Lesson 3-1 Representing Relations Lesson 3-2 Representing Functions Lesson 3-3 Linear Functions Lesson 3-4 Arithmetic Sequences Lesson 3-5 Describing Number Patterns Chapter Menu

Lesson 3-1 Ideas/Vocabulary Represent relations as sets of ordered pairs, tables, mappings, and graphs. Find the inverse of a relation. mapping inverse Lesson 3-1 Ideas/Vocabulary

Represent a Relation A. Express the relation {(4, 3), (–2, –1), (–3, 2), (2, –4), (0, –4)} as a table, a graph, and a mapping. Table List the x-coordinates in the first column and the corresponding y-coordinates in the second column. Animation: Relation Lesson 3-1 Example 1a

Graph Graph each ordered pair on a coordinate plane. Represent a Relation Graph Graph each ordered pair on a coordinate plane. Lesson 3-1 Example 1a

Represent a Relation Mapping List the x-values in set X and the y-values in set Y. Draw an arrow from the x-value to the corresponding y-value. Lesson 3-1 Example 1a

Represent a Relation B. Determine the domain and range for the relation {(4, 3), (–2, –1), (–3, 2), (2, –4), (0, –4)}. Answer: The domain for this relation is {–3, –2, 0, 2, 4}. The range is {–4, –1, 2, 3}. Lesson 3-1 Example 1b

A. Express the relation {(3, –2), (4, 6), (5, 2), (–1, 3)} as a mapping. B C D A. C. B. D. CYP 1-1

B. Determine the domain and range of the relation {(3, –2), (4, 6), (5, 2), (–1, 3)}.

Determine the domain and range of the relation. Use a Relation A. OPINION POLLS The table shows the percent of students satisfied with their grades at the time of the survey. Determine the domain and range of the relation. Answer: The domain is {1996, 1999, 2002, 2005}. The range is {21, 32, 51, 60}. Lesson 3-1 Example 2a

Use a Relation B. Graph the data. The values of the x-axis need to go from 1996 to 2005. It is not practical to begin the scale at 0. Begin at 1996 and extend to 2005 to include all of the data. The units can be 1 unit per grid square. The values on the y-axis need to go from 21 to 60. In this case it is possible to begin the scale at 0. Begin at 0 and extend to 70. You can use units of 10. Lesson 3-1 Example 2b

C. What conclusions might you make from the graph of the data? Use a Relation C. What conclusions might you make from the graph of the data? Answer: Satisfaction increased from 1996 to 2002, but decreased from 2002 to 2005. Lesson 3-1 Example 2c

ENDANGERED SPECIES The table shows the approximate world population of the Indian Rhinoceros from 1986 to 2005. Determine the domain and range for the data provided in the table. D = {1000, 1700, 1900, 2100, 2400}; R = {1986, 1990, 1994, 1998, 2005} D = {5}; R = {4} D = {1986, 1990, 1994, 1998, 2005}; R = {1000, 1700, 1900, 2100, 2400} D = {2005}; R = {2400}

B. Graph the data. A B C D A. C. B. D. CYP 2-2

C. What conclusions might you make from the graph of the data? The population of the Indian rhinoceros has been decreasing since 1986. The population of the Indian rhinoceros has been increasing since 1986. The population of the Indian rhinoceros has stayed the same since 1986. The Indian rhinoceros has become extinct.

Lesson 3-1 Key Concept 1

Inverse Relation Express the relation shown in the mapping as a set of ordered pairs. Then write the inverse of the relation. Relation: Notice that both 7 and 0 in the domain are paired with 2 in the range. Answer: {(5, 1), (7, 2), (4, –9), (0, 2)} Inverse: Exchange X and Y in each ordered pair to write the inverse relation. Answer: {(1, 5), (2, 7), (–9, 4), (2, 0)} Lesson 3-1 Example 3

Express the relation shown in the mapping as a set of ordered pairs Express the relation shown in the mapping as a set of ordered pairs. Then write the inverse of the relation. A. Relation {(2, 3), (1, –4), (2, 5)} Inverse {(3, 2), (–4, 1), (5, 2)} B. Relation {(3, 2), (–4, 1), (5, 2)} Inverse {(–3, –2), (4, –1), (–5, –2)} C. Relation {(3, 2), (–4, 1), (5, 2)} Inverse {(2, 3), (1, –4), (2, 5)} D. Relation {(3, 2), (–4, 1), (5, 2)}

End of Lesson 3-1

Five-Minute Check (over Lesson 3-1) Main Ideas and Vocabulary Targeted TEKS Key Concept: Functions Example 1: Identify Functions Example 2: Equations as Functions Example 3: Function Values Example 4: Nonlinear Function Values Example 5: Representations of Functions Lesson 3-2 Menu

Lesson 3-2 Ideas/Vocabulary Determine whether a relation is a function. Find functional values. function vertical line test function notation function value Lesson 3-2 Ideas/Vocabulary

Lesson 3-2 Key Concepts 1

A. Determine whether the relation is a function. Explain. Identify Functions A. Determine whether the relation is a function. Explain. Answer: This is a function because the mapping shows each element of the domain paired with exactly one member of the range. Lesson 3-2 Example 1a

Animation: Is This a Function? Identify Functions B. Determine whether the relation is a function. Explain. Answer: This table represents a function because the table shows each element of the domain paired with exactly one element of the range. Animation: Is This a Function? Lesson 3-2 Example 1b

A. Is this relation a function? Explain. Yes; for each element of the domain, there is only one corresponding element in the range. Yes; because it can be represented by a mapping. No; because it has negative x-values. No; because both –2 and 2 are in the range.

B. Is this relation a function? Explain. No; because the element 3 in the domain is paired with both 2 and –1 in the range. No; because there are negative values in the range. Yes; because it is a line when graphed. Yes; because it can be represented in a chart.

Equations as Functions Determine whether x = –2 is a function. Graph the equation. Since the graph is in the form Ax + By = C, the graph of the equation will be a line. Place your pencil at the left of the graph to represent a vertical line. Slowly move the pencil to the right across the graph. At x = –2 this vertical line passes through more than one point on the graph. Answer: The graph does not pass the vertical line test. Thus, the line does not represent a function. Lesson 3-2 Example 2

Determine whether 3x + 2y = 12 is a function. Yes No Not enough information

Function Values A. If f(x) = 3x – 4, find f(4). f(4) = 3(4) – 4 Replace x with 4. = 12 – 4 Multiply. = 8 Subtract. Answer: f(4) = 8 Lesson 3-2 Example 3a

f(–5) = 3(–5) – 4 Replace x with –5. = –15 – 4 Multiply. Function Values B. If f(x) = 3x – 4, find f(–5). f(–5) = 3(–5) – 4 Replace x with –5. = –15 – 4 Multiply. = –19 Subtract. Answer: f(–5) = –19 Lesson 3-2 Example 3b

A. If f(x) = 2x + 5, find f(3). A A. 8 B B. 7 C C. 6 D D. 11 Lesson 3-2 Example 3CYP-A

B. If f(x) = 2x + 5, find f(–8). A. –3 B. –11 C. 21 D. –16 A B C D Lesson 3-2 Example 3CYP-B

Nonlinear Function Values PHYSICS The function h(t) = 160t + 16t2 represents the height of an object ejected downward from an airplane at a rate of 160 feet per second. A. Find the value h(3). h(3) = 160(3) + 16(3)2 Replace t with 3. = 480 + 144 Multiply. = 624 Simplify. Answer: h(3) = 624 Lesson 3-2 Example 4

Nonlinear Function Values PHYSICS The function h(t) = 160t + 16t2 represents the height of an object ejected downward from an airplane at a rate of 160 feet per second. B. Find the value h(2z). h(2z) = 160(2z) + 16(2z)2 Replace t with 2z. = 320z + 64z2 Multiply. Answer: h(2z) = 320z + 64z2 Lesson 3-2 Example 4

The function h(t) = 180 – 16t2 represents the height of a ball thrown from a cliff that is 180 feet above the ground. A. Find the value h(2). 164 ft 116 ft 180 ft 16 ft

The function h(t) = 180 – 16t2 represents the height of a ball thrown from a cliff that is 180 feet above the ground. B. Find the value h(3z). 180 – 16z2 ft 180 ft 36 ft 180 – 144z2 ft

Representations of Functions The algebraic form of a function is m = 5d, where d is the number of dollars customers of Mike’s Car Rental donate to a charity and m is the donation made by Mike’s Car Rental. Which of the following represents the same function? A. For every $2 donated, Mike’s Car Rental donates $7. B. f(d) = 5m C. D. Lesson 3-2 Example 5

Representations of Functions Read the test item The independent variable is d and the dependent variable is m. Solve the test item Choice A represents the function m = 3.5d. This is incorrect because it should be m = 5d. Choices B and C represent the function , which is incorrect. In choice D, the graph represents the function m = 5d, which is correct. The answer is D. Lesson 3-2 Example 5

B. For every two hours the pool is rented, the cost to the PTA is $10. A. The algebraic form of a function is c = 5h, where h is the number of hours a pool is rented and c is the cost to the PTA. Which of the following represents the same function? A B C D A. f(h) = 5c B. For every two hours the pool is rented, the cost to the PTA is $10. C. f(h) = 2c D. For every two hours the pool is rented, the cost to the PTA is $5. Lesson 3-2 CYP 5

End of Lesson 3-2

Five-Minute Check (over Lesson 3-2) Main Ideas and Vocabulary Targeted TEKS Key Concept: Standard Form of a Linear Equation Example 1: Identify Linear Equations Example 2: Analyze Graphs Example 3: Analyze Tables Example 4: Graph by Making a Table Example 5: Graph by Using Intercepts Lesson 3-3 Menu

Lesson 3-3 Ideas/Vocabulary Identify linear equations, intercepts, and zeros. Graph linear equations. linear equation standard form x-intercept y-intercept zero Lesson 3-3 Ideas/Vocabulary

Lesson 3-3 Key Concept 1

Identify Linear Equations A. Determine whether 5x + 3y = z + 2 is a linear equation. If so, write the equation in standard form. First rewrite the equation so that the variables are on the same side of the equation. 5x + 3y = z + 2 Original equation 5x + 3y – z = z + 2 – z Subtract z from each side. 5x + 3y – z = 2 Simplify. Since 5x + 3y – z has three different variables, it cannot be written in the form Ax + By = C. Answer: This is not a linear equation. Lesson 3-3 Example 1a

Identify Linear Equations B. Determine whether is a linear equation. If so, write the equation in standard form. Rewrite the equation so that both variables are on the same side of the equation. Original equation Subtract y from each side. Simplify. Lesson 3-3 Example 1b

Identify Linear Equations To write the equation with integer coefficients, multiply each term by 4. Original equation Multiply each side of the equation by 4. 3x – 4y = 32 Simplify. The equation is now in standard form where A = 3, B = –4, and C = 32. Answer: This is a linear equation. Lesson 3-3 Example 1b

A. Determine whether y = 4x – 5 is a linear equation A. Determine whether y = 4x – 5 is a linear equation. If so, write the equation in standard form. linear equation; y = 4x – 5 not a linear equation linear equation; 4x – y = 5 linear equation; 4x + y = 5

B. Determine whether 8y –xy = 7 is a linear equation B. Determine whether 8y –xy = 7 is a linear equation. If so, write the equation in standard form. not a linear equation linear equation; 8y – xy = 7 linear equation; 8y = 7 + xy linear equation; 8y – 7 = xy

A. Determine the x-intercept, y-intercept, and zero. Analyze Graphs WATER STORAGE A valve on a tank is opened and the water is drained, as shown in the graph. A. Determine the x-intercept, y-intercept, and zero. Answer: x-intercept = 4; y-intercept = 200; zero of the function = 4 Lesson 3-3 Example 2a

B. Describe what the intercepts mean. Analyze Graphs WATER STORAGE A valve on a tank is opened and the water is drained, as shown in the graph. B. Describe what the intercepts mean. Answer: The x-intercept 4 means that after 4 minutes, there are 0 gallons of water left in the tank. The y-intercept of 200 means that at time 0, or before any water was drained, there were 200 gallons of water in the tank. Lesson 3-3 Example 2b

BANKING Janine has money in a checking account BANKING Janine has money in a checking account. She begins withdrawing a constant amount of money each month as shown in the graph. A. Determine the x-intercept, y-intercept, and zero. 10; 250; 10 10; 10; 250 250; 10; 250 5; 10; 5

B. In relation to the previous problem, describe what the x-intercept of 10 means? It is the amount of money before anything was withdrawn. That after 10 months, there is no money left in the account. It is the amount of money in the account after 6 months. This cannot be determined.

A. Determine the x-intercept, y-intercept, and zero. Analyze Tables ANALYZE TABLES A box of peanuts is poured into bags a the rate of 4 ounces per second. The table shows the function relating to the weight of the peanuts in the box and the time in seconds the peanuts have been pouring out of the box. A. Determine the x-intercept, y-intercept, and zero. Answer: x-intercept = 500; y-intercept = 2000; zero of the function = 500 Lesson 3-3 Example 3a

Interactive Lab: Graphing Relations and Functions Analyze Tables B. Describe what the intercepts in the previous problem mean. Answer: The x-intercept 500 means that after 500 seconds, there are 0 ounces of peanuts left in the box. The y-intercept of 2000 means that at time 0, or before any peanuts were poured, there were 2000 ounces of peanuts in the box. Interactive Lab: Graphing Relations and Functions Lesson 3-3 Example 3a

ANALYZE TABLES Jules has a food card for Disney World ANALYZE TABLES Jules has a food card for Disney World. The table shows the function relating the amount of money on the card and the number of times he has stopped to purchase food. A. Determine the x-intercept, y-intercept, and zero. 5; 125; 5 5; 5; 125 125; 5; 125 5; 10; 15

B. In relation to the previous problem, describe what the y-intercept of 125 means? It represents the time when there is no money left on the card. It represents the original value of the card. At time 0, or before any food stops, there was $125 on the card. This cannot be determined.

Graph by Making a Table Graph y = 2x + 2. Select values from the domain and make a table. Then graph the ordered pairs. The domain is all real numbers, so there are infinite solutions. Draw a line through the points. Answer: Lesson 3-3 Example 4

Is this graph the correct graph for y = 3x – 4? yes no not enough information to determine

Graph by Using Intercepts Graph 4x – y = 4 using the x-intercept and the y-intercept. To find the x-intercept, let y = 0. 4x – y = 4 Original equation 4x – 0 = 4 Replace y with 0. 4x = 4 Divide each side by 4. x = 1 To find the y-intercept, let x = 0. 4x – y = 4 Original equation 4(0) – y = 4 Replace x with 0. –y = 4 Divide each side by –1. y = –4 Lesson 3-3 Example 5

Animation: Graph Linear Equations Graph Using Intercepts Graph 4x – y = 4. Answer: The x-intercept is 1, so the graph intersects the x-axis at (1, 0). The y-intercept is –4, so the graph intersects the y- axis at (0, –4). Plot these points. Then draw a line that connects them. Animation: Graph Linear Equations Lesson 3-3 Example 5

Is this graph the correct graph for 2x + 5y = 10? yes no not enough information to determine

End of Lesson 3-3

Five-Minute Check (over Lesson 3-3) Main Ideas and Vocabulary Targeted TEKS Key Concept: Arithmetic Sequence Example 1: Identify Arithmetic Sequences Key Concept: Writing Arithmetic Sequences Example 2: Extend a Sequence Key Concept: nth Term of an Arithmetic Sequence Example 3: Write an Equation for a Sequence Lesson 3-4 Menu

Lesson 3-4 Ideas/Vocabulary Recognize arithmetic sequences. Extend and write formulas for arithmetic sequences. sequence terms arithmetic sequence common difference Lesson 3-4 Ideas/Vocabulary

Lesson 3-4 Key Concepts 1

Identify Arithmetic Sequences A. Determine whether –15, –13, –11, –9, ... is an arithmetic sequence. Explain. Answer: This is an arithmetic sequence because the difference between terms is constant. Lesson 3-4 Example 1a

Identify Arithmetic Sequences B. Determine whether , , , , … is an arithmetic sequence. Explain. Answer: This is not an arithmetic sequence because the difference between terms is not constant. Lesson 3-4 Example 1b

A. Determine whether 2, 4, 8, 10, 12, … is an arithmetic sequence. Cannot be determined. This is not an arithmetic sequence because the difference between terms is not constant. This is an arithmetic sequence because the difference between terms is constant.

B. Determine whether … is an arithmetic sequence. Cannot be determined This is not an arithmetic sequence because the difference between terms is not constant. This is an arithmetic sequence because the difference between terms is constant. CYP 3-4-1

Lesson 3-4 Key Concept 2

Find the common difference by subtracting successive terms. Extend a Sequence TEMPERATURE The arithmetic sequence –8, –11, –14, –17, … represents the daily low temperature in ºF. Find the next three terms. Find the common difference by subtracting successive terms. The common difference is –3. Lesson 3-4 Example 2

Answer: The next three terms are –20, –23, –26. Extend a Sequence Subtract 3 from the last term of the sequence to get the next term in the sequence. Continue subtracting 3 until the next three terms are found. Answer: The next three terms are –20, –23, –26. Lesson 3-4 Example 2

TEMPERATURE The arithmetic sequence 58, 63, 68, 73, … represents the daily high temperature in ºF. Find the next three terms. A. 78, 83, 88 B. 76, 79, 82 C. 73, 78, 83 D. 83, 88, 93 A B C D Lesson 3-4 Example 2CYP

Lesson 3-4 Key Concept 3

A. Write an equation for the nth term of the sequence. Extend a Sequence MONEY The arithmetic sequence 1, 10, 19, 28, … represents the total number of dollars Erin has in her account after her weekly allowance is added. A. Write an equation for the nth term of the sequence. In this sequence, the first term, a1, is 1. Find the common difference. The common difference is +9. Lesson 3-4 Example 3a

Use the formula for the nth term to write an equation. Extend a Sequence Use the formula for the nth term to write an equation. an = a1 + (n –1)d Formula for the nth term an = 1 + (n –1)(9) a1 = 1, d = 9 an = 1 + 9n – 9 Distributive Property an = 9n – 8 Simplify. Lesson 3-4 Example 3a

Extend a Sequence Check: For n = 1, 9(1) – 8 = 1. For n = 3, 9(3) – 8 = 19, and so on. Answer: an = 9n – 8 Lesson 3-4 Example 3a

B. Find the 12th term in the sequence. Extend a Sequence B. Find the 12th term in the sequence. Replace n with 12 in the equation written in part A. an = 9n – 8 Equation for the nth term a12 = 9(12) – 8 Replace n with 12. a12 = 100 Simplify Answer: 100 Lesson 3-4 Example 3b

C. Graph the first five terms of the sequence. Extend a Sequence C. Graph the first five terms of the sequence. Answer: The points fall on a line. The graph of an arithmetic sequence is linear. Lesson 3-4 Example 3c

MONEY The arithmetic sequence 2, 7, 12, 17, 22, … represents the total number of pencils Claire has in her collection after she goes to her school store each week. A. Write an equation for the nth term of the sequence. an = 2n + 7 an = 5n + 2 an = 2n + 5 an = 5n – 3

MONEY The arithmetic sequence 2, 7, 12, 17, 22, … represents the total number of pencils Claire has in her collection after she goes to her school store each week. B. Find the 12th term in the sequence. 12 57 52 62

MONEY The arithmetic sequence 2, 7, 12, 17, 22, … represents the total number of pencils Claire has in her collection after she goes to her school store each week. C. Does this graph show the first five terms of the sequence? yes no cannot determine

End of Lesson 3-4

Five-Minute Check (over Lesson 3-4) Main Ideas Targeted TEKS Example 1: Proportional Relationships Example 2: Nonproportional Relationships Lesson 3-5 Menu

Lesson 3-5 Ideas/Vocabulary Write an equation for a proportional relationship. Write an equation for a nonproportional relationship. Lesson 3-5 Ideas/Vocabulary

A.3 The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. (B) Look for patterns and represent generalizations algebraically. A.5 The student understands that linear functions can be represented in different ways and translates among their various representations. (C) Use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions. Lesson 3-5 TEKS

Proportional Relationships A. ENERGY The table shows the number of miles driven for each hour of driving. Graph the data. What conclusion can you make about the relationship between the number of hours driving h and the numbers of miles driven m? Answer: The graph shows a linear relationship between the number of hours driving and the number of miles driven. Lesson 3-5 Example 1a

Proportional Relationships B. Write an equation to describe this relationship. Look at the relationship between the domain and the range to find a pattern that can be described as an equation. Lesson 3-5 Example 1b

Proportional Relationships Since this is a linear relationship, the ratio of the range values to the domain values is constant. The difference of the values for h is 1, and the difference of the values for m is 50. This suggests that m = 50h. Check to see if this equation is correct by substituting values of h into the equation. Lesson 3-5 Example 1b

Proportional Relationships Check If h = 1, then m = 50(1) or 50. If h = 2, then m = 50(2) or 100. If h = 3, then m = 50(3) or 150. If h = 4, then m = 50(4) or 200. The equation checks. Answer: m = 50h Since this relation is also a function, we can write the equation as f(h) = 50h, where f(h) represents the number of miles driven. Lesson 3-5 Example 1b

A. Graph the data in the table A. Graph the data in the table. What conclusion can you make about the relationship between the number of miles walked and the time spent walking? There is a linear relationship between the number of miles walked and time spent walking. There is a nonlinear relationship between the number of miles walked and time spent walking. There is not enough information on the table to determine a relationship. There is an inverse relationship between miles walked and time spent walking.

B. Write an equation to describe the relationship between hours and miles walked. A. m = 3h B. m = 2h C. m = 1.5h D. m = 1h A B C D Lesson 3-5 Example 1CYP-B

Nonproportional Relationships Write an equation in function notation for the relation graphed below. Make a table of ordered pairs for several points on the graph. Lesson 3-5 Example 2

Nonproportional Relationships The difference in the x values is 1, and the difference in the y values is 3. The difference in y values is three times the difference of the x values. This suggests that y = 3x. Check this equation. Check If x = 1, then y = 3(1) or 3. But the y value for x = 1 is 1. This is a difference of –2. Try some other values in the domain to see if the same difference occurs. y is always 2 less than 3x. Lesson 3-5 Example 2

Nonproportional Relationships This pattern suggests that 2 should be subtracted from one side of the equation in order to correctly describe the relation. Check y = 3x – 2. If x = 2, then y = 3(2) – 2 or 4. If x = 3, then y = 3(3) – 2 or 7. Answer: y = 3x – 2 correctly describes this relation. Since the relation is also a function, we can write the equation in function notation as f(x) = 3x – 2. Lesson 3-5 Example 2

Write an equation in function notation for the relation that is graphed. f(x) = x + 2 f(x) = 2x f(x) = 2x + 2 f(x) = 2x + 1