Understanding Functions: Sequences, Mappings, Graphs, and Equations, Study notes of Reasoning

Download Understanding Functions: Sequences, Mappings, Graphs, and Equations and more Study notes Reasoning in PDF only on Docsity! Week of: May 11 to May 17, 2020 Grade: 8 Content: Math Learning Objective Blurb: Greetings 8th graders! We hope you are safe and well with your families! This week, students learn about relations and functions. Students analyze mappings, sets of ordered pairs, sequences, tables, graphs and equations and determine which are functions. We’ve included some video links to help you if you get stuck! This work will not be graded, just do your best and have fun! Carnegie Learning: Use with Carnegie Resources provided below: Video 1: Derby Days- Slope Intercept Form of a line: https://vimeo.com/406211962 Video 2: Derby Days- Slope Intercept Form of a line: https://vimeo.com/407205566 Printable Resources: Skills Practice: Module 2, Topic 2, Lesson 4 Derby Days see below Family Guide below Practice Activities: On-Line All students now have access to an on-line program called Mathia! Mathia- If you are already in Mathia, please continue to work in the program. If you are new to Mathia: Please see the log-in information attached. Video Links: Khan Academy: Refresh your memory with any or all of the following: What is a Function? https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/8th- functions-and-function-notation/v/what-is-a-function TOPIC 3: Family Guide • M2-177 Carnegie Learning Family Guide Grade 8 Module 2: Developing Function Foundations TOPIC 3: INTRODUCTION TO FUNCTIONS In this topic, students explore functions in terms of sequences, mappings, sets of ordered pairs, graphs, tables, verbal descriptions, and equations. Because students have a strong foundation in writing equations of lines, they can construct equations for linear functions. Students learn the formal defi nition of a function and analyze functions and relations represented in a wide variety of ways. Finally, students further investigate the focus function: the linear function. Where have we been? Throughout elementary school, students described patterns and explained features of the pattern. They have also formed ordered pairs with terms of two sequences and compared the terms. Therefore, sequences are used as the entry point for this topic. Where are we going? The study of functions is a predominant topic in high school mathematics. As students move into high school, they will develop and use formal notation (e.g., f(x)) to denote and operate with functions. In high school, students will use sequences as a launching point for linear and exponential functions. Using the Vertical Line Test to Determine if a Relation Is a Function A standard test to determine whether a graphed relation is a function is called the vertical line test. If you draw a vertical line anywhere on the graph and cross more than one point, the relation is not a function. The graph shown illustrates a relation that is not a function. x8 6 8 94 4 62 2 73 510 y 9 5 7 3 1 0 C03_SE_FG_M02_T3.indd M2-177 3/16/17 12:45 PM M2-206 • TOPIC 3: Introduction to Functions Getting Started What’s My Rule? Rules can be used to generate sequences of numbers. They can also be used to generate (x, y) ordered pairs. 1. Write an equation to describe the relationship between each independent variable x and the dependent variable y. Explain your reasoning. a. x y 26 212 23 0 0 12 3 24 b. x y 1 22 5 210 21 2 210 20 c. x y 210 9 22 1 0 21 5 4 d. x y 0 2 4 4 5 4.5 20 12 2. Create your own table and have a partner determine the equation you used to build it. You can sketch the graph to help determine the equation. C03_SE_M02_T03_L03.indd 206 4/7/17 1:17 PM Use braces, { }, to denote a set. LESSON 3: One or More Xs to One Y • M2-207 As you learned previously, ordered pairs consist of an x-coordinate and a y-coordinate. You also learned that a series of ordered pairs on a coordinate plane can represent a pattern. You can also use a mapping to show ordered pairs. A mapping represents two sets of objects or items. Arrows connect the items to represent a relationship between them. When you write the ordered pairs for a mapping, you are writing a set of ordered pairs. A set is a collection of numbers, geometric figures, letters, or other objects that have some characteristic in common. 1. Write the set of ordered pairs that represent a relationship in each mapping. a. 1 2 3 4 1 3 5 7 b. 1 2 3 4 5 1 3 5 7 c. 1 2 3 4 5 1 3 5 7 d. 2 4 6 8 7 9 2 20 2. Create a mapping from the set of ordered pairs. a. {(5, 8), (11, 9), (6, 8), (8, 5)} b. {(3, 4), (9, 8), (3, 7), (4, 20)} Functions as Mappings from One Set to Another ACTIVIT Y 3.1 C03_SE_M02_T03_L03.indd 207 4/7/17 1:17 PM WORKED EXAMPLE Each mapping represents a function because no input, or domain value, is mapped to more than one output, or range value. M2-208 • TOPIC 3: Introduction to Functions 3. Write the set of ordered pairs to represent each table. a. Input Output 210 220 25 210 0 0 5 10 10 20 b. x y 20 210 10 25 0 0 10 5 20 10 The mappings and ordered pairs shown in Questions 1 through 3 form relations. A relation is any set of ordered pairs or the mapping between a set of inputs and a set of outputs. The first coordinate of an ordered pair in a relation is the input, and the second coordinate is the output. A function maps each input to one and only one output. In other words, a function has no input with more than one output. The domain of a function is the set of all inputs of the function. The range of a function is the set of all outputs of the function. Notice the use of set notation when writing the domain and range. The range is {1, 3, 5, 7}. The range is {1, 3, 7}. 1 2 3 4 1 3 5 7 1 2 3 4 1 3 7 In each mapping shown, the domain is {1, 2, 3, 4}. C03_SE_M02_T03_L03.indd 208 4/7/17 1:17 PM LESSON 3: One or More Xs to One Y • M2-211 Functions as Mapping Inputs to Outputs ACTIVIT Y 3.2 You have determined if sets of ordered pairs represent functions. In this activity you will examine different situations and determine whether they represent functional relationships. Read each context and decide whether it fits the definition of a function. Explain your reasoning. 1. Input: Sue writes a thank-you note to her best friend. Output: Her best friend receives the thank-you note in the mail. 2. Input: A football game is being telecast. Output: It appears on televisions in millions of homes. 3. Input: There are four puppies in a litter. Output: One puppy was adopted by the Smiths, another by the Jacksons, and the remaining two by the Fullers. 4. Input: The basketball team has numbered uniforms. Output: Each player wears a uniform with her assigned number. 5. Input: Beverly Hills, California, has the zip code 90210. Output: There are 34,675 people living in Beverly Hills. 6. Input: A sneak preview of a new movie is being shown in a local theater. Output: 65 people are in the audience. C03_SE_M02_T03_L03.indd 211 4/7/17 1:17 PM M2-212 • TOPIC 3: Introduction to Functions 7. Input: Tara works at a fast food restaurant on weekdays and a card store on weekends. Output: Tara’s job on any one day. 8. Input: Janelle sends a text message to everyone in her contact list on her cell phone. Output: There are 41 friends and family on Janelle’s contact list. Determining Whether a Relation Is a Function ACTIVIT Y 3.3 Analyze the relations in each pair. Determine which relations are functions and which are not functions. Explain how you know. 1. Mapping A Mapping B 10 11 12 13 1000 2000 3000 10 11 12 13 1000 2000 3000 C03_SE_M02_T03_L03.indd 212 4/7/17 1:17 PM LESSON 3: One or More Xs to One Y • M2-213 2. Table A Table B Input Output 22 4 21 1 0 0 1 1 2 4 x y 2 24 1 21 0 0 1 1 2 4 3. Sequence A Sequence B 7, 10, 13, 16, 19, … 10, 30, 10, 30, 10, … 4. Set A Set B {(2, 3), (2, 4), (2, 5), (2, 6), (2, 7)} {(2, 1), (3, 1), (4, 1), (5, 1), (6, 1)} 5. Scenario A Input: The morning announcements are read over the school intercom system during homeroom period. Scenario B Input: Each student goes through the cafeteria line. Output: All students report to homeroom at the start of the school day to listen to the announcements. Output: Each student selects a lunch option from the menu. C03_SE_M02_T03_L03.indd 213 4/7/17 1:17 PM NOTES M2-216 • TOPIC 3: Introduction to Functions Functions as Equations ACTIVIT Y 3.5 So far, you have determined whether a mapping, context, or a graph represents a function. You can also determine whether an equation is a function. WORKED EXAMPLE The given equation can be used to convert yards to feet. Let x represent the number of yards, and let y represent the number of feet. y 5 3x To test whether this equation is a function, first, substitute values for x into the equation, and then determine if any x-value can be mapped to more than one y-value. If each x-value has exactly one y-value, then it is a function. Otherwise, it is not a function. In this case, every x-value can be mapped to only one y-value. Each x-value is multiplied by 3. Some examples of ordered pairs are (2, 6), (10, 30), and (5, 15). Therefore, this equation is a function. x y 5 3x 1 3 3 9 4 12 8 24 It is not possible to test every possible input value in order to determine whether or not the equation represents a function. You can graph any equation to see the pattern and use the vertical line test to determine if it represents a function. C03_SE_M02_T03_L03.indd 216 4/7/17 1:17 PM LESSON 3: One or More Xs to One Y • M2-217 If you do not recognize the graph of the equation, use a graphing calculator to see the pattern. 1. Determine whether each equation is a function. List three ordered pairs that are solutions to each. Explain your reasoning. a. y 5 5x 1 3 b. y 5 x2 c. y 5 |x| d. x2 1 y2 5 1 e. y 5 4 f. x 5 2 2. Explain what is wrong with Taylor's reasoning. If two different inputs go to the same output, it can still be a function. Taylor The equation y2 = x represents a function. x y 4 2 9 3 25 5 C03_SE_M02_T03_L03.indd 217 4/7/17 1:17 PM NOTES M2-218 • TOPIC 3: Introduction to Functions TALK the TALK Function Organizer 1. Complete the graphic organizer for the concept of function. Write a definition for function in your own words. Then, create a problem situation that can be represented using a function. Finally, create a table of ordered pairs and sketch a graph to represent the function. Function Definition Problem Situation Graph Table/ Ordered Pairs C03_SE_M02_T03_L03.indd 218 4/7/17 1:17 PM LESSON 3: One or More Xs to One Y • M2-221 Review Tell whether each graph is discrete or continuous. Also, tell whether each graph is increasing, decreasing, both, or neither. 1. x 43 1 2 3 4 5 6 7 8 9 1021 y 5 6 7 8 9 10 2. x 43 1 2 3 4 5 6 7 8 9 1021 y 5 6 7 8 9 10 Determine the slope and y-intercept of the linear relationship described by each equation. 3. y 5 x __ 2 1 5 4. y 5 x __ 4 Calculate the slope of the line represented by each table. 5. 6. x y 2 21 3 1.5 4 4 5 6.5 x y 2 8 4 2 6 24 9 213 C03_SE_M02_T03_L03.indd 221 4/7/17 1:17 PM C03_SE_M02_T03_L03.indd 222 4/7/17 1:17 PM