for a tai ride is $.50 for the first /8 of a mile, and $0.5 for each additional /8 of a mile rounded to the nearest /8 mile. a. Find a function, C = f, that represents the cost of the trip as a function of its length. Your domain should start at 0 and go up to one mile in /8 mile intervals. b. What is the cost for a 5/8 mile ride? c. How far can ou go for $.00?. A parking garage in Manhattan charges in this wa: For each hour or part of an hour, the garage charges $0 per hour, with a dail maimum of $50 per da. a. How much will a customer pa he/she parks for hours?.5 hours? b. 4 hours? c. Write a piecewise function that has as its input the number of hours parked and outputs the total price paid b the customer.

29 4. The minimum pament on a credit card is based on the total amount owed. A credit card compan uses the following rules: For a bill less than $0 the entire amount is due. For a bill of at least $0 but less than $500, the minimum due is $0. There is a minimum of $0 due on a bill of at least $500 but less than $000, A minimum of $50 due on a bill of a least $000, but less than $500, and A minimum of $70 is due on bills $500 or more. Find the function F that describes the minimum pament on a bill of X dollars. 5. A paperback sells for $. The author is paid roalties of 0% on the first 0,000 copies sold, and 5% on an additional copies. a. When the 6,000 th book is sold, how much will the author earn on that sale? b. Also, what will the author s total roalties be at that point? c. When the,000 th book is sold, how much will the author earn on that sale? d. Also, what will the author s total roalties be at that point? e. Let be the number of copies sold. Write a piecewise function for R the total roalt earned in terms of. f. How man copies have to be sold in order for the author to have earned $0,000? 6. You are a buer for a grocer store and ou are asked to purchase potatoes for the grocer store. The distributor of potatoes tells ou that ou bu up to 50 bushels of potatoes, ou will pa $40 per bushel; and for each bushel ou purchase above 50 bushels, ou will pa $0 per bushel. a. How much will our grocer store pa in total ou decide to purchase 40 bushels? b. 60 bushels? 00 bushels? c. Write a function which has as its input values -values the number of bushels of potatoes purchased and outputs the total amount of mone that our grocer store will pa for the potatoes.

30 AFM - - Unit GRAPHING - - TEST REVIEW SHEET # I. Graph [ ]

31 7. 6 II. Short Answer 8. Determine all of the asmptotes of 4 f Given f 5 4, find f. 0. Given the following, describe the transformations to their parent graph. b. a. c. 4. Given the following function, graph each transformation. f = a. f b. f + c. -f d. f

32 AFM - - Unit GRAPHING - - TEST REVIEW SHEET # I. Ident the parent graph and describe the transformation.. f II. What restrictions, an, must be placed on the domain of each function? 5. f. f. 4 f 4. f 5 5. f 6. f 7. f 8. f 6 III. Write the equation for the graphs of the following functions.. ; sht the graph up 5 units. ; sht the graph to the left units. ; shrink verticall b /5 4. ; sht the graph down units 5. ; flip over the -ais 6. ; stretch verticall b 7. ; stretch horizontall b 8. ; sht the graph up 4 units and left 5 units 9. ; sht the graph to the left 5 units and flip over the -ais

33 0. ; sht the graph to the left units and down unit. ; stretch verticall b and flip over the -ais. ; sht the graph to the right 4 units and up units. ; sht the graph to the left units, down 4 units, and flip over the -ais 4. ; sht the graph to the left units and up unit 5. ; shrink verticall b /4 and sht to the right units 6. [] ; sht the graph to the right 6 units and up units IV. Write the equation of the graph