Linear Functions Module Overview ................................................................................................................................................. 3 Topic A: Linear Sequences (F-IF.A.1, F-IF.A.2, F-IF.A.3, F-IF.B.6, F-BF.A.1a, F-LE.A.1,

F-LE.A.2, F-LE.A.3) ................................................................................................................................ 12 Lesson 1: Integer Sequences—Should You Believe in Patterns? .......................................................... 14 Topic B: Functions and Their Graphs (F-IF.A.1, F-IF.A.2, F-IF.B.4, F-IF.B.5, F-IF.C.7a) ....................................... 80 Lesson 8: Why Stay with Whole Numbers? ........................................................................................... 82 Lessons 9–10: Representing, Naming, and Evaluating Functions ......................................................... 92 Lesson 11: The Graph of a Function .................................................................................................... 116 Lesson 12: The Graph of the Equation 𝑦 = 𝑓(𝑥) ................................................................................ 134 Lesson 13: Interpreting the Graph of a Function ................................................................................ 153 Lesson 14: Linear and Exponential Models—Comparing Growth Rates ............................................ 167 Prentice Hall (Math A) Chapter 5 – Graphs of Linear Equations/Functions 5-1: Slope 5-2: Rate of Change 5-4: Slope-Intercept Form 5-5: Equation of a Line 5-6: Scatter Plots and Linear Equations 5-7: Ax + By = c Form 5-8: Parallel & Perpendicular Lines Chapter 6 – Systems of Linear Equations/Inequalities 6-1: Solving Linear Systems by Graphing 6-6: Graphing Systems of Linear Inequalities

ALGEBRA I • MODULE 3

New York State Common Core

Mathematics Curriculum

Module 3: Linear and Exponential Functions

1

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Algebra I • Module 3

Linear Functions

OVERVIEW In earlier grades, students defined, evaluated, and compared functions and used them to model relationships between quantities (8.F.A.1, 8.F.A.2, 8.F.A.3, 8.F.B.4, 8.F.B.5). In this module, students extend their study of functions to include function notation and the concepts of domain and range. They explore many examples of functions and their graphs, focusing on the contrast between linear and exponential functions. They interpret functions given graphically, numerically, symbolically, and verbally; translate between representations; and understand the limitations of various representations.

In Topic A, students explore arithmetic and geometric sequences as an introduction to the formal notation of functions (F-IF.A.1, F-IF.A.2). They interpret arithmetic sequences as linear functions with integer domains and geometric sequences as exponential functions with integer domains (F-IF.A.3, F-BF.A.1a). Students compare and contrast the rates of change of linear and exponential functions, looking for structure in each, and distinguishing between additive and multiplicative change (F-IF.B.6, F-LE.A.1, F-LE.A.2, F-LE.A.3).

In Topic B, students connect their understanding of functions to their knowledge of graphing from Grade 8. They learn the formal definition of a function and how to recognize, evaluate, and interpret functions in abstract and contextual situations (F-IF.A.1, F-IF.A.2). Students examine the graphs of a variety of functions and learn to interpret those graphs using precise terminology to describe such key features as domain and range, intercepts, intervals where the function is increasing or decreasing, and intervals where the function is positive or negative (F-IF.A.1, F-IF.B.4, F-IF.B.5, F-IF.C.7a).

In Topic C, students extend their understanding of piecewise functions and their graphs including the absolute value and step functions. They learn a graphical approach to circumventing complex algebraic solutions to equations in one variable, seeing them as 𝑓(𝑥) = 𝑔(𝑥) and recognizing that the intersection of the graphs of 𝑓(𝑥) and 𝑔(𝑥) are solutions to the original equation (A-REI.D.11). Students use the absolute value function and other piecewise functions to investigate transformations of functions and draw formal conclusions about the effects of a transformation on the function’s graph (F-IF.C.7, F-BF.B.3).

Finally, in Topic D, students apply and reinforce the concepts of the module as they examine and compare exponential, piecewise, and step functions in a real-world context (F-IF.C.9). They create equations and functions to model situations (A-CED.A.1, F-BF.A.1, F-LE.A.2), rewrite exponential expressions to reveal and relate elements of an expression to the context of the problem (A-SSE.B.3c, F-LE.B.5), and examine the key features of graphs of functions, relating those features to the context of the problem (F-IF.B.4, F-IF.B.6).

The Mid-Module Assessment follows Topic B. The End-of-Module Assessment follows Topic D.

FUNCTION:

A function is a correspondence between two sets, 𝑋 and 𝑌, in which each element of 𝑋 is matched to one and only one element of 𝑌. The set

𝑋 is called the domain of the function.

The notation 𝑓: 𝑋 → 𝑌 is used to name the function and describes both 𝑋 and 𝑌. If 𝑥 is an element in the domain 𝑋 of

a function 𝑓: 𝑋 → 𝑌, then 𝑥 is matched to an element of 𝑌 called 𝑓(𝑥). We say 𝑓(𝑥) is the value in 𝑌 that denotes the output or image of 𝑓

corresponding to the input 𝑥.

The range (or image) of a function 𝑓: 𝑋 → 𝑌 is the subset of 𝑌, denoted 𝑓(𝑋), defined by the following property: 𝑦 is

an element of 𝑓(𝑋) if and only if there is an 𝑥 in 𝑋 such that 𝑓(𝑥) = 𝑦.

PART A (ENY M3 Lesson 9-11) 1. Let 𝑋 = {1, 2, 3, 4} and 𝑌 = {5, 6, 7, 8, 9}. 𝑓 and 𝑔 are defined below. 𝑓: 𝑋 → 𝑌 𝑔: 𝑋 → 𝑌 𝑓 = {(1,7), (2,5), (3,6), (4,7)} 𝑔 = {(1, 5), (2, 6), (1, 8), (2,9), (3,7)}

a. Is 𝑓 a function? If yes, what is the domain, and what is the range? If no, explain why 𝑓 is not a function. b. Is 𝑔 a function? If yes, what is the domain and range? If no, explain why 𝑔 is not a function. c. What is 𝑓(2)? d. If 𝑓(𝑥) = 7, then what might 𝑥 be?

2. Study the 4 representations of a function below. How are these representations alike? How are they different?

TABLE:

FUNCTION: DIAGRAM:

Let 𝑓: {0, 1, 2, 3, 4, 5} → {1, 2, 4, 8, 16, 32} such that 𝑥 ↦ 2𝑥.

SEQUENCE:

Let 𝑎𝑛+1 = 2𝑎𝑛, 𝑎0 = 1 for 0 ≤ 𝑛 ≤ 4 where 𝑛 is an integer.

3. Let 𝑋 = {0, 1, 2, 3, 4, 5}. Complete the following table using the definition of 𝑓. 𝑓: 𝑋 → 𝑌 Assign each 𝑥 in 𝑋 to the

expression 2𝑥.

𝑥 0 1 2 3 4 5

𝑓(𝑥)

What are 𝑓(0), 𝑓(1), 𝑓(2), 𝑓(3), 𝑓(4), and 𝑓(5)?

What is the range of 𝑓?

4. Let 𝑓(𝑥) = 6𝑥 − 3, and let 𝑔(𝑥) = 0.5(4)𝑥. Find the value of each function for the given input. a. 𝑓(0) j. 𝑔(0) b. 𝑓(−10) k. 𝑔(−1) c. 𝑓(2) l. 𝑔(2) d. 𝑓(0.01) m. 𝑔(−3) e. 𝑓(11.25) n. 𝑔(4)

f. 𝑓(−√2) o. 𝑔(√2)

g. 𝑓 (5

3) p. 𝑔 (

1

2)

h. 𝑓(1) + 𝑓(2) q. 𝑔(2) + 𝑔(1) i. 𝑓(6) − 𝑓(2) r. 𝑔(6) − 𝑔(2)

5. Since a variable is a placeholder, we can substitute letters that stand for numbers in for 𝑥. Let 𝑓(𝑥) = 6𝑥 − 3, and let 𝑔(𝑥) =0.5(4)𝑥, and suppose 𝑎, 𝑏, 𝑐, and ℎ are real numbers. Find the value of each function for the given input.

a. 𝑓(𝑎) h. 𝑔(𝑏) b. 𝑓(2𝑎) i. 𝑔(𝑏 + 3) c. 𝑓(𝑏 + 𝑐) j. 𝑔(3𝑏) d. 𝑓(2 + ℎ) k. 𝑔(𝑏 − 3) e. 𝑓(𝑎 + ℎ) l. 𝑔(𝑏 + 𝑐) f. 𝑓(𝑎 + 1) − 𝑓(𝑎) m. 𝑔(𝑏 + 1) − 𝑔(𝑏) g. 𝑓(𝑎 + ℎ) − 𝑓(𝑎)

6. What is the range of each function given below? a. Let 𝑓(𝑥) = 9𝑥 − 1. b. Let 𝑔(𝑥) = 32𝑥.

Input 0 1 2 3 4 5

Output 1 2 4 8 16 32

0

1

2

3

4

5

1

2

4

8

16

32

𝑓

c. Let 𝑓(𝑥) = 𝑥2 − 4.

d. Let ℎ(𝑥) = √𝑥 + 2. e. Let 𝑎(𝑥) = 𝑥 + 2 such that 𝑥 is a positive integer. f. Let 𝑔(𝑥) = 5𝑥 for 0 ≤ 𝑥 ≤ 4.

7. Provide a suitable domain and range to complete the definition of each function. a. Let 𝑓(𝑥) = 2𝑥 + 3. b. Let 𝑓(𝑥) = 2𝑥. c. Let 𝐶(𝑥) = 9𝑥 + 130, where 𝐶(𝑥) is the number of calories in a sandwich containing 𝑥 grams of fat. d. Let 𝐵(𝑥) = 100(2)𝑥, where 𝐵(𝑥) is the number of bacteria at time 𝑥 hours over the course of one day.

8. Sketch the graph of the functions defined by the following formulas, and write the graph of 𝑓 as a set using set-builder notation. (Hint:

Assume the domain is all real numbers unless specified in the problem.)

a. 𝑓(𝑥) = 𝑥 + 2

b. 𝑓(𝑥) = 3𝑥 + 2

c. 𝑓(𝑥) = 3𝑥 − 2

d. 𝑓(𝑥) = −3𝑥 − 2

e. 𝑓(𝑥) = −3𝑥 + 2

f. 𝑓(𝑥) = −13

𝑥 + 2, −3 ≤ 𝑥 ≤ 3

g. 𝑓(𝑥) = (𝑥 + 1)2 − 𝑥2, −2 ≤ 𝑥 ≤ 5

h. 𝑓(𝑥) = (𝑥 + 1)2 − (𝑥 − 1)2, −2 ≤ 𝑥 ≤ 4

9. The figure shows the graph of 𝑓(𝑥) = −5𝑥 + 𝑐. (ENY M3-11)

a. Find the value of 𝑐.

b. If the graph of 𝑓 intersects the 𝑥-axis at 𝐵, find the coordinates of 𝐵.

PART B: For each Y = MX + B form, identify the slope (M) and y-intercept.

Write the equation of a line with the given slope and y-intercept.

Find the slope and y-intercept for each graph. Then write the equation in Y=MX+B form.

PART C: For each equation, graph the line on the axes provided. 2.

3) 4)

5) 6)

PART D: Write the equation of a line with the given point and slope.

Write the equation of a line given two points.

PART E: Identify the slope parallel and perpendicular to the graph of each equation.

PART F: Use the formula 𝑟𝑖𝑠𝑒

𝑟𝑢𝑛 to determine the slope of each line.

7) 8) 9)

PART G: Use the formula

12

12

xx

yy

to calculate the slope of each set of points.

PART H

PART I - Solve the linear system of equations by graphing.

PART J - Graph each of the following linear inequalities.

PART K (PH – Add’l Practice)

VERBAL PROBLEMS PACKET 2 Linear Functions & Systems p7

PART L – Model each problem with a linear equation or function. Identify the slope, x-intercept, and y-intercept.

Sketch the graph of the function.

1) Y is 3 more than twice x. 2) Y is 3 less than x. 3) Y is 5 more than one-half x. 4) Y is one-third x. 5) Y is 7 more than x. 6) Twice y increased by 4 times x is 8.

7) The sum of x and y is 10. 8) The sum of x and y is -4. 9) Twice the sum of x and y is 10.

10) Twice x increased by 4 is y. 11) One-third x decreased by 10 is y. 12) One-half y increased by 2 is 4.

13) The line parallel to y = 4x – 2 which passes through (1, 6). 14) The line parallel to y = -3x + 2 which passes through (3, -7).

15) The line parallel to y = -x which passes through (2, 0).

16) The line parallel to y = -x + 1 which passes through (5, -4).

17) The line parallel to y = -(1/2)x which passes through (2, 0).

18) The line parallel to y = -(1/3)x + 7 which passes through (6, 9).

19) The line parallel to y = -(2/5)x - 1 which passes through (-5, 6). 20) The line perpendicular to y = -x + 3 which passes through (4, 11).

21) The line perpendicular to y = -(1/2)x - 1 which passes through (4, 10).

22) The line perpendicular to y = -(2/3)x + 3 which passes through (-8, 2).

23) The line perpendicular to y = -(3/4)x + 3 which passes through (9, 1).

24) The line perpendicular to y = -5x + 3 which passes through (10, 10). 25) The line perpendicular to y = 3x + 3 which passes through (12, 2).

Model each scenario with a system of linear equations. Sketch each equation using its x and y-intercepts. Identify the

solution graphically or algebraically.

26) Twice a number x increased by 5 is y. Three times a number x decreased by 3 is y. Find x and y.

27) Three times a number x increased by 2 is y. Two times a number x decreased by 5 is y. Find x and y. 28) Four times a number x increased by 2 is y. Five times a number x decreased by 2 is y. Find x and y.

29) A number x increased by 10 is y. Three times a number x decreased by 2 is y. Find x and y.

30) Four times a number x increased by 7 is y. Twice a number x decreased by 5 is y. Find x and y.

31) The sum of twice x and three times y is 8. The sum of x and y is 3. Find x and y.

32) The sum of four times x and two times y is 10. The sum of x and y is 4. Find x and y.

33) The sum of three times x and three times y is 12. The sum of x and twice y is 6. Find x and y. 34) The sum of three times x and three times y is 3. The sum of three times x and twice y is 2. Find x and y.

35) The sum of x and four times y is 1. The sum of two times x and three times y is 7. Find x and y.

Identify the solution to each problem.

36) An exam has 20 questions: 5-point multiple choice (x) and 10-point open response (y) questions. The total points on the exam is 100 points. How many multiple choice and open-response questions are there?

37) An exam has 20 questions: 4-point multiple choice (x) and 6-point open response (y) questions. The total points

on the exam is 100 points. How many multiple choice and open-response questions are there?

38) An exam has 25 questions: 3-point multiple choice (x) and 8-point open response (y) questions. The total points on

the exam is 100 points. How many multiple choice and open-response questions are there?

39) An exam has 20 questions: 4-point multiple choice (x) and 6-point open response (y) questions. The total points on the exam is 100 points. How many multiple choice and open-response questions are there?

40) Concert ticket sale revenues are represented by the equations 5x + 2y = 48 and 3x + 2y = 32. If x represents the

cost for each adult ticket and y represents the cost for each student ticket, what is the cost for each adult ticket?