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# Week 2, Section 2.1 READING notes and PRACTICE PROBLEMS 1101

UGA

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This 3 page Class Notes was uploaded by Leslea Motley on Monday August 22, 2016. The Class Notes belongs to 1101 at University of Georgia taught by Erik Miller in Fall 2016. Since its upload, it has received 12 views. For similar materials see Mathematical Modeling in Math at University of Georgia.

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Date Created: 08/22/16

Section 2.1. Recommended Problems (pg. 27): 1, 3, 6. Functions in the Real World • Function: rule associating the values of one quantity (dependent variable) with the values of another (independent variable); in other words, for every single value of the independent variable there is precisely one associated value for the dependent variable o Depicted in one/four ways: 1. Formulas/equations 2. Graphs 3. Tables 4. Words A. Representing Functions with Formulas and Equations • Think of this as “a relationship between two quantities that is given as a formula” • EXAMPLES: 2 o Formula for area of a circle: A = π r o F = (9/5)C + 32 - Formula comparing Farenheit to Celsius B. Representing Functions with Graphs • Valuable because they display accurate information as well as describing an overview of the behavior of that quantity (aka – trends, patterns) • NOT every graph represents a function – it is possible that a graph does not satisfy the condition that there is precisely one related value for the DV that relates to the value of the IV. C. Representing Functions with Tables D. Representing Functions with Words • Common words/phrases used: o Related to o Depends on o Measurement of something overtime is a function of something else o One quantity is a multiple of another quantity o One quantity is proportional to another quantity E. Why Study Functions? • When a situation involves two quantities – several questions may arise: o “Is there a functional relationship between the two quantities? Or, is one quantity a function of another?” o “If there is a relationship, can we find a formula for it?” o “Can we construct a table or graph relating the two quantities? Especially, if we cannot find a formula.” o “How can the knowledge of a function aid in understanding the relationship between the two quantities or allow us to make predictions or informed decisions about one of the variables based on the other?” F. Connecting Between the Different Representations • Formula = symbolic representation; graph = geometric; table = numerical 1) Formula à Table o Construct the table by substituting different values for the independent variable and calculating the corresponding value of the dependent variable o This process leads to the formation of a table that is associated with the function given by a formula in an identical way 2) Formula à Graph o Create a table – values in the table correspond to points on the graph 3) Graph à Table OR Table à Graph o Read points on graph to produce table o Plot the values on table as points on a graph, and connect them with the appropriate, usually-smooth curve 4) Table à Formula OR Graph à Formula o Further discussed in next chapter PROBLEMS 1. Which of the following relationships are functions and which are not? Explain your reasoning. For those that are functions, identify which of the two quantities depends on the other – again, explain reasoning. (Relationship does not represent a function; Relationship represents a function) a) The number of miles driven in a car versus the number of gallons of gas used *This is a function because it describes the relationship between two quantities. The number of gallons of gas used depends on the miles driven in a car. b) The speed of a four-legged animal and its weight *The speed of an animal should not be directly related to the animal’s weight. c) The major-league baseball player who has a certain number of home runs at the end of the season. *This relationship involves one person and one number, therefore, it cannot represent a relationship between two quantities. d) The student who has a specific score on the SAT test in a particular year. *This relationship involves only one value NOT a relationship between quantities. e) The amount of rain that falls on any particular day of the year in Seattle *This relationship compares the amount of rain to the day of the year, therefore, by comparing two quantities – it is a function. The amount of rainfall may depend on the day of the year. f) The day of the year on which given amounts of snow, in inches, fall in Buffalo. *This describes a list of numbers not a relationship between quantities. 3. Consider the scenario: “You left home to run to the local gym. You began at a constant rate of speed, but sped up once you realized how energetic you felt. About half way there, you began to tire so you slowed down.” Sketch a graph of your distance from home as a function of time 6. Which tables of values represent functions and which do not? Explain your reasoning. A. x y B. x y 0 12 5 21 C. 1 17 8 34 2 23 10 63 12 52 3 17 4 11 11 40 x y 5 4 8 30 8 4 3 14 3 30 8 16 15 9 7 16 12 30 A. This table represents a function because there is one value of y corresponding to each value of x. B. This table does NOT represent a function because the x-value, 8, has two corresponding y-values. C. This table does NOT represent a function because there are two identical x- values, and two identical y-values.

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