Functions Math B Honors
Module #7 Student Edition 2017-2018
Created in collaboration with Utah Middle School Math Project A University of Utah Partnership Project
San Dieguito Union High School District
SDUHSD Math B Honors Module #7 – STUDENT EDITION 2017-2018
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Table of Contents MODULE #7: FUNCTIONS STANDARDS FOR MATHEMATICAL PRACTICE: A GUIDE FOR STUDENTS AND PARENTS .................................... 5 SECTION 7.1: EXPLORE LINEAR AND NON-LINEAR FUNCTIONS* ....................................................... 6 7.1A LESSON: INTRODUCTION TO FUNCTIONS* ............................................................................................ 7 7.1B LESSON: FUNCTION RULES AND FUNCTION NOTATION* ...................................................................... 12 7.1B EXTENSION: VIDEO COMPANIES ........................................................................................................ 17 7.1C LESSON: LINEAR AND NON-LINEAR PATTERNS * ................................................................................. 18 7.1D LESSON: COMPARING LINEAR AND NON-LINEAR FUNCTIONS* ............................................................. 21 7.1D EXTENSION: BRIDGES AND HIGHWAYS .............................................................................................. 26 SECTION 7.2: ANALYZE FUNCTIONAL AND NON-FUNCTIONAL RELATIONSHIPS* .......................... 27 7.2A LESSON: ABSOLUTE VALUE FUNCTIONS ............................................................................................ 28 7.2A EXTENSION: POOL TABLE ................................................................................................................. 33 7.2B LESSON: QUADRATIC FUNCTIONS (PART 1) ....................................................................................... 34 7.2B EXTENSION: BRAKING DISTANCE....................................................................................................... 37 7.2C LESSON: QUADRATIC FUNCTIONS (PART 2) ....................................................................................... 38 7.2C EXTENSION: LINEAR AND NON-LINEAR FUNCTIONS ............................................................................ 41 7.2D LESSON: FEATURES OF GRAPHS AND INTERVAL NOTATION* ............................................................... 43 7.2E LESSON: STORIES AND GRAPHS* ...................................................................................................... 49 7.2E EXTENSION: THE HURDLES RACE ..................................................................................................... 53 7.2E TASK: GRAPHING STORIES ............................................................................................................... 55
* Denotes a lesson that was adapted from Utah Middle School Math Project © Utah Middle School Math Project & University of Utah http://utahmiddleschoolmath.org/ This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 Generic License http://creativecommons.org/licenses/by-nc/2.5/ This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/legalcode
SDUHSD Math B Honors Module #7 – STUDENT EDITION 2017-2018
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Module #7: Functions Online support for this module can be found at http://goo.gl/s2vDtM (case sensitive) or using the QR code below. This website includes copies of student lessons, homework, and instructional support videos.
Common Core Standard(s):
Curriculum Support Website
8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. 8.F.3 Interpret the equation as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function , giving the area of a square as a function of its side length, is not linear because its graph contains the points (1, 1), (2, 4) and (3, 9), which are not on a straight line. 8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
Academic Vocabulary: function, input, output, domain, range, function notation, independent variable, dependent variable, linear function, absolute value function, quadratic function, non-linear function, increasing, decreasing, constant, discrete, continuous
Module Overview: In this module, students transition from solving an equation for an unknown number, to exploring the concept of a “function” that describes a relationship between two variables. In a function, the emphasis is on the relationship between two varying quantities, where one value (the output) depends on another value (the input). We start the module with an introduction to the concept of function and provide students with the opportunity to explore functional relationships algebraically, graphically, numerically in tables, and through verbal descriptions. We then make the distinction between linear, absolute value, quadratic, and other nonlinear functions. Students analyze the characteristics of the graphs, tables, equations, and contexts of linear and nonlinear functions, solidifying the understanding that linear functions grow by equal differences over equal intervals. Finally, students use functions to model relationships between quantities that are linearly related.
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Connections to Content: Prior Knowledge: Up to this point, students have been working with linear equations. They know how to solve, write, and graph equations. In this module, students make the transition to functions. Functions describe situations where one quantity determines another. In this module, the goal is to have students understand the relationship between the two quantities and to construct a function to model the relationship between two quantities that are linearly related. Future Knowledge: This module develops an understanding of what a function is, and gives students the opportunity to interpret functions represented in different ways, identify the key features of functions, and construct functions for quantities that are linearly related. This work is fundamental to future coursework where students will apply these concepts, skills, and understandings to additional families of functions.
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Standards for Mathematical Practice: A Guide for Students and Parents The Standards for Mathematical Practices are central to the Common Core. These practices build fluency and help students become better decision-makers and problem solvers. The practices reflect the most advanced and innovative thinking on how students should interact with math content. Students and parents will develop skill with these standards by asking some of these questions:
Make Sense of Problems and Persevere in Solving Them.
What is the problem that you are solving for? Can you think of a problem that you recently solved that is similar to this one? How will you go about solving the problem?(i.e. What’s your plan?) Are you progressing towards a solution? How do you know? Should you try a different solution plan? How can you check your solution using a different method?
Construct Viable Arguments and Critique the Reasoning of Others.
Can you write or recall an expression or equation to match the problem situation? What do the numbers or variables in the equation refer to? What’s the connection among the numbers and variables in the equation?
Reason Abstractly and Quantitatively.
Tell me what your answers(s) mean(s) How do you know that your answer is correct? If I told you I think the answer should be (a wrong answer), how would you explain to me why I’m wrong?
Model with Mathematics.
Do you know a formula or relationship that fits this problem situation? What’s the connection among the numbers in the problem? Is your answer reasonable? How do you know? What do(es) the number(s) in your solution refer to?
Use Appropriate Tools Strategically.
What tools could you use to solve this problem? How can each one help you? Which tool is most useful for this problem? Explain your choice. Why is this tool (the one selected) better to use than (another tool mentioned)? Before you solve the problem, can you estimate the solution?
Attend to Precision.
What do the symbols that you used mean? What units of measure are you using (for measurement problems) Explain to me what (term from the lesson) means.
Look For and Make Use of Structure.
What do you notice about the answers to the exercises you’ve just completed? What do different parts of the expression or equation you are using tell you about possible correct answers?
Look for and Express Regularity in Repeated Reasoning.
What shortcut can you think of that will always work for these kinds of problems? What pattern(s) do you see? Can you make a generalization?
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Section 7.1: Explore Linear and Non-Linear Functions* Section Overview: This section begins by using a context to introduce a relation that represents a function and one that does not represent a function. By analyzing several contextual examples, students derive their own definition of a function. In the next section, the vending machine analogy is used to help students further their understanding of a function and input and output values. Students write functions in function notation and learn that the input is the domain and the output is the range. Students formally build and model functions in later sections and determine if a relation is a function from different representations (i.e. table, graph, mapping, story, patterns, equations, and ordered pairs). Students then focus on the characteristics that separate linear from nonlinear functions. Finally, they analyze different representations of a function to determine whether or not the representations suggest a linear relationship between two variables.
Concepts and Skills to be mastered: By the end of this section students should be able to:
Understand that a function is a rule that assigns to each input exactly one output. Understand that f(x) is function notation, is read “f of x”, and indicates that the equation is a function. Identify if a relation represents a function given different representations (i.e., table, graph, mapping, story, patterns, equations, and ordered pairs). Distinguish between linear and nonlinear functions given a context, table, graph, or equation.
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7.1A Lesson: Introduction to Functions* Name:
Period:
1. Jason is spending the week fishing at the Springville Fish Hatchery. Each day, he catches 2 fish for each hour he spends fishing. a. Determine the independent and dependent variables for this situation. Define your variables and write an equation.
b. Complete table of values and create a graph for this situation. Label each axis. Should the graph be discrete or continuous? Explain.
10
x:
9
y:
8
3
7
1
6 5
4
4 3
0
2
4
1 0
1
2
3
4
5
6
7
8
9 10
c. Why is this situation graphed only in the first quadrant? The situation above is an example of a function. In this situation, we would say that the number of fish caught is a function of the number of hours spent fishing or is a function of . 2. Sean is also spending the week fishing, but he is fishing in the Bear River. Each day he records how many hours he spends fishing, and how many fish he caught. The table of values shows this relationship. a. Label each column in the table and create a graph for this situation. Label each axis. x:
y: 10
1
4
9
0
0
8
2
5
6
3
1
5
3
8
3
5
5
2
7
4
1 0
1
2
3
4
5
6
7
8
9 10
b. Is this relation a function? Explain.
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3. Compare and contrast the relationship for Jason’s week spent fishing and Sean’s week spent fishing. Make a conjecture (an educated guess) about what kind of relationship is a function and what kind of relationship is not a function.
A mapping is a representation of a relation, and can also represent a function. A mapping is a diagram in which each input value is paired with exactly one output value. 4. Vanessa is buying gumballs at Vincent’s Drug Store. The mapping below shows the relationship between number of pennies, x, she puts into the machine, and the number of gumballs she gets out, y.
a. Complete the table of values and create a graph for this situation. Label each axis. Should the graph be discrete or continuous? Explain.
Number of pennies (x)
Number of gumballs (y)
12 11 10 9 8 7 6 5 4
b. Write an equation that models this relationship.
3 2 1 0
1
2
3
4
5
6
7
8
c. Is this relation a function? Explain.
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5. Kevin is across town at Marley’s Drug Store. The table relates the number of pennies he puts into the machine and how many gumballs he get out.
Number of pennies (x) 1
Number of gumballs (y) 2
a. Create a mapping and graph for this situation. Label each axis.
2
3
2
4
3
6
12 11 10 9 8 7 6 5 4 3 2 1
b. Is this relation a function? Explain.
0
1
2
3 4 5 6
7 8
6. Cody is at Ted’s Drug Store. The graph relates the number of pennies he puts into the machine on different occasions and how many gumballs he gets out. a. Explain how this gumball machine works.
b. Create a mapping for this situation.
c. Is this relation a function? Explain.
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A function is a relation in which each input value, x, corresponds to exactly one unique output value, y. Referring back to Jason and Sean’s fishing trips, Jason’s relation is a function because for each hour he spends fishing, there is a unique number of fish caught. In other words, the number of fish caught depends on the number of hours spent fishing. Sean’s relation is not a function because in three hours, he caught 1 fish and also 8 fish. For Sean’s relation, when time, x, is 3, there are two different y-values, 1 and 8. It is not possible to determine the number of fish Sean caught based on the number of hours he fished. For Sean, the number of fish is not a function of the number of hours spent fishing. Referring back to the gumball machines at the three different drugstores, the gumball machine at Vincent’s Drug Store represents a function because each penny inserted into the gumball machine generates a unique amount of gumballs. If you know how many pennies are inserted into the gumball machine at Vincent’s, you can determine how many gumballs will come out. However, the gumball machines at both Marley’s and Ted’s Drug Stores are not functions because there is not a unique number of gumballs generated based on the number of pennies you put in. You are unable to determine the number of gumballs that will come out based on how many pennies are put into the machine.
7. The cost for entry into a local amusement park is $45. Once inside, you can ride an unlimited number of rides. a. Determine the independent and dependent variables for this situation.
b. Complete the table of values and create a mapping and graph for this situation. Label and number each axis.
x:
y: 0 4 8 10
c. Should the graph be discrete or continuous? Explain.
d. Does this situation represent a function? Explain.
e. Write an equation that represents this situation.
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8. Ashley says that all linear equations are functions. Is she correct? Explain why or why not and justify your reasoning by sketching two graphs.
9. What type of line does not represent a function?
The vertical line test is a visual way to determine if a curve is a graph of a function or not. A relation is a function if there are no vertical lines that intersect the graph at more than one point. Use the vertical line test to determine if the following graphs represent functions.
10.
11.
12.
13.
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7.1B Lesson: Function Rules and Function Notation* Name:
Period:
Another way to think about the x and y variables is as input and output values. To better understand how input and output values are related to a function, we’ll investigate the following Vending Machine Analogy. When you buy snacks from a vending machine, you push a button (input) and out comes your candy (output). Let’s pretend that G4 corresponds to a Snickers bar. If you input G4, you would expect to get a Snickers bar as your output. If you entered G4 and sometimes the machine spits out a Snickers and other times it spits out a Kit Kat bar, you would say the machine is “not functioning.” One input, G4, corresponds to two different outputs, Snickers and Kit Kat. This situation is represented by the following mapping:
What would a mapping look like for a vending machine that is “functioning” properly?
Input
Output
When the vending machine is “functioning” properly, each input corresponds to exactly one output. The snack that comes out of the machine is dependent on the button you push. We call this variable the dependent variable. The button you push is the independent variable. Let’s look at one more scenario with the vending machine. There are times that different inputs will lead to the same output. With some vending machines, companies may stock popular items in multiple locations in the machine. This can be represented by the mapping to the right. Does this mapping represent a function? Explain why or why not.
In a function, the set of input values is defined as the domain. The set of output values is defined as the range.
When the specific input, or x-value, is inserted into the function or rule, the output is a function of x. The output is represented as f(x), which means “the function evaluated at x” and is read “f of x.” This is called function notation.
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In the table below, each input value, x, corresponds to an output value, y or . Imagine the input value going into a function machine and coming out a different number, the output value. Something is happening to that number in the function machine. Directions: Each table of values given below represents a function. State the domain, range, and function rule. Then explain what the function rule represents. 1.
Input x -5 -3 1
Output f(x) 15 17 21
3 5
23 25
a. What values represent the domain?
b. What values represent the range?
c. What function rule represents the table of values? Write the rule in function notation.
d. Explain in words what the function rule represents.
2.
Input x -9 0 3
Output f(x) -3 0 1
12 30
4 10
a. What values represent the domain?
b. What values represent the range?
c. What function rule represents the table of values? Write the rule in function notation.
d. Explain in words what your function rule represents.
Directions: Use the words input and output to describe the function rule for each of the following relations. Write the function rule as an equation using function notation. If the relation is not a function, explain why. 3.
Input x 12 -9 4 -8 20
Output f(x) 7 -14 -1 -13 15
4.
Input x 1 -4 5
Output f(x) 4 -11 16
-8 3
-23 10
Function description:
Function description:
Function rule:
Function rule
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5.
Input x 2 5 2 4 2
6.
Output f(x) 3 -9 7 9 0
Input x
Output f(x)
-2 3 -1 -5 4
-8 27 -1 -125 64
Function description:
Function description:
Function rule:
Function rule:
7. Are all functions linear? Explain why or why not.
Directions: Given the function
, evaluate the function for each value of x.
8.
9.
(
)
10.
( )
11. Examining your values for problems #8-10, state the domain and range. Domain:
Range:
12. What value of x makes f(x) = 0?
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Directions: Let 13.
[
and
. Find the following values. Show all of your work.
]
15.
14.
[
16.
( )
]
( )
Functions can also describe non-numeric relations. Study each non-numeric relation and its mapping below. Does the relation represent a function? Explain your reasoning. 17. Student’s name versus the color of their shirt. Name
Color
Sam Joe Luis Mia
Red Blue Green
18. Where a person lives versus the team they root for in college football. Live California Nevada Arizona
SDUHSD Math B Honors Module #7 – STUDENT EDITION 2017-2018
Team Aztecs Bruins Utes Sun Devils
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19. Make a mapping of students’ names (first name and last name) in your classroom vs. their birthday. Collect data from 10 students. Name
Birthday
Is this relation a function? Justify your answer.
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7.1B Extension: Video Companies Name:
Period:
Joe works at a video streaming company that has two monthly plans to choose from: Plan 1
Plan 2
Flat rate of $7/month plus $2.50 for each video viewed
$4 per video viewed
1. What is the independent variable? What is the dependent variable?
2. Write an equation for each plan for one month. Define your variables.
3. Which situation represents a function? Explain why.
4. How much would 3 videos cost for each plan?
5. How much would 5 videos cost for each plan?
6. Compare the two plans and explain what advice you would give customers trying to decide which plan is best for them, based on how many videos they usually watch each month.
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7.1C Lesson: Linear and Non-linear Patterns * Name:
Period:
Complete Foods, a local grocery store, has hired two different companies to design a display for food items that are on sale each week. They currently have a display that is 6 boxes wide. They would like the center part of the display to be taller than the outside pieces of the display to showcase their “mega deal of the week.” The following are the designs that two different companies submitted to Complete Foods, using the current display as their starting point. Design Team 1:
Stage 2
Current Display (Stage 1)
Stage 3
1. Draw Stage 4 of this design. Explain your process.
2. Can the relationship between stage number and number of boxes in a stage in this pattern be modeled by a linear function? Create a table, graph and equation to support your answer. Table of Values
Graph
Equation
y
Number of Boxes Number of boxes
Stage
16 14 12 10 8 6 4 2 0
1
2 3 4 stage number
5
Is the relation a function based on the table of values?
Is the relation a function based on the graph?
Is the relation linear based on the table of values?
Is the relation linear based on the graph?
x
Does the equation represent a linear relation?
3. Justify that the pattern represents linear growth by using the model.
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Design Team 2:
Current Display (Stage 1)
Stage 2
Stage 3
4. Draw Stage 4 of this design. Explain your process.
5. Can the relationship between stage number and number of boxes in a stage in this pattern be modeled by a linear function? Create a table and graph to support your answer.
Table of Values Stage
Graph
Number of Boxes
Is the relation a function based on the table of values?
Is the relation a function based on the graph?
Is the relation linear based on the table of values?
Is the relation linear based on the graph?
6. Justify that the pattern does not represent linear growth by using the model.
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7. Owen earns quarters each day that he makes his bed in the morning. On the first day, Owen’s mom gives him 2 quarters. On the second day, Owen’s mom gives him 4 quarters. On the third day, he receives 6 quarters, and 8 quarters on the fourth day. Owen makes his bed every day and this pattern continues. The model shows how many quarters Owen earns each day (each box represents 1 quarter). Consider the relationship between the number of quarters received on a given day and the day number. a. What is the independent variable? Dependent variable?
b. Is the data linear? Explain your reasoning.
c. Define your variables and write an equation to model the relationship between the number of quarters received and the day number.
d. Consider the relationship between the total number of quarters Owen has earned and the day number. Complete the table of values.
Day
e. Is the data for total number of quarters linear? Explain your reasoning.
# of quarters added that day
Total # of Quarters
1 2 3 4 5
8.
Carbon-14 has a half-life of 5,730 years. The table below shows the amount of carbon-14 that will remain after a given number of years. Consider the relationship between number of years and amount of carbon-14 remaining. a. What is the independent variable? Dependent variable?
b. Is the data linear? Explain your reasoning.
# of years
Milligrams of Carbon-14
0
8
5,730
4
11,460
2
17,190
1
22,920
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7.1D Lesson: Comparing Linear and Non-Linear Functions* Name:
Period:
Directions: Complete the table of values and sketch a graph of the given function. In the tables, round to the nearest tenth when needed. State whether or not the given function is linear and justify your reasoning. Function
Table of Values x
1.
Sketch of Graph y
y
-2 Linear Function? Explain why or why not.
-1 x
0 1 2
2.
| | Linear Function? Explain why or why not.
x
y
y
-2 -1 x
0 1 2
x
3. Linear Function? Explain why or why not.
y
y
-2 -1 x
0 1 2
4.
√
x
y
y
0 Linear Function? Explain why or why not.
1 2
x
3 4
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5.
x Linear Function? Explain why or why not.
y
y
0 1 2
x
3 4
x
6.
y
y
-2 Linear Function? Explain why or why not.
-1 0 x
1 2
7. Using the table of values, describe the qualities you see in the equations of non-linear functions
8. Circle the letter next to the table if the data represents a linear function.
A
x 0 1 2 3
y -5 0 5 10
x 2.1 2.2 2.3 2.4
y 4 5 6 7
E
B
x 0 1 2 3
y 4 8 16 32
x 1 3 6 10
y 2 4 6 8
F
C
G
x 0.2 0.4 0.6 0.8
y 0.2 0.4 0.6 0.8
x 5 5 5 5
y -10 -5 0 5
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D
H
x 10 30 50 70
y -20 -40 -60 80
x 4 8 12 16
y 2 1 0 -1
22
6. Consider the perimeter of a square as a function of the side length of the square. a. What are the independent and dependent variables?
b. Complete the graph and table for this function. side length
y 24
perimeter
22 20 18
1
Perimeter
2 3 4
16 14 12 10
5
8 6 4
c. Write an equation to model the perimeter, P, as a function of side length, s.
2 0
1
2
3
4
5
6
7
x
Side length
d. What does the point (10, 40) represent in this context?
e. Find another ordered pair that the graph would pass through.
f. Is this function linear? Explain your reasoning using the graph, table of values, and equation.
Graph
Table of Values
Equation
g. Is this relation proportional? Explain why or why not.
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10. Consider the area of a square as a function of the side length of the square. a. Draw pictures to represent the first four stages of these squares. The first two have been drawn for you.
Side length = 1
Side length = 2
Stage 3
Stage 2
Stage 1
y
b. Complete the graph and table for this function. Side length
Stage 4
28
Area
26 24
1
22 20 18
2 3 Area
4 5
16 14 12 10
c. Write an equation to model the area, A, as a function of side length, s.
8 6 4 2 0
1
2
3
4
5
6
7
8
x
Side length d. Would this graph pass through the point (8, 64)? Justify your answer.
e. What does the point (8, 64) represent in this context?
f. List three more ordered pairs that this graph would pass through.
g. Is this function linear? Explain your reasoning using the graph, table of values, and equation. Graph
Table of Values
SDUHSD Math B Honors Module #7 – STUDENT EDITION 2017-2018
Equation
24
Directions: Examine the given situation and complete the following. 11. Mr. Cortez drove for 5 hours. At the end of 2 hours, he had driven 90 miles. After 5 hours, he had driven 225 miles. a. State the independent and dependent variables.
b. Complete the table of values. Can time vs. distance driven be modeled by a linear function? Explain why or why not.
x:
y: 2 3
135 180
5
c. How fast is Mr. Cortez driving?
8
360
12. A tennis tournament starts with 128 players and after round 1, there are 64 players left. After each round, half the players have lost and are eliminated from the tournament. Therefore, in round 2, there are 32 players, in round 3, there are 16 players and so on. a. State the independent and dependent variables.
b. Complete the graph and label each axis. Can round number vs. number of players be modeled by a linear function? Explain why or why not.
140 120 100 80 60
40 20 0
1
2
3
4
5
6
7
13. Your family buys a printer for $100 and the ink cartridges cost $25 each. a. Can the number of ink cartridges purchased vs. total cost be modeled by a linear function? Explain why or why not.
b. Define your variables and write an equation that represents this situation.
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7.1D Extension: Bridges and Highways Name:
Period:
Bridges and highways often have expansion joints, which are small gaps in the roadway between one bridge section and the next. The gaps are put there so that the bridge will have room to expand when the weather gets hot. Suppose a bridge has a gap of 1.3 cm when the temperature is , and that the gap narrows to 0.9 cm when the temperature warms to . 1. What is the independent and dependent variables for this situation? Explain.
2. Assuming the gap width varies linearly with the temperature, write a function equation in slopeintercept form that relates the gap width, , to temperature, . Show all of your work.
3. How wide would the gap be at Gap at
? How wide would the gap be at
? Show all of your work.
Gap at
4. At what temperature would the gap close completely? If this situation was graphed, what would this value represent on the graph?
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Section 7.2: Analyze Functional and Non-Functional Relationships* Section Overview: In this section, students will analyze absolute value and quadratic functions. They will apply different transformations to alter functions, and understand that absolute value functions are represented by a “V” shaped graph and quadratic functions are represented by a parabola. Students will learn key features of graphs of functions and interval notation and apply this knowledge to describe functional relationships between two quantities. Students will analyze distance-time graphs.
Concepts and Skills to be mastered: By the end of this section students should be able to:
Understand absolute value and quadratic functions. Match the representations (table, graph, equation, and context) of linear and other nonlinear situations. Identify and interpret key features of a graph that models a relationship between two quantities. Sketch a graph that displays key features of a function that has been described in context.
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7.2A Lesson: Absolute Value Functions Name:
Period: | | to complete the following.
1. Use the equation
a. Complete the table of values. x
b. Create a mapping for
| |
y
-4 1 0 1 4
| |
c. Graph the equation
d. Explain in words why the graph of a “V” shape graph.
| | and
2. The graphs of
|
| | creates
| are shown below. Complete the following.
a. Explain in words how the graphs are similar.
b. Explain in words how the graphs are different.
c. How is
|
| shifted from
| |?
SDUHSD Math B Honors Module #7 – STUDENT EDITION 2017-2018
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| | will differ from the graph of
3. Without graphing, make a prediction about how the graph of | |
| | and
4. Use the equations
| | to complete the following. | |
a. Complete the table of values. x
-4
1
-2
0
0
-1
2
-4
4
|
5. Without graphing, make a prediction about how the graph of | |
6. Use the equation
|
| to complete the following.
a. Complete the table of values for
|
|
b. On the same coordinate plane you used above, graph
x
y
4
b. Graph each equation on the same coordinate plane. Label your graphs.
c. Was your prediction correct?
| |
x
y
| will differ from the graph of
-4
-2
0
2
4
y |
|
c. Was your prediction correct? SDUHSD Math B Honors Module #7 – STUDENT EDITION 2017-2018
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7. Without graphing, make a prediction about how the graph of | |
8. Use the equation
| |
-2
-1
will differ from the graph of
to complete the following. | |
a. Complete the table of values for x
| |
0
1
2
b. On the coordinate plane below, graph Label your line.
| |
𝒚
|𝒙|
y
c. Was your prediction correct?
| |
9. Use the equation
to complete the following. | |
a. Complete the table of values for x
-2
-1
0
1
. 2
y | |
b. Graph
on the same coordinate plane used above. | |
c. Explain in words how the graph of
|
10. Use the equation
-3
-1
| |.
| to complete the following. |
a. Complete the table of values for x
differs from the graph of
1
|.
3
5
y
b. Graph
|
|on the same coordinate plane used above.
c. Explain in words how the graph of
|
| differs from the graph of
SDUHSD Math B Honors Module #7 – STUDENT EDITION 2017-2018
| |.
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| | The general form for any absolute value function is the graph of an absolute function as compared to the graph of
How do the values of | |?
and
affect
11. Complete the tables below. When the value of negative
is
When the value of
When the value of positive
is positive
is
When the value of increases
When the value of
a. What does the
control in the graph of
|
|
?
b. What does the
control in the graph of
|
|
?
c. What does the
control in the graph of
|
|
?
When the value of decreases
is negative
Directions: Describe in words how the given absolute value graph differs from the graph of Explain any transformations that might take place. 12.
13.
|
| |.
|
| |
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14.
| |
15.
|
|
Directions: Write an equation of the absolute value graph with the given description. 16. The equation of a graph shifted left 5 units and 9 units down from
| |.
17. The equation of a graph shifted up 4 units from
| |
18. The equation of a graph shifted 8 units up from
| |, opens down and is wider.
19. The equation of a graph shifted right 2 units from
.
| | and is steeper.
20. The equation of a graph shifted right 6 units and 8 units down from
SDUHSD Math B Honors Module #7 – STUDENT EDITION 2017-2018
|
|
.
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7.2A Extension: Pool Table Name
Period:
On Saturday night, Jackson met up with his friend Sean to play pool. On the first game, Jackson is hoping to sink the five ball in the bottom left pocket. He decides to use absolute value functions to determine if he will make the shot. Directions: Use the pool table picture below to complete the following. 1. Jackson first hits the ball at (-0.5, 4) and banks it off the side of the pool table at (-1.25, 5). He wants the | | ball to go in the pocket at (-5, 0). Using the the general form , write an equation for the path of the ball. Show all of your work.
\
2. Does Jackson make the shot? Explain why or why not.
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7.2B Lesson: Quadratic Functions (Part 1) Name:
Period: adapted from Illuminations.nctm.org
The Steel Dragon in Japan was built in 2000 and is one of the longest and tallest roller coasters in the world. Use the function to determine the height of the Steel Dragon as it falls from its tallest drop. The variable h represents the height above ground (in feet) and t represents the time the coaster has been falling (in seconds) 1. Create a table of values to determine how long it takes the Steel Dragon to reach the bottom. Write your data as an ordered pair. Time in seconds
Substitute value of t into the equation
Height above ground in feet
Ordered pair
400
(0,400)
0 h = 400 1 2 3 4 5
2. Graph the ordered pairs.
3. After how many seconds does the Steel Dragon reach the bottom? How do you know?
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4. Determine the average velocity of the Steel Dragon by using the function:
The equation represents a quadratic function. A quadratic function is a function that can be described by an equation in the form In a quadratic function, the greatest power of the variable is 2. The graph of a quadratic function is a parabola. Directions: Complete the table and graph the given quadratic functions. 5.
x
y
6.
x
2
2
1
1
0
0
-1
-1
-2
-2
y
7. The graphs are symmetric across which line?
8. Explain in words how symmetry is shown in the tables of values.
9. Why does this symmetry occur? Explain using the equations.
10. Explain in words how the graph of changes occur?
compares to the graph of
SDUHSD Math B Honors Module #7 – STUDENT EDITION 2017-2018
. Why do these
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11. Without graphing, explain in words how the graph of of .
would differ from the graph
Directions: Complete the table and graph the given quadratic functions. x
12.
y
13.
x
2
2
1
1
0
0
-1
-1
-2
-2
y
14. Complete the table of values for each function. Graph each function on the same coordinate plane. Label each graph as linear, absolute value, or quadratic. | | x 4 2 1 0 -1
y
x
y
x
2
3
1
1
0
0
-1
-1
-2
-3
y
a. State the point of intersection
b. Verify algebraically that your point of intersection is correct.
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7.2B Extension: Braking Distance Name:
Period:
Suppose that a Toyota Prius has a braking distance that can be computed with the following quadratic function,
, where
represents miles per hour and
represents braking distance (in feet).
1. What is the braking distance, in feet, of this car going 30 mph?
2. What is the braking distance, in feet, of this car going 60 mph?
3. What is the braking distance, in feet, of this car going 90 mph?
4. Suppose that a car took 500 feet to brake. Use your computations above to make a prediction about how fast the car was going when the brakes were applied. Justify your prediction.
5. Using your data from above, complete the table of values and graph the distance as a quadratic function on a coordinate plane. Label your ordered pairs.
mph
distance
30
distance
60 90 500
6. Evaluating your graph, how fast was the car going if the driver needed 500 feet to brake? Was your prediction correct?
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7.2C Lesson: Quadratic Functions (Part 2) Name: 1. The graphs of
Period: and
are shown below.
a. How does the graph graph ?
compare to the
b. Predict what the graph in comparison to the graph
would look like
c. Complete the table and graph the function on the same coordinate plane. Was your prediction correct?
x 3
y
4 5 6 7
Directions: Complete the table and graph for the given quadratic function. 2.
x
y
3.
x
2
2
1
1
0
0
-1
-1
-2
-2
SDUHSD Math B Honors Module #7 – STUDENT EDITION 2017-2018
y
38
The general form for any quadratic function is the graph of a quadratic function as compared to the graph of
How do the values of ?
and
affect
4. Complete the tables below. When the value of positive
is
When the value of
When the value of is negative
is positive
When value of
and the increases
When the value of
a. What does the
control in the graph of
?
b. What does the
control in the graph of
?
c. What does the
control in the graph of
?
5. Without graphing, explain in words how the graph of
and the decreases
is negative
compares to the graph of
6. Without graphing, explain in words how the graph of
7. Without graphing, explain in words how the graph of
When value of
compares to the graph of
compares to the graph of
8. Write the equation of a quadratic function that is narrower, opens down, and shifts 2 units to the right from the graph of .
9. Write the equation of a quadratic function that is shifted 4 units to the left and 5 units down from the graph of
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10. Write the equation of a quadratic function that opens down and is shifted 3 units to the right from the graph of
11. Write the equation of a quadratic function that is shifted 8 units up and is wider than the graph of
Directions: For each representation, determine if the function is linear, absolute value, quadratic, or none. Justify your reasoning. 12. Josh is draining a swimming pool at a constant rate of 5 gallons per minute.
14.
Height
Shoe size
62
6
74
13
70
9
67
11
16. A person’s height as a function of a person’s age (from age 0 to 100).
13.
Time -3 -1
Feet 18 10
0 1
9 10
3
18
15. Scott tossed a water balloon in the air, releasing it at 3 feet above the ground. The water balloon hits the ground after 3 seconds.
17.
x
y
2
4
1
2
0
0
-1
2
2
4
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7.2C Extension: Linear and Non-Linear Functions Name:
Period:
Directions: Determine if each situation given represents a linear or non-linear function. Complete the missing information for each situation. Situation 1. It is 10 degrees below zero at 7am. The temperature increases 2 degrees per hour.
Table and Equation Hours after 7 am 0
Graph
Temp. °F
4 Is this situation linear or non-linear? Explain.
8 Equation:
2. Joe throws a whiffle ball out of a window that is 4 meters high. The ball hits the ground 2 seconds after being thrown. Is this situation linear or non-linear? Explain.
Time
Meters
0 1 1.5 2
Equation:
3. You have 1 virus on your computer. The number of viruses quadruples each week.
Is this situation linear or non-linear? Explain.
# of weeks 0
# of viruses
1 2 3 Equation: y = 4x
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4. You have 4 songs on your ipod playlist. You download 3 songs per day.
# of days
# of songs
0 2 3
Is this situation linear or non-linear? Explain.
4 Equation:
5. The number of cell phone users in Circle Town starts at 100 and increases by 75% each year. Is this situation linear or non-linear? Explain.
# of years 0
# of users
1 2 3 Equation: y = (100)(1.75)x
6. Josh is draining a swimming pool at a constant rate of 5 gallons per minute. The swimming pool starts with 200 gallons of water. Is this situation linear or non-linear? Explain.
Minutes
Gallons
0 4 8 10 Equation:
In how many minutes will the swimming pool be completely drained?
How is this value shown on the graph?
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7.2D Lesson: Features of Graphs and Interval Notation* Name:
Period:
Directions: Sort graphs A through P into the groups below by writing the letter of the graph in the correct description of what is happening in the graph.
Function is increasing on the entire graph
Function has a minimum point
Function is constant for an extended amount of x-values
Function is increasing on some parts of the graph, decreasing on some parts of the graph, and constant for an extended amount of x-values on some parts of the graph
Graph is Discrete
Graph is Continuous
Non-linear function
Multiple linear functions on the same graph
Linear function
Directions: Define the following terms in your own words. 1. Increasing graph-
2. Decreasing graph-
3. Constant graph-
4. Discrete graph-
5. Continuous graph-
6. Minimum point-
7. Maximum point-
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8. What type of slope in a linear function represents an increasing graph? Decreasing graph? Constant graph? Increasing Graph
Decreasing Graph
Constant Graph
9. Draw an example of a function that increases on some parts of the graph, has a minimum point, and decreases on some parts of the graph.
10. Draw an example of a function that decreases on some parts of the graph and is constant for an extended amount of x-values on some parts of the graph.
11. Draw an example of a function that is discrete and increasing and non-linear.
12. Draw an example of a function that is continuous decreasing and linear.
13. Use the graph to complete the following. a. For what x-values is the graph increasing?
b. For what x-values is the graph decreasing?
c. For what x-values is the graph constant?
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y
14. Draw a graph that is decreasing, constant, and then increasing.
8
a. State the interval(s) where your graph is decreasing.
6 4 2
b. State the interval(s) where your graph is constant.
-8
-6
-4
-2
2 -2
4
6
8
x
-4
c. State the interval(s) where your graph is infinitely increasing.
-6 -8
Directions: State the x- and y-intercepts for each graph as ordered pairs. State the intervals of increase, decrease and constant in interval notation. 15. x-intercept(s):
y-intercept(s):
interval of increase:
interval of decrease:
interval of constant:
16. x-intercept(s): y-intercept(s):
interval of increase:
interval of decrease:
interval of constant:
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Sorting Graphs for Lesson 5.2D A
B
C
D
E
F
SDUHSD Math B Honors Module #7 – STUDENT EDITION 2017-2018
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G
I
K
H
J
L
SDUHSD Math B Honors Module #7 – STUDENT EDITION 2017-2018
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M
O
N
P
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7.2E Lesson: Stories and Graphs* Name:
Period:
Directions: Match the story with the correct graph. 1. Lucy walked away from her house at a steady rate. On each graph, distance represents the distance from Lucy’s house.
2. Sunny walked toward the swimming pool at a steady rate. On each graph, distance represents the distance from the swimming pool.
3. Anna ran towards the beach and then walked away. On each graph, distance represents the distance from the beach.
4. Max stood still as the others were running.
5. Why are the graphs above only graphed in the 1st quadrant?
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6. Use the distance-time graph to answer the following questions. a. What is the speed during the first 20 seconds?
b. Starting at 0 seconds, what is the distance after 50 seconds?
c. What is the speed from 60 to 80 seconds?
d. When was the object traveling the fastest? Explain.
7. Draw a possible distance-time graph for the story described below. Label your starting point, ending point and the parts of the graph that represent the given story. Story: Stacy walks at a constant rate from her house to the bus stop. She sees the bus coming, so she runs at a constant rate to catch it. She gets to the bus stop, but misses the bus. After a short rest, Stacy turns around and walks back home at a constant rate.
a. What does the graph look like when Stacy walks from her house to the bus stop? Explain why.
b. What does the graph look like when Stacy walks home?
c. What happens to the line if Stacy runs?
d. What happens to the line if Stacy walks?
e. What happens to the line when Stacy stands still?
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Directions: Create a story that represents the graph. 8. Cynthia is doing research on how hot coffee is when it is served. At one coffee shop, the coffee was served at 180˚ F. Cynthia bought a coffee and then left it on the counter to cool. The temperature in the room was 70˚ F. The graph shows the temperature of the coffee (in ˚F) as a function of time (in minutes) since it was served.
Story (include statements about temperature):
9. A toy rocket is launched straight up in the air from the ground. It leaves the launcher with an initial velocity of 96 ft. /sec. The graph shows the height of the rocket in feet with respect to time in seconds. Story:
.
10. B-Bytes is releasing the most anticipated new Xbox game of the summer. The graph shows the total number of games sold as a function of the number of days since the game was released. Story:
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Directions: Sketch a graph that represents the given story.
Distance from school
11. Zach walks home from school. When the bell rings, Zach runs to his locker to grab his books and starts walking home. When he is about halfway home, he realizes that he forgot his math book so he turns around and runs back to school. After retrieving his math book, he realizes that he is going to be late so he sprints home. Sketch a graph of Zach’s distance from school as a function of time since the bell rang.
Time
420 360 Revenue ($)
12. Solitude is offering a ski clinic for teens. The cost of the class is $30 per student. A minimum of 5 students must sign up in order for Solitude to hold the class. The maximum number of students who can participate in the class is 12. Sketch a graph that shows the revenue Solitude will bring in dependent on the number of students that take the class.
300 240 180 120
60 0
2 4 6 8 10 12 14 16 # of Students
13. A parking garage charges $5 per hour and a flat rate of $40 for 8 or more hours. Sketch a graph of the total cost depending on how many hours a car is in the garage.
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7.2E Extension: The Hurdles Race Name:
Period: The Hurdles Race
Distance (meters)
400
A
300
B 200
C D
100
15
30
45
60
75
90
105
120
135
150
Time (seconds) Directions: Use the graph to complete the race commentary below.
Ladies and Gentlemen, boys and girls, welcome to this year’s women’s Olympic qualifying 400 meter hurdles. In lane A we have Ann Packer, in lane B we have Betty Cuthbert, in lane C we have Christine Ohuruogu, and in lane D we have Deedee Trotter. This is going to be a close race. The racers are set on the blocks. Ready, set, and they are off. 1)___________ shot out of the block at 2)_______meters per sec and is creating a large lead over the other 3 runners. Let’s see if she can keep up her speed, wait, oh no…15 seconds into the race, it looks like we have a runner down, I am not sure if 3)________ was pushed, or accidently 4)___________________. 5)____________ has taken over the lead; she is running 6)____________ meters per second. She is going so fast I do not think anyone will catch her. There is a close race between 2nd and 3rd place now. In 2nd place is 7)______________; she is running 8)______ meters per second. And closely behind her is 9)__________________, who is running 10)_____ m/s. Wait one minute, 11)____________ has just started to pick her speed, she is now running at 12)______meters per second. Will she be able to hold this speed for the last 13)_____ meters of the race? It seems that she started her final sprint a little early into the race. Here she goes 14) _____ meters into the race and she has now taken over 2nd place. Oh no, they both need to not trip over 15) ___________ who is still hurt in the track. Whoa, I am glad they were able to get around her. Will our current second place runner 16)______________ be able to catch up with 17)___________? SDUHSD Math B Honors Module #7 – STUDENT EDITION 2017-2018
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She is running faster than her, but she has a lot of ground to make up. They are coming down the final stretch. Crossing the finish line first at 18)_____seconds is 19)_________. In second with a time of 20)______seconds is 21)_____________. Finally in third place with a time of 22)______seconds is 23) _________. Poor 24) _________ was never able to get back into the race.
Directions: Write your corresponding answers in the table below. 1.
13.
2.
14.
3.
15.
4.
16.
5.
17.
6.
18.
7.
19.
8.
20.
9.
21.
10.
22.
11.
23.
12.
24.
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7.2E Task: Graphing Stories Name:
Period: task adapted from Dan Meyer and buzzmath, Graphing Stories
Directions: Working in groups of 3 or 4, watch each video and create a graph to represent the situation.
#1 Distance from Home Plate Label each axis and number the y-axis Video: graphingstories.com/21f
#2 Video: Weight of Cups Label each axis and number the y-axis Video: graphingstories.com/0xa
What is the approximate slope value? What is the maximum point? What does the point represent in the context?
What is the weight at 8 seconds? What valid equation could represent this story?
#3 Video: Balloon Length Label each axis and number the y-axis Video: graphingstories.com/RUV
Explain what is happening when the graph is constant.
#4 Video: Water Volume Label each axis and number the y-axis Video: graphingstories.com/6NN
What is the approximate slope value?
Why does this graph decrease at the end?
What valid equation could represent this story?
Does this graph represent a function?
Is this story proportional?
Does this graph represent a linear function? SDUHSD Math B Honors Module #7 – STUDENT EDITION 2017-2018
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