3.6.1 Building Functions from Context. Warm Up

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1 Name: # Honors Coordinate Algebra: Period Ms. Pierre Date: Building Functions from Context Warm Up 1. Willem buys 4 mangoes each week, and mango prices vary from week to week. Write an equation that represents the cost of the mangoes. If each mango costs $1 this week, what is the total cost of his mangoes for this week? 2. Kerin drives at a speed of 55 miles per hour on the highway for her job. Write an equation that represents the distance she travels. If one day she drives for 6 hours, how many miles did she travel?

2 3. Mr. Stevens teaches 4 math classes. Depending on absences, the number of students in each class varies. Write an equation that represents the number of students Mr. Stevens teaches in a day. If there are 30 students in each class and one day all of the students were present, how many students did Mr. Stevens teach that day? 4. Jessica reads approximately 12 pages of her novel each hour. Depending on extracurricular activities and homework, the time that Jessica has to read varies. Write an equation that represents the number of pages Jessica reads. If Jessica read for 3 hours yesterday, approximately how many pages did she read?

3 Verbal descriptions of mathematical patterns and situations can be represented using equations and expressions. A variable is a letter used to represent a value or unknown quantity that can change or vary in an expression or equation. An expression is a combination of variables, quantities, and mathematical operations; 4, 8x, and b are all expressions. An equation is an expression set equal to another expression; a = 4, = x + 9, and (2 + 3) 1 = 2c are all equations. Drawing a model can help clarify a situation. When examining a pattern, look for changes in quantities. A function is a relation between two variables, where one is independent and the other is dependent. For each independent variable there is only one dependent variable. One way to generalize a functional relationship is to write an equation. A linear function can be represented using a linear equation. A linear equation relates two variables, and both variables are raised to the 1st power; the equation s = 2r 7 is a linear equation. The slope-intercept form of a linear equation is y = mx + b. The form of a linear function is similar, f (x) = mx + b, where x is the independent quantity, m is the slope, b is the y-intercept, and f (x) is the function evaluated at x or the dependent quantity. The slope, or the measure of the rate of change of one variable with respect to another variable, between any two pairs of independent and dependent quantities is constant if the relationship between the quantities is linear. Consecutive terms in a pattern have a common difference if the pattern is linear. An exponential function can be represented using an exponential equation. An exponential equation relates two variables, and a constant in the equation is raised to a variable; the equation w = 3 v is an exponential equation. The general form of an exponential equation is y = ab x. The form of an exponential function is similar, f (x) = ab x, where a and b are real numbers. Terms have a common ratio if the pattern is exponential. An explicit equation describes the nth term of a pattern, and is the algebraic representation of a relationship between two quantities. An equation that represents a function, such as f (x) = 2x, is one type of explicit equation. Evaluating an equation for known term numbers is a good way to determine if an explicit equation correctly describes a pattern. Key Concepts A situation that has a mathematical pattern can be represented using an equation. A variable is a letter used to represent an unknown quantity. An expression is a combination of variables, quantities, and mathematical operations. An equation is an expression set equal to another expression. An explicit equation describes the nth term in a pattern. A linear equation relates two variables, and each variable is raised to the 1st power. The general equation to represent a linear function is f (x) = mx + b, where m is the slope and b is the y-intercept. An exponential equation relates two variables, and a constant in the equation is raised to a variable.

4 The general equation to represent an exponential function is f (x) = ab x, where a and b are real numbers. Consecutive dependent terms in a linear function have a common difference. If consecutive terms in a linear pattern have an independent quantity that increases by 1, the common difference is the slope of the relationship between the two quantities. Use the slope of a linear relationship and a single pair of independent and dependent values to find the linear equation that represents the relationship. Use the general equation f (x) = mx + b, and replace m with the slope, f (x) with the dependent quantity, and x with the independent quantity. Solve for b. Consecutive dependent terms in an exponential function have a common ratio. Use the common ratio to find the exponential equation that describes the relationship between two quantities. In the general equation f (x) = ab x, b is the common ratio. Let a 0 be the value of the dependent quantity when the independent quantity is 0. The general equation to represent the relationship would be: f (x) = a 0 b x. Let a 1 be the value of the dependent quantity when the independent quantity is 1. The general equation to represent the relationship would be: f (x) = a 1 b x 1. A model can be used to analyze a situation. Example 1 The starting balance of Anna s account is $1,250. She takes $30 out of her account each month. How much money is in her account after 1, 2, and 3 months? Find an equation to represent the balance in her account at any month.

5 Example 2 Consider that the first figure below has two 180º angles, one on each side of the line segment. Each of these angles is then bisected or cut in half. This pattern continues, and the first 4 figures in the pattern are shown. Write an equation to represent the relationship between the figure number and the number of angles in the figure. f x = 2 2!!! = 2!

6 Example 3 A video arcade charges an entrance fee, then charges a fee per game played. The entrance fee is $5, and each game costs an additional $1. Find the total cost for playing 0, 1, 2, or 3 games. Describe the total cost with an explicit equation.

7 Guided Practice Write an explicit equation to represent each pattern below. 1. ) Mr. Ramos notices a pattern in the number of people attending the weekly student government meetings. For weeks 1, 2, 3, 4, and 5, the number of students attending the meeting was 31, 43, 55, 67, and 79, respectively. 2.) Angelo sells cookies in packages, where each package contains the same number of cookies. The total amounts of cookies he has after 1, 2, 3, 4, and 5 packages are sold are 110, 88, 66, 44, and 22, respectively.

8 3.) As a treat, Nia eats a portion of a chocolate bar each day. She eats the same portion of the remaining bar each day. On day 0, the bar of chocolate starts with 32 pieces. After 1 day, 16 pieces remain. After days 2, 3, and 4, there are a total of 8, 4, and 2 pieces remaining. 4.) Given the diagram that follows, describe the number of blocks in Figure x. Figure 1 Figure 2 Figure 3

9 Independent Practice Problem-Based Task 3.6.1: Texting for the Win Lucas s friend Isabel is performing in a singing competition. The winner of the competition will be determined by call-in votes. To help Isabel earn votes, Lucas sends text messages to 8 of his friends. He then asks each of his friends to send texts to 8 more friends, and asks his friends to ask each of their friends to send 8 texts. He is hoping this pattern will continue and that many people will receive text messages telling them to vote for Isabel. Call each set of text messages a round of text messages, where Lucas s messages are the first round, Lucas s friends messages are the second round, and so on. Assuming that no one sends texts to the same person, how many texts will be sent after x rounds of text messages? a. Draw a diagram to model the situation. b. How many texts were sent the first round? c. How many texts were sent the second round? d. Determine how many texts were sent in rounds 3, 4, and 5.

10 How many texts will be sent in round x? In other words, what is the explicit definition of the number of texts sent in any round, x? Homework Write an explicit equation to represent each pattern below. 1. ) A rural school uses a phone tree to reach parents when the school is closed. Each parent calls multiple parents to notify them of the school closing. These parents then each call multiple parents, and so on. The diagram below shows the number of parents called after each round of calls. Each dot represents a parent. Find an explicit equation to represent the number of parents called in any round x. Round 1 Round 2 Round 3

11 2.) A hotel charges a room fee per night, plus an additional fee if more than one guest is staying in a room. Good Nights hotel charges $150 per night for a room, plus $25 per guest if more than one guest is staying in a room. Find an explicit equation to represent the nightly cost for any number of guests. 3.) The population of a city is growing. Each year, the population increases by approximately 10%, or 0.10 times the previous year s population. The population this year is 10,000. Find an explicit equation to represent the population of the town in any year. Consider that year 0 is this year.