Introduction Verbal descriptions of mathematical patterns and situations

Introduction 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, 8 x, and b + 102 are all expressions. An equation is an expression set equal to another expression; a = 4, 1 + 23 = x + 9, and (2 + 3)1 = 2 c are all equations. 1 3. 6. 1: Building Functions from Context

Introduction, continued 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 1 st power; the equation s = 2 r – 7 is a linear equation. 2 3. 6. 1: Building Functions from Context

Introduction, continued 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. 3 3. 6. 1: Building Functions from Context

Introduction, continued 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. 4 3. 6. 1: Building Functions from Context

Introduction, continued 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) = 2 x, 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. 5 3. 6. 1: Building Functions from Context

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. 3. 6. 1: Building Functions from Context 6

Key Concepts, continued • A linear equation relates two variables, and each variable is raised to the 1 st 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. 7 3. 6. 1: Building Functions from Context

Key Concepts, continued • 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. 8 3. 6. 1: Building Functions from Context

Key Concepts, continued • 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. 9 3. 6. 1: Building Functions from Context

Key Concepts, continued • 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. 10 3. 6. 1: Building Functions from Context

Common Errors/Misconceptions • only examining the relationship between two terms to determine the general rule for a pattern • confusing recursive and explicit equations • incorrectly evaluating a recursive or explicit equation when determining if an equation matches a situation 11 3. 6. 1: Building Functions from Context

Guided Practice 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. 12 3. 6. 1: Building Functions from Context

Guided Practice: Example 1, continued 1. Use the description of the account balance to find the balance after each month. Anna’s account has $1, 250. After 1 month, she takes out $30, so her account balance decreases by $30: $1250 – $30 = $1220. The new starting balance of Anna’s account is $1, 220. After 2 months, she takes out another $30. Subtract this $30 from the new balance of her account: $1220 – $30 = $1190. 13 3. 6. 1: Building Functions from Context

Guided Practice: Example 1, continued The new starting balance of Anna’s account is $1, 190. After 3 months, she takes out another $30. Subtract this $30 from the new balance of her account: $1190 – $30 = $1160. 14 3. 6. 1: Building Functions from Context

Guided Practice: Example 1, continued 2. Determine the independent and dependent quantities. The month number is the independent quantity, since the account balance depends on the month. The account balance is the dependent quantity. 15 3. 6. 1: Building Functions from Context

Guided Practice: Example 1, continued 3. Determine if there is a common difference or common ratio that describes the change in the dependent quantity. Organize your results in a table. Enter the independent quantity in the first column, and the dependent quantity in the second column. The balance at zero months is the starting balance of the account, before any money has been taken out. Because the independent quantity is changing by one unit, analyzing the differences between the dependent quantities will determine if there is a common difference between the dependent quantities. 3. 6. 1: Building Functions from Context 16

Guided Practice: Example 1, continued Month Account balance in dollars ($) 0 1250 1 1220 1250 – 1220 = – 30 2 1190 1220 – 1190 = – 30 3 1160 1190 – 1160 = – 30 Difference The account balance has a common difference; it decreases by $30 for every 1 month. The relationship between the month and the account balance can be represented using a linear function. 17 3. 6. 1: Building Functions from Context

Guided Practice: Example 1, continued 4. Use the common difference to write an explicit equation. The general form of a linear function is: f (x) = mx + b, where m is the slope and b is the y-intercept. The common difference between the dependent terms in the pattern is the slope of the relationship between the independent and dependent quantities. Replace m with the slope, and replace x and f (x) with an independent and dependent quantity pair in the relationship, such as (1, 1220). 18 3. 6. 1: Building Functions from Context

Guided Practice: Example 1, continued Solve for b. 1220 = (– 30) • (1) + b 1250 = b f (x) = – 30 x + 1250 19 3. 6. 1: Building Functions from Context

Guided Practice: Example 1, continued 5. Evaluate the equation to verify that it is correct. Organize your results in a table. Use the explicit equation to find each term. The terms that are calculated should match the terms in the original list. Month, x Account balance, f(x), in dollars ($) 0 (– 30) • (0) + 1250 = 1250 1 (– 30) • (2) + 1250 = 1220 2 (– 30) • (2) + 1250 = 1190 3 (– 30) • (3) + 1250 = 1160 20 3. 6. 1: Building Functions from Context

Guided Practice: Example 1, continued The pairs of dependent and independent quantities match the ones in the original pattern, so the explicit equation is correct. The balance in Anna’s account can be represented using the equation f (x) = – 30 x + 1250. ✔ 21 3. 6. 1: Building Functions from Context

Guided Practice: Example 1, continued 22 3. 6. 1: Building Functions from Context

Guided Practice 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. 23 3. 6. 1: Building Functions from Context

Guided Practice: Example 3, continued 1. Use the description of the costs to find the total costs. If no games are played, then only the entrance fee is paid. The total cost for playing 0 games is $5. If 1 game is played, then the entrance fee is paid, plus the cost of one game. If each game is $1, the cost of one game is $1. The total cost is $5 + $1 = $6. 24 3. 6. 1: Building Functions from Context

Guided Practice: Example 3, continued If 2 games are played, then the entrance fee is paid, plus the cost of two games. If each game is $1, the cost of two games is $1 • 2 = $2. The total cost is $5 + $2 = $7. If 3 games are played, then the entrance fee is paid, plus the cost of three games. If each game is $1, the cost of three games is $1 • 3 = $3. The total cost is: $5 + $3 = $8. 25 3. 6. 1: Building Functions from Context

Guided Practice: Example 3, continued 2. Identify the independent and dependent quantities. The total cost is dependent on the number of games played, so the number of games is the independent quantity and the total cost is the dependent quantity. 26 3. 6. 1: Building Functions from Context

Guided Practice: Example 3, continued 3. Determine if there is a common difference or a common ratio between the dependent terms. There appears to be a common difference between the dependent terms. Use a table to find the difference between the dependent quantities. Subtract the current term from the previous term. 27 3. 6. 1: Building Functions from Context

Guided Practice: Example 3, continued Games Cost in dollars ($) Difference 0 5 1 6 6– 5=1 2 7 7– 6=1 3 8 8– 7=1 The common difference between the dependent terms is $1. 28 3. 6. 1: Building Functions from Context

Guided Practice: Example 3, continued 4. Use the common difference to write an explicit equation. The general form of a linear function is: f (x) = mx + b, where m is the slope and b is the y-intercept. The common difference between the dependent terms in the pattern is the slope of the relationship between the independent and dependent quantities. Replace m with the slope, and replace x and f (x) with an independent and dependent quantity pair in the relationship, such as (1, 6). Solve for b. 6 = (1) • (1) + b 5=b 3. 6. 1: Building Functions from Context 29

Guided Practice: Example 3, continued 5. Evaluate the equation to verify that it is correct. Organize your results in a table. Use the explicit equation to find each term. The terms that are calculated should match the terms in the original list. Games Cost in dollars ($) 0 1 • (0) + 5 = 5 1 1 • (1) + 5 = 6 2 1 • (2) + 5 = 7 3 1 • (3) + 5 = 8 30 3. 6. 1: Building Functions from Context

Guided Practice: Example 3, continued The pairs of independent and dependent quantities match the ones in the original pattern, so the explicit equation is correct. The total cost of any number of games, x, can be represented using the equation: f (x) = x + 5. ✔ 31 3. 6. 1: Building Functions from Context

Guided Practice: Example 3, continued 32 3. 6. 1: Building Functions from Context