Complete Unit 2
Package
HighSchoolMathTeachers.com©2017
Table of Contents
Unit 2 Pacing Chart -------------------------------------------------------------------------------------------- 1
Algebra 1 Unit 2 Skills List ---------------------------------------------------------------------------------------- 4
Unit 2 Lesson Plans -------------------------------------------------------------------------------------------- 5
Day 21 Bellringer -------------------------------------------------------------------------------------------- 29
Day 21 Activity -------------------------------------------------------------------------------------------- 31
Day 21 Practice -------------------------------------------------------------------------------------------- 34
Day 21 Exit Slip -------------------------------------------------------------------------------------------- 37
Day 22 Bellringer -------------------------------------------------------------------------------------------- 39
Day 22 Practice -------------------------------------------------------------------------------------------- 41
Day 22 Exit Slip -------------------------------------------------------------------------------------------- 44
Day 23 Bellringer -------------------------------------------------------------------------------------------- 46
Day 23 Practice -------------------------------------------------------------------------------------------- 48
Day 23 Exit Slip -------------------------------------------------------------------------------------------- 50
Day 24 Bellringer -------------------------------------------------------------------------------------------- 52
Day 24 Activity -------------------------------------------------------------------------------------------- 54
Day 24 Exit Slip -------------------------------------------------------------------------------------------- 59
Day 25 Bellringer -------------------------------------------------------------------------------------------- 61
Week 5 Assessment -------------------------------------------------------------------------------------------- 63
Day 26 Bellringer -------------------------------------------------------------------------------------------- 69
Day 26 Activity -------------------------------------------------------------------------------------------- 72
Day 26 Exit Slip -------------------------------------------------------------------------------------------- 73
Day 27 Bellringer -------------------------------------------------------------------------------------------- 75
Day 27 Activity -------------------------------------------------------------------------------------------- 77
Day 27 Practice -------------------------------------------------------------------------------------------- 79
Day 27 Exit Slip -------------------------------------------------------------------------------------------- 82
Day 28 Bellringer -------------------------------------------------------------------------------------------- 84
Day 28 Activity -------------------------------------------------------------------------------------------- 86
Day 28 Exit Slip -------------------------------------------------------------------------------------------- 95
Day 29 Bellringer -------------------------------------------------------------------------------------------- 97
Day 29 Activity -------------------------------------------------------------------------------------------- 99
Day 30 Bellinger -------------------------------------------------------------------------------------------- 103
Week 6 Assessment -------------------------------------------------------------------------------------------- 105
Day 31 Bellringer -------------------------------------------------------------------------------------------- 111
Day 31 Activity -------------------------------------------------------------------------------------------- 113
Day 31 Practice -------------------------------------------------------------------------------------------- 116
Day 31 Exit Slip -------------------------------------------------------------------------------------------- 122
Day 32 Bellringer -------------------------------------------------------------------------------------------- 124
Day 32 Activity -------------------------------------------------------------------------------------------- 126
Day 32 Practice -------------------------------------------------------------------------------------------- 129
Day 32 Exit Slip -------------------------------------------------------------------------------------------- 134
Day 33 Bellringer -------------------------------------------------------------------------------------------- 136
Day 33 Activity -------------------------------------------------------------------------------------------- 138
Day 33 Practice -------------------------------------------------------------------------------------------- 143
Day 33 Exit Slip -------------------------------------------------------------------------------------------- 150
Day 34 Bellringer -------------------------------------------------------------------------------------------- 152
Day 34 Activity -------------------------------------------------------------------------------------------- 154
Day 34 Practice -------------------------------------------------------------------------------------------- 159
Day 34 Exit Slip -------------------------------------------------------------------------------------------- 166
Day 35 Bellringer -------------------------------------------------------------------------------------------- 168
Week 7 Assessment -------------------------------------------------------------------------------------------- 170
Unit 2 Test -------------------------------------------------------------------------------------------- 176
CCSS Algebra 1 Pacing Chart – Unit 2
HighSchoolMathTeachers © 2017 Page 1
Unit Week Day CCSS Standards Mathematical Practices Objective I Can Statements
2 – Linear Equations
5 – Slope 21
CCSS.MATH.CONTENT.HSF.IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*
CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.
The student can calculate the average rate of change for a linear function given the equation or a table.
I can calculate the average rate of change for a linear function given the equation or a table.
2 – Linear Equations
5 – Slope 22 CCSS.MATH.CONTENT.HSF.LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context.
CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
The student will be able to interpret the slope and x- and y- intercepts in a linear function in terms of a context.
I can interpret the slope and x- and y- intercepts in a linear function in terms of a context.
2 – Linear Equations
5 – Slope 23
CCSS.MATH.CONTENT.HSN.Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP4 Model with mathematics.
Given a graph, the student will be able to draw conclusions and make inferences.
Given a graph, I can draw conclusions and make inferences.
2 – Linear Equations
5 – Slope 24
CCSS.MATH.CONTENT.HSS.ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
The student will be able to explain what the slope and y intercept means in the context of the situation.
I can explain what the slope and y intercept means in the context of the situation.
2 – Linear Equations
5 – Slope 25 Assessment Assessment Assessment Assessment
CCSS Algebra 1 Pacing Chart – Unit 2
HighSchoolMathTeachers © 2017 Page 2
2 – Linear Equations
6 – Linear Functions and their Inverses
26
CCSS.MATH.CONTENT.HSF.IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*
CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
The student will be able to graph lines expressed in slope-intercept form or standard form, by hand.
I can graph lines expressed in slope-intercept form or standard form, by hand.
2 – Linear Equations
6 – Linear Functions and their Inverses
27
CCSS.MATH.CONTENT.HSF.BF.B.4.A Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1.
CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
The student will be able to find the inverse of a given linear function.
I can find the inverse of a linear function.
2 – Linear Equations
6 – Linear Functions and their Inverses
28
CCSS.MATH.CONTENT.HSF.LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.
The student will be able to construct linear functions including arithmetic sequences from a table, graph or situation.
I can construct linear functions including arithmetic sequences from a table, graph or situation.
2 – Linear Equations
6 – Linear Functions and their Inverses
29
CCSS.MATH.CONTENT.HSA.REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
The student will be able to understand, apply, and explain the results of using inverse operations.
I can understand, apply, and explain the results of using inverse operations.
2 – Linear Equations
6 – Linear Functions and their Inverses
30 Assessment Assessment Assessment Assessment
CCSS Algebra 1 Pacing Chart – Unit 2
HighSchoolMathTeachers © 2017 Page 3
2 – Linear Equations
7 – Translating Graphs
31
CCSS.MATH.CONTENT.HSF.BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x),f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
Students will be able to relate the vertical translation of a linear function to its y-intercept.
I can relate the vertical translation of a linear function to its y-intercept.
2 – Linear Equations
7 – Translating Graphs
32
CCSS.MATH.CONTENT.HSF.BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x),f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an ex
CCSS.MATH.PRACTICE.MP6 Attend to precision.
Students will be able to find the value of k given f(x) replaced by f(x) + k on a graph of a linear or exponential function.
I can find the value of k given f(x) replaced by f(x) + k on a graph of a linear or exponential function.
2 – Linear Equations
7 – Translating Graphs
33
CCSS.MATH.CONTENT.HSF.BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x),f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an ex
CCSS.MATH.PRACTICE.MP4 Model with mathematics.
Students will be able to make perform vertical translations on linear and exponential graphs.
I can perform vertical translations on linear and exponential graphs.
2 – Linear Equations
7 – Translating Graphs
34
CCSS.MATH.CONTENT.HSF.BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x),f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an ex
CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
Students will be able to describe what will happen to a function when f(x) is replaced by f(x) + k for different values of k.
I can describe what will happen to a function when f(x) is replaced by f(x) + k for different values of k.
2 – Linear Equations
7 – Translating Graphs
35 Assessment Assessment Assessment Assessment
Algebra 1 Unit 1 Skills List
HighSchoolMathTeachers © 2017 Page 4
Algebra 1 Unit 2 Skills List
Number Unit Week CCSS Skill
10 2 4/7 F.BF.3 Translate a graph in function notation
11 2 5 F.IF.6 Calculate Slope
12 2 5 S.ID.7 Interpret meaning of the slope and
intercepts
13 2 6 F.BF.2 Construct an arithmetic sequence
14 2 6 F.BF.4 Find the inverse of a function
Unit 2 Lesson Plan
HighSchoolMathTeachers ©2017 Page 5
Unit 2 – Linear Equations
Course: Algebra 1
Topic: 5 – Slope
Day: 21
Common Core State Standard: CCSS.MATH.CONTENT.HSF.IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*
Mathematical Practice: CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.
Objective: The student can calculate the average rate of change for a linear function given the equation or a table.
I can statement: I can calculate the average rate of change for a linear function given the equation or a table.
Procedures: 1. Students will complete the Week 5 Bellringer (Day 21). 2. Students will work with partners and complete the Day 21 Slope worksheet 1. 3. The Day-21-Slope Presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the Week 5 Exit Slip (Day 21) before le+aving for the day. 5. Use the Day 21 Slope Worksheet 2 as individual practice or homework.
Materials: Week 5 Bellringer (Day 21) Day 21 Slope Worksheet Day-21-Slope Presentation Week 5 Exit Slip (Day 21) Day 21 Slope Worksheet 2
Unit 2 Lesson Plan
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Accommodations/Special Circumstances: Technology: Function Matching - http://illuminations.nctm.org/Activity.aspx?id=3520
Reflection:
Extra/Additional Resources:
Unit 2 Lesson Plan
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Unit 2 – Linear Equations
Course: Algebra 1
Topic: 5 – Slope
Day: 22
Common Core State Standard: CCSS.MATH.CONTENT.HSF.LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context.
Mathematical Practice: CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
Objective: The student will be able to interpret the slope and x- and y- intercepts in a linear function in terms of a context.
I can statement: I can interpret the slope and x- and y- intercepts in a linear function in terms of a context.
Procedures: 1. Students will complete the Week 5 Bellringer (Day 22). 2. Students will work with partners and complete the Shower Vs. Bath Lesson. 3. The Day 22 Presentation Shower vs Bath will be used to look for misconceptions and encourage discussion. 4. Students will complete the Week 5 Exit Slip (Day 22) before leaving for the day. 5. Use the Day 22 Shower VS Bath Handout as individual practice or homework.
Materials: Week 5 Bellringer (Day 22) Shower vs. Bath Day 22 Presentation Shower vs Bath Week 5 Exit Slip (Day 22) Day 22 Shower vs Bath Handout
Accommodations/Special Circumstances:
Technology:
Unit 2 Lesson Plan
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Reflection:
Extra/Additional Resources:
Unit 2 Lesson Plan
HighSchoolMathTeachers ©2017 Page 9
Unit 2 – Linear Equations
Course: Algebra 1
Topic: 5 – Slope
Day: 23
Common Core State Standard: CCSS.MATH.CONTENT.HSN.Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP4 Model with mathematics.
Objective: Given a graph, the student will be able to draw conclusions and make inferences.
I can statement: Given a graph, I can draw conclusions and make inferences.
Procedures: 1. Students will complete the Week 5 Bellringer (Day 23). 2. Students will work with partners and complete the Rate of Change of Beaker. 3. The Day-23-Presentation- Understanding the rate of change when it is not constant will be used to look for misconceptions and encourage discussion. 4. Students will complete the Week 5 Exit Slip (Day 23) before leaving for the day. 5. Use the Day 23 Beaker Activity Handout as individual practice or homework.
Materials: Week 5 Bellringer (Day 23) Rate of Change Beaker Day-23-Presentation- Understanding the rate of change when it is not constant Week 5 Exit Slip (Day 23) Day 23 Beaker Activity Handout
Unit 2 Lesson Plan
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Accommodations/Special Circumstances: Technology: NCTM Function Match - http://illuminations.nctm.org/Activity.aspx?id=3520
Reflection:
Extra/Additional Resources:
Unit 2 Lesson Plan
HighSchoolMathTeachers ©2017 Page 11
Unit 2 – Linear Equations
Course: Algebra 1
Topic: 5 – Slope
Day: 24
Common Core State Standard: CCSS.MATH.CONTENT.HSS.ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP4 Model with mathematics.
Objective: The student will be able to explain what the slope and y intercept means in the context of the situation.
I can statement: I can explain what the slope and y intercept means in the context of the situation.
Procedures: 1. Students will complete the Week 5 Bellringer (Day 24). 2. Students will work with partners and complete the Day 24 Review Handout. 3. The Day-24-Presentation- Putting it all together will be used to look for misconceptions and encourage discussion. 4. Students will complete the Week 5 Exit Slip (Day 24) before leaving for the day. 5. Use the Day 24 Three_ways_for_finding_slope_foldable as individual practice or homework.
Materials: Week 5 Bellringer (Day 24) Day 24 Review Handout Day-24-Presentation- Putting it all together Day 24 Three ways for finding slope foldable Week 5 Exit Slip (Day 24)
Accommodations/Special Circumstances:
Technology:
Unit 2 Lesson Plan
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Reflection:
Extra/Additional Resources:
Unit 2 Lesson Plan
HighSchoolMathTeachers ©2017 Page 13
Unit 2 – Linear Equations
Course: Algebra 1
Topic: 6 – Linear Functions and their Inverses
Day: 26
Common Core State Standard: CCSS.MATH.CONTENT.HSF.IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*
Mathematical Practice: CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
Objective: The student will be able to graph lines expressed in slope-intercept form or standard form, by hand.
I can statement: I can graph lines expressed in slope-intercept form or standard form, by hand.
Procedures: 1. Students will complete the Week 6 Bellringer (Day 26). 2. Students will work with partners and complete the Day 26 Cricket Activity. 3. The Day-26-Presentation- Graphing linear functions will be used to look for misconceptions and encourage discussion. 4. Students will complete the Week 6 Exit Slip (Day 26) before leaving for the day. 5. Use the Day-26-Practice -Graphing linear functions as individual practice or homework.
Materials: Week 6 Bellringer (Day 26) Day 26 Cricket Activity Day-26-Presentation- Graphing linear functions Week 6 Exit Slip (Day 26) Day-26-Practice -Graphing linear functions
Accommodations/Special Circumstances:
Technology:
Unit 2 Lesson Plan
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Reflection:
Extra/Additional Resources:
Unit 2 Lesson Plan
HighSchoolMathTeachers ©2017 Page 15
Unit 2 – Linear Equations
Course: Algebra 1
Topic: 6 – Linear Functions and their Inverses
Day: 27
Common Core State Standard: CCSS.MATH.CONTENT.HSF.BF.B.4.A Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1.
Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
Objective: The student will be able to find the inverse of a given linear function.
I can statement: I can find the inverse of a linear function.
Procedures: 1. Students will complete the Week 6 Bellringer (Day 27). 2. Students will work with partners and complete the Day 27 Cricket Activity. 3. The Day 27 – Inverse Functions will be used to look for misconceptions and encourage discussion. 4. Students will complete the Week 6 Exit Slip (Day 27) before leaving for the day. 5. Use the Day 27 Introduction to Inverses as individual practice or homework.
Materials: Week 6 Bellringer (Day 27) Day 27 Cricket Activity Day 27 – Inverse Functions Week 6 Exit Slip (Day 27) Day 27 Introduction to Inverses
Accommodations/Special Circumstances: Technology:
Unit 2 Lesson Plan
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Reflection:
Extra/Additional Resources:
Unit 2 Lesson Plan
HighSchoolMathTeachers ©2017 Page 17
Unit 2 – Linear Equations
Course: Algebra 1
Topic: 6 – Linear Functions and their Inverses
Day: 28
Common Core State Standard: CCSS.MATH.CONTENT.HSF.LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Mathematical Practice: CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.
Objective: The student will be able to construct linear functions including arithmetic sequences from a table, graph or situation.
I can statement: I can construct linear functions including arithmetic sequences from a table, graph or situation.
Procedures: 1. Students will complete the Week 6 Bellringer (Day 28). 2. Students will work with partners and complete the Day 28 Arithmetic Sequences. 3. The Day 28 Arithmetic Sequences PPT will be used to look for misconceptions and encourage discussion. 4. Students will complete the Week 6 Exit Slip (Day 28) before leaving for the day. 5. Use the Day-28-Practice -Arithmetic sequence as individual practice or homework.
Materials: Week 6 Bellringer (Day 28) Day 28 Arithemetic Sequences Day 28 Arithemetic Sequences Presentation Week 6 Exit Slip (Day 28) Day-28-Practice -Arithmetic sequence
Accommodations/Special Circumstances: Technology:
Unit 2 Lesson Plan
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Reflection:
Extra/Additional Resources:
Unit 2 Lesson Plan
HighSchoolMathTeachers ©2017 Page 19
Unit 2 – Linear Equations
Course: Algebra 1
Topic: 6 – Linear Functions and their Inverses
Day: 29
Common Core State Standard: CCSS.MATH.CONTENT.HSA.REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Mathematical Practice: CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
Objective: The student will be able to understand, apply, and explain the results of using inverse operations.
I can statement: I can understand, apply, and explain the results of using inverse operations.
Procedures: 1. Students will complete the Week 6 Bellringer (Day 29). 2. Students will work with partners and complete the Day 29 Put it all together. 3. The Day-29-Presentation- Graphing linear functions, inverses and arithmetic series will be used to look for misconceptions and encourage discussion. 4. Students will complete the Week 6 Exit Slip (Day 29) before leaving for the day. 5. Use the Day-29-Practice -Put all together as individual practice or homework.
Materials: Week 6 Bellringer (Day 29) Day 29 Put it all together Day-29-Presentation- Graphing linear functions, inverses and arithmetic series Week 6 Exit Slip (Day 29) Day-29-Practice -Put all together
Unit 2 Lesson Plan
HighSchoolMathTeachers ©2017 Page 20
Accommodations/Special Circumstances:
Technology:
Reflection:
Extra/Additional Resources:
Unit 2 Lesson Plan
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Unit 2 – Linear Equations
Course: Algebra 1
Topic: 7 – Translating Graphs
Day: 31
Common Core State Standard: CCSS.MATH.CONTENT.HSF.BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x),f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Mathematical Practice: CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
Objective: Students will be able to relate the vertical translation of a linear function to its y-intercept.
I can statement: I can relate the vertical translation of a linear function to its y-intercept.
Procedures: 1. Students will complete the Day 31-Bellringers-Vertical and horizontal transformations. 2. Students will work with partners and complete the Day 31 Activity Translating functions. 3. The Day-31-Presentation- Translating functions will be used to look for misconceptions and encourage discussion. 4. Students will complete the Day-31-Exit-Slip-Translating functions before leaving for the day. 5. Use the Day-31-Practice- Translating functions as individual practice or homework.
Materials: Day 31-Bellringers-Vertical and horizontal transformations Day 31 Activity Translating functions Day-31-Presentation- Translating functions Day-31-Exit-Slip-Translating functions Day-31-Practice- Translating functions
Unit 2 Lesson Plan
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Accommodations/Special Circumstances: Technology:
Reflection:
Extra/Additional Resources:
Unit 2 Lesson Plan
HighSchoolMathTeachers ©2017 Page 23
Unit 2 – Linear Equations
Course: Algebra 1
Topic: 7 – Translating Graphs
Day: 32
Common Core State Standard: CCSS.MATH.CONTENT.HSF.BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x),f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an ex
Mathematical Practice: CCSS.MATH.PRACTICE.MP6 Attend to precision.
Objective: Students will be able to find the value of k given f(x) replaced by f(x) + k on a graph of a linear or exponential function.
I can statement: I can find the value of k given f(x) replaced by f(x) + k on a graph of a linear or exponential function.
Procedures: 1. Students will complete the Day 32-Bellringers-Stretching and compressing graphs . 2. Students will work with partners and complete the Day-32-Activity-Stretching and compressing graphs . 3. The Day-32-Presentation- Stretching and compressing graphs will be used to look for misconceptions and encourage discussion. 4. Students will complete the Day-32-Exit slip- Stretching and compressing graphs before leaving for the day. 5. Use the Day-32-Practice- Stretching and compressing graphs as individual practice or homework.
Materials: Day 32-Bellringers-Stretching and compressing graphs Day-32-Activity-Stretching and compressing graphs Day-32-Presentation- Stretching and compressing graphs Day-32-Exit slip- Stretching and compressing graphs Day-32-Practice- Stretching and compressing
Unit 2 Lesson Plan
HighSchoolMathTeachers ©2017 Page 24
Accommodations/Special Circumstances: Technology:
Reflection:
Extra/Additional Resources:
Unit 2 Lesson Plan
HighSchoolMathTeachers ©2017 Page 25
Unit 2 – Linear Equations
Course: Algebra 1
Topic: 7 – Translating Graphs
Day: 33
Common Core State Standard: CCSS.MATH.CONTENT.HSF.BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x),f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an ex
Mathematical Practice: CCSS.MATH.PRACTICE.MP4 Model with mathematics.
Objective: Students will be able to make perform vertical translations on linear and exponential graphs.
I can statement: I can perform vertical translations on linear and exponential graphs.
Procedures: 1. Students will complete the Day 33-Bellringers- Reflection about the x and y axis . 2. Students will work with partners and complete the Day-33-Activity- Reflection about the x and the y axis. 3. The Day-33-Presentation- Reflection about the x and y axis will be used to look for misconceptions and encourage discussion. 4. Students will complete Day-33-Exit slip- Reflection about the x and the y axis graphs before leaving for the day. 5. Use the Day-33-Practice- Reflection about the x and y axis as individual practice or homework.
Materials: Day 33-Bellringers- Reflection about the x and y axis Day-33-Activity- Reflection about the x and the y axis Day-33-Presentation- Reflection about the x and y axis Day-33-Exit Slip- Reflection about the x and y axis Day-33-Practice- Reflection about the x and y axis.
Unit 2 Lesson Plan
HighSchoolMathTeachers ©2017 Page 26
Accommodations/Special Circumstances: Technology:
Reflection:
Extra/Additional Resources:
Unit 2 Lesson Plan
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Unit 2 – Linear Equations
Course: Algebra 1
Topic: 7 – Translating Graphs
Day: 34
Common Core State Standard: CCSS.MATH.CONTENT.HSF.BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x),f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an ex
Mathematical Practice: CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
Objective: Students will be able to describe what will happen to a function when f(x) is replaced by f(x) + k for different values of k.
I can statement: I can describe what will happen to a function when f(x) is replaced by f(x) + k for different values of k.
Procedures: 1. Students will complete the Day 34-Bellringers-Summary-transformations of parent function . 2. Students will work with partners and complete the Day-34-Activity-Summary- transformation of parent function. 3. The Day-34-Presentation- Summary-transformations of parent function will be used to look for misconceptions and encourage discussion. 4. Students will complete Day-34-Exit-Slip-Summary-Transformation of parent function graphs before leaving for the day. 5. Use the Day-34-Practice- Summary-Transformations of parent function as individual practice or homework.
Materials: Day 34-Bellringers-Summary-transformations of parent function Day-34-Activity-Summary- transformation of parent function Day-34-Presentation-Summary-Transformation of parent function Day-34-Exit-Slip-Summary-Transformation of parent function Day-34-Prac
Unit 2 Lesson Plan
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Accommodations/Special Circumstances: Technology:
Reflection:
Extra/Additional Resources:
Day 21 Bellringer Name ____________________________________
HighSchoolMathTeachers ©2017 Page 29
Graph the functions
1. 𝑓(𝑥) = 2𝑥 − 6
2. 𝑓(𝑥) = −1
3𝑥 + 5
3. 𝑓(𝑥) = 5 + 𝑥
4. 𝑓(𝑥) = .5𝑥 + 1
Day 21 Bellringer Name ____________________________________
HighSchoolMathTeachers ©2017 Page 30
Answer Key
Day 21
1.
2
3.
4.
Day 21 Activity Name ___________________________________
HighSchoolMathTeachers ©2017 Page 31
Given the following lines find their slopes.
Line 1________________ Line 3_______________
Line 2________________ Line 4_______________
Line 5________________ Line 7__________________
Line 6________________ Line 8__________________
Line 2
Line 1
Line 3
Line 4
Line 7
Line 8
Line 5
Line 6
Day 21 Activity Name ___________________________________
HighSchoolMathTeachers ©2017 Page 32
Given a point and a slope draw the line.
9. slope: -2 10. slope: 3/5
point (-4, 1) point (-2,0)
11. slope : 0 12. slope: undefined
point (0, 1) point (2,5)
Day 21 Activity Name ___________________________________
HighSchoolMathTeachers ©2017 Page 33
Answer Key
Line 1: __1/2___
Line 2: __-2/3__
Line 3: __1/2___
Line 4: __-3____
Line 5: __-1____
Line 6: __1_____
Line 7: _undefined_
Line 8: __0____
9.
10.
11.
12.
Day 21 Practice Name ________________________
HighSchoolMathTeachers ©2017 Page 34
Given 2 points find the slope using the slope formula. Show all work.
12
12
xx
yym
1. (1, 2) 𝑎𝑛𝑑 (7, 9) 2. (−5, 3) 𝑎𝑛𝑑 (−1, 0)
3. (5, −1) 𝑎𝑛𝑑 (0, 3) 4. (6, 2) 𝑎𝑛𝑑 (6, −5)
5. (12, 5) 𝑎𝑛𝑑 (9, 8) 6. (−3, −7) 𝑎𝑛𝑑 (−8, −1)
Day 21 Practice Name ________________________
HighSchoolMathTeachers ©2017 Page 35
7. (2, −5) 𝑎𝑛𝑑 (7, −5) 8. (2, ¾) 𝑎𝑛𝑑 (4, ¼)
9. (1
2,
2
3) 𝑎𝑛𝑑 (0,
1
3) 10. (3, −5) 𝑎𝑛𝑑 (0, 0)
Day 21 Practice Name ________________________
HighSchoolMathTeachers ©2017 Page 36
Answer Key
1.
2.
3.
4. Unidentified
5.
6.
7. 0
8.
9.
10.
Day 21 Exit Slip Name ____________________________________
HighSchoolMathTeachers ©2017 Page 37
Day 21
Identify the slope and the y intercept from the following graphs.
1. Slope _________________
y-intercept ____________
2. Slope _________________
y-intercept ____________
3. Slope _________________
y-intercept ____________
Day 21 Exit Slip Name ____________________________________
HighSchoolMathTeachers ©2017 Page 38
Answer Key
Day 21
1. Slope: 1
4
y-intercept: 4
2. Slope: 3
2
y-intercept: -1
3. Slope: - 2
3
y-intercept: 3
Day 22 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 39
1. Find the slope of the line that intersects (7, 2) and (4, 8).
2. Find the slope through the line 2𝑥 + 4𝑦 = 8
3. Find the slope of the line below.
4. Find y so that the line containing (5, 1) and (6, y) has a slope of -3.
Day 22 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 40
Day 22
1. -2
2. -1/2
3. 1
4. 𝑦 = −2
Day 22 Practice Name ___________________________
HighSchoolMathTeachers©2017 Page 41
1. What was your initial reaction to the question, “What do you think is cheaper, a shower or a bath”?
Explain.
2. Make a list of all the information you think you will need to solve the problem.
3. In the space below keep a record of each step as you solve the problem. Show your calculations and
write a sentence explaining what each answer represents.
Step 1:
Step 2:
Day 22 Practice Name ___________________________
HighSchoolMathTeachers©2017 Page 42
Step 3:
Step 4:
Day 22 Practice Name ___________________________
HighSchoolMathTeachers©2017 Page 43
ANSWER KEY & TEACING NOTES
1. Time to discuss. Poll the class, list reasons on the board
2. HOW LONG, HOW MUCH WATER, HOW MUCH DOES THE WATER COST
Guiding questions: Keep in mind which is cheaper is what we are trying to solve. Is it important to know
the size of the tub? How long is the average shower? How long is the average bath? Does it matter
how full you make the tub?
3.
Step 1: HOW LONG
What does the time on the screen mean?
00:02:23:74 means 2 minutes, 23.74 seconds (precision: what units do you need?)
Shower time: 23.74 sec = .40 min. (2.40 min) or 120 + 23.74 = 143.74 sec
Bath time: 10.15 sec = .17 min (8.17 min) or 480 + 10.15 = 490.15 sec
Step 2: HOW MUCH WATER
Shower: 5.36 gal Bath 45.22 gal
Step 3: HOW MUCH DOES THE WATER COST
In Livonia: $5.024/100 cu ft. x 0.13368 cu ft. /gal = $0.006716/gal
Shower: $0.036 Bath: $0.304
Step 4: HOW LONG OF A SHOWER CAN YOU TAKE TO EQUAL THE COST OF A BATH?
Shower rate: 26.82 sec/gal
How many seconds to use 45.22 gallons?
45.22 gal x 26.82 sec/1gal = 1212.8 sec
What units make the most sense for your answer, seconds or minutes?
1212.8 sec x 1min/60 sec = 20.2 minutes
Day 22 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2017 Page 44
Day 22
Suppose that the water level of a river is 34 feet and that it is receding at a rate of
0.5 foot per day. Write an equation for the water level, L, after d days. In how
many days will the water level be 26 feet?
Day 22 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2017 Page 45
Answer Key
Day 22
𝑓(𝑥) = 34 − 0.5𝑥
16 days
Day 23 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 46
1. A rope is tied around two trees. The lower end of the rope is 5 feet off the ground
and its higher end is 9 feet off the ground. The two trees are 2 feet far away from
each other. What is the slope of the rope?
2. The average lifespan of American women has been tracked, and the model for the
data is y = 0.2t + 73, where t = 0 corresponds to 1960. Explain the meaning of the
slope and y-intercept.
3. Todd had 5 gallons of gasoline in his motorbike. After driving 100 miles, he had 3
gallons left. What is the slope and the y-intercept of the function?
4. Jamie received $25 from her favorite aunt for her birthday. She wants to save her
money to purchase an e-reader. She decided to save the $25 and add $5 from her
own money each week. The amount of money that she has saved is a function of
time. What is the slope of the function?
Day 23 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 47
Answer Key
Day 23
1. 2
2. Every year, the average lifespan of American women increased by 0.2 years, in 1960 the average
lifespan of an American woman was 73 years.
3. -1/50, 5
4. 𝑓(𝑥) = 5𝑥 + 25, 5
Day 23 Practice Name ____________________________________
HighSchoolMathTeachers©2017 Page 48
Make a hypothesis: If you turn the tap on and record the time it takes to fill the beaker up to the last marking on the glass, will the height change at a constant rate or will it change throughout? Explain. If the rate of change of the height of the water changes, when will the rate be the largest, and when will it be lowest? Procedure: Turn the tap on and record the time it takes to fill the beaker up to the last marking on the glass, using a constant stream of water. Record this three times and average your times. What does this represent?
Fill up both beakers, using this stream of water. Was your hypothesis correct? If not, what did you notice?
Beaker #1 Beaker #2 Beaker #3
Trial 1
Trial 2
Trial 3
Average
Day 23 Practice Name ____________________________________
HighSchoolMathTeachers©2017 Page 49
Calculations: If you were to fill up the cylindrical beaker with the stream of water, how fast would the height change when it was half full? When it was entirely full? If you were to fill up the conical beaker with the stream of water how fast would the height change when it was half full? When was it was entirely full?
If you were to fill up the beaker flask with the stream of water how fast would the height change when it was half full? When was it was entirely full?
Day 23 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2017 Page 50
Day 23
Draw a graph representing the amount of time the constant stream of water is running with the height
on the beaker.
Day 23 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2017 Page 51
Day 23
Day 24 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 52
Draw a graph to represent the situation.
1. The amount of people in a grocery store over a 24 hour day
2. The height of a child as he plays on a slide at the playground
3. The speed of a car as it drives through a busy town
4. The height of a plant over 12 months
Day 24 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 53
Day 24
Answer Key
Answer may vary
Day 24 Activity Name ____________________________________
HighSchoolMathTeachers©2017 Page 54
Slope if given a graph
Complete a t-table for each equation. Then graph.
1. y = 3x + 1 2. x = 4 3. x + 2y = 4
Slope if given a graph
4. Graph: y = 4
3x 2
Positive Slope- Line goes
up and to the right
Slope = ___________
x y
Day 24 Activity Name ____________________________________
HighSchoolMathTeachers©2017 Page 55
5. Graph: 4x – 2y = 8 (Find y first)
Negative Slope- Line goes
up and to the left
Slope= _____________
Special Lines
6. Graph: y = 1 7. Graph: x = 3
Horizontal Line Vertical Line
Slope= ____________ Slope=____________
Slope if Given 2 Points
run
rise =
).(..
).(..
xxinchange
yyinchange
m =
12
12
xx
yy
Find the slope of the line given 2 points.
8. (7, 4), (5, 8) 9. (–3, 0), (6, 3)
x y
x y x y
Day 24 Activity Name ____________________________________
HighSchoolMathTeachers©2017 Page 56
10. (–2, 1), (–2, –6) 11. (-3, -4), (-9, -7)10
12. (9, -4), (-4, -4) 13. (2, 1), (2, 4)
Writing an Equation of a Line
Write the equation of the given line.
14.) 15.)
m = b = ( , ) m = b = ( , )
y = y =
16.) 17.)
m = b = ( , ) m = b = ( , )
y = y =
Day 24 Activity Name ____________________________________
HighSchoolMathTeachers©2017 Page 57
Identify Slope and y-intercept
18. y = 3
1x 19. y = 5 20. x = 4
m = _____ m = _____ m = _____
b = ( , ) b = ( , ) b = ( , )
Graph using slope and y-intercept
21. y = 2
3x – 1 22. 2x + y = 4
Find the slope of a line from Patterns
23. 6, 9, 12, …
24. 3, 1, –1, –3, …
25. 3, 6, 12, 24, …
26. 7, 5, 6, 4, 5, 3…
Day 24 Activity Name ____________________________________
HighSchoolMathTeachers©2017 Page 58
Find the slope:
Find Slope: x
y
(Change in y’s and x’s)
or 12
12
xx
yy
27. Write an equation for cost.
# socks 1 2 3 4 5
cost 3 6 9 12 15 m = __________
28.
x 1 2 3 4 5
y 3 5 7 9 11 m = __________
29. Find the equation of the line that has a slope of 6 and passes through the point
(-3,5).
30. Find the slope of the line whose equation is y - 3x = 7.
Day 24 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2017 Page 59
Day 24
Austin collected 100 pounds of aluminum cans to recycle. He plans to collect an
additional 25 pounds each week. Write and graph the equation for the total
pounds, P, of aluminum cans after w weeks. What does the slope and y-intercept
represent? How long will it take Rufus to collect 400 pounds of cans?
Day 24 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2017 Page 60
Answer Key Day 24 𝑓(𝑤) = 25𝑤 + 100 The slope is 25 and represents the change per week. The y-intercept is 100 and represents the initial number of pounds collected.
Day 25 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 61
1. The water level of a river is 28 feet and that it is receding at a rate of 0.5 foot per day. Write a function for the water level, L, after d days. In how many days will the water level be 16 feet?
2. For babysitting, Nicole charges a flat fee of $5, plus $8 per hour. Write a function for the cost, C, after h hours of babysitting. How much money will she make if she babysits 5 hours?
3. Rufus collected 100 pounds of aluminum cans to recycle. He plans to collect an additional 25 pounds each week. Write a function for the total pounds, P, of aluminum cans after w weeks. How long will it take Rufus to collect 400 pounds of cans?
4. A canoe rental service charges a $25 transportation fee and $10 dollars an hour to rent a canoe.
Write a function representing the cost, y, of renting a canoe for x hours. What is the cost of renting
the canoe for 6 hours?
Day 25 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 62
Day 25
Answer Key
1. 𝑓(𝑑) = 28 − 0.5𝑑, 24 2. 𝑓(ℎ) = 8ℎ + 5, 45 3. 𝑓(𝑤) = 25𝑤 + 100, 12 4. 𝑓(𝑥) = 10𝑥 + 25, 85
63
High School Math Teachers
Algebra 1
Weekly Assessment Package
Week 5
HighSchoolMathTeachers©2017
64
Week #5
1. Given 𝑓(𝑥) = 𝑥2 − 2𝑥 + 9, find:
a. 𝑓(2) =
b. 𝑓(−3) =
c. 𝑓(1/2) =
2. Find the slope of the graph between the
two points.
a. (4, 3), (8, -5)
b. (3/4, 5/2), (2/3, -1/4)
c. (5, 8), (5, 10)
3. You have $22.50 in your piggy bank. You choose to buy two cookies every day at lunch
for yourself and your sweetheart. They cost $0.75 for both cookies. Create an equation,
table, and graph for this situation.
Equation: _________________________
Table: Graph:
65
Week #5 Continued
4. The below table provides some U.S. Population data from 1982 to 1988:
Year Population
(thousands) Change in Population
(thousands) 1982 231,664 --- 1983 233,792 2128 1984 235,825 2033 1985 237,924 2099 1986 240,133 2209 1987 242,289 2156 1987 244,499 2210
If we were to model the relationship between the U.S. population and the year, would a linear function be appropriate? Explain why or why not.
Mike decides to use a linear function to model the relationship. He chooses 2139, the average of the values in the 3rd column, for the slope. What meaning does this value have in the context of this model?
Use Mike's model to predict the U.S. population in 1992.
5. As I fill the following beaker with water at a
constant rate, graph the height of the water in
relation to time.
6. Suppose 𝑓 is a function. If 12 =𝑓(−9), give the coordinates of a point on the graph of f.
If 16 is a solution of the equation 𝑓(𝑤) = 6, give a point on the graph of f.
66
Unit 2 - KEYS
Weekly Assessments
67
Week #5 - KEY
1. Given 𝑓(𝑥) = 𝑥2 − 2𝑥 + 9, find:
a. 𝑓(2) = 9
b. 𝑓(−3) = 24
c. 𝑓(1/2) = 8.25
2. Find the slope of the graph between the two
points.
a. (4, 3), (8, -5) -1/2 b. (3/4, 5/2), (1/2, -1/4) 11 c. (5, 8), (5, 10) undefined
3. You have $22.50 in your piggy bank. You choose to buy two cookies every day at lunch for yourself and
your sweetheart. They cost $.75 for both cookies. Create an equation, table, and graph for this situation.
Equation: y = 22.5 - 0.75x
Table:
x 0 5 10 15 y 22.50 18.75 15.00 11.25
68
Week #5 Key Continued
4. The below table provides some U.S. Population data from 1982 to 1988:
Year Population
(thousands) Change in Population
(thousands) 1982 231,664 --- 1983 233,792 2128 1984 235,825 2033 1985 237,924 2099 1986 240,133 2209 1987 242,289 2156 1987 244,499 2210
If we were to model the relationship between the U.S. population and the year, would a linear function be appropriate? Explain why or why not.
Yes the function is linear, because the change of population stays relatively the same each year.
Mike decides to use a linear function to model the relationship. He chooses 2139, the average of the values in the 3rd column, for the slope. What meaning does this value have in the context of this model?
The number 2139 tells us the amount that the population increases each year.
Use Mike's model to predict the U.S. population in 1992.
5*2139 + 244,499 = 255,194 http://illustrativemathematics.org/illustrations/353 5. As I fill the following beaker with water at a
constant rate, graph the height of the water in
relation to time.
6. Suppose 𝑓 is a function.
a. If 12 = 𝑓(−9), give the coordinates of a point on the graph of f. (-9, 12)
b. If 16 is a solution of the equation𝑓(𝑤) = 6, give a point on the graph of f. (16, 6)
Day 26 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 69
Day 26
1. Solve for 𝑦 : 6𝑥 + 2𝑦 = 12
2. Solve for 𝑦 ∶ 𝑥 – 3𝑦 = 17
3. Solve for 𝑦 ∶ 3𝑥 − 5𝑦 = 12
4. Solve for 𝑦: − 5𝑥 – 𝑦 = 12
Day 26 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 70
Answer key
Day 26
1. y=-3x+6 2. y=-17/ 3 + 1/3 x 3. y=-12/5 + 3/5x 4. y=-12-5x
Day 26 Cricket Activity Name ____________________________________
HighSchoolMathTeachers©2017 Page 71
Cricket Chirps Vs. Temperature
𝑦 = 𝑥 + 40
𝑦 = temperature in degrees Fahrenheit
𝑥 = cricket chirps per minute
Fill in the table of values below and then draw the equation of the line on the graph.
𝑥 0 10 20 30
𝑦 = 𝑥 + 40
(𝑥, 𝑦)
Use the graph to answer these questions:
1. What would the temperature be if there are 20 chirps per minute? 2. If the temperature was 80 ⁰F, how many chirps per minute would there be? 3. Describe the relationship between temperature and chirps per minute.
Day 26 Cricket Activity Name ____________________________________
HighSchoolMathTeachers©2017 Page 72
Temperature Conversion Graph
You can use this formula to convert from degrees Celsius to degrees Fahrenheit:
𝑦 = 1.8𝑥 + 32
y = temperature in degrees Fahrneheit
x = temperature in degrees Celcius
Fill in the table of values below and then draw the equation of the line on the graph.
𝑥 0 10 20 30
𝑦 = 1.8𝑥 + 32
(𝑥, 𝑦)
Use the graph to answer
these questions:
1. What is zero degrees Celsius in Fahrenheit?
2. What is zero degrees Fahrenheit in degrees Celsius?
3. What temperature in degrees Celsius is the same in degrees Fahrenheit?
Day 26 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2017 Page 73
Day 26
Create a table and graph the data given in the following function.
𝑓(𝑥) = 3.5𝑥 + 1.5
Day 26 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2017 Page 74
Answer Key Day 26
𝑥 -2 -1 0 1 2
𝑓(𝑥) -5.5 -2 1.5 5 8.5
Day 27 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 75
Day 27
1. Write a linear function to describe each set of data.
Current (ampere), x 0 1 2 3
Volt (ohms), y 6 4 2 0
2. Write a linear function to describe each set of data.
Mass (grams), x 2 3 4 5
Force (newton), y 5 0 -5 24
3. Write a linear function to describe each set of data.
Speed (m/s), x 2 4 6 8
Distance (meters), y 8 11 14 17
4. Write a linear function to describe each set of data.
Time (sec), x 0 1 2 3
Acceleration (m/s2), y 12 16 20 24
Day 27 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 76
Answer Key Day 27
1. y=-2x+6 2. y=5x+15 3. y=3/2x+5 4. y=4x+12
Day 27 Activity Name ____________________________________
HighSchoolMathTeachers©2017 Page 77
FLOWCHART
1. Design a flowchart explaining how to find the inverse of a function.
2. Design a flowchart explaining how to find the function if you know its inverse.
Be sure to include the following details:
Answer the question, “Do all functions have an inverse?”
Include at least three examples within your flowchart.
Use appropriate notation and use it correctly.
Use appropriate terminology effectively.
You may find it useful to use Microsoft Word or Glogster. Both of these programs (along
with many others) have flowchart clipart built in to their software.
Day 27 Activity Name ____________________________________
HighSchoolMathTeachers©2017 Page 78
Stick a "y" in for the "𝑓 𝑥 " 𝑔𝑢𝑦:
Switch the X and Y (because every (x, y) has (y, x) partner.
Solve for Y:
Stick in the inverse notation, 𝑓−1 𝑥
Answer Key
Day 27 Practice Name ____________________________________
HighSchoolMathTeachers©2017 Page 79
1. Suppose you are given the following directions: • From home, go north on Rt 23 for 5 miles • Turn east (right) onto Orchard Street • Go to the 3rd traffic light and turn north (left) onto Avon Drive • Tracy’s house is the 5th house on the right.
If you start from Tracy’s house, write down the directions to get home.
•
•
•
•
How did you come up with the directions to get home from Tracy’s?
2. Suppose you are given the following algorithm: • Starting with a number, add 5 to it
• Divide the result by 3
The final result is 10.
Working backwards knowing this result, find the original number. Show your work.
Write a function f(x), which when given a number x (the original number) will model the
operations given above.
𝑓(𝑥) = ___________________________
Day 27 Practice Name ____________________________________
HighSchoolMathTeachers©2017 Page 80
Write a function g(x), which when given a number x (the final result), will model the
backward algorithm that you came up with above.
𝑔(𝑥) = ___________________________
Using the functions you discovered above, fill in the table.
𝑥 𝑦 = 𝑓(𝑥) 𝑧 = 𝑔(𝑦) 𝑓(𝑧)
10
1
4
19
What patterns have you noticed in the columns (outputs)?
In this scenario, 𝒇(𝒙) and 𝒈(𝒙) are inverses of each other because g(x) will undo the
actions of 𝒇(𝒙). Thus, we could write 𝒈(𝒙) as 𝒇−𝟏(𝒚) described as “f inverse.”
Day 27 Practice Name ____________________________________
HighSchoolMathTeachers©2017 Page 81
Answer Key 1. Suppose you are given the following directions:
• From home, go north on Rt 23 for 5 miles • Turn east (right) onto Orchard Street • Go to the 3rd traffic light and turn north (left) onto Avon Drive • Tracy’s house is the 5th house on the right.
If you start from Tracy’s house, write down the directions to get home
• From Tracy’s house, go south for 5 houses on Avon Drive.
• Turn west (right) onto Orchard Street.
• Go for three traffic lights and turn south onto RT 23.
• Go for 5 miles and you will be at home.
How did you come up with the directions to get home from Tracy’s?
Just follow the inverse directions. Or draw the directions and go back from the Tracy’s house to the
home.
2. Suppose you are given the following algorithm:
• Starting with a number, add 5 to it
• Divide the result by 3
The final result is 10.
Working backwards knowing this result, find the original number. Show your work.
Multiply the final result by 3.
10 ∙ 3 = 30
Subtract 5 from it.
30 − 5 = 25
The original number is 25.
Write a function f(x), which when given a number x (the original number) will model the operations
given above.
𝑓(𝑥) =𝑥+5
3
Write a function g(x), which when given a number x (the final result), will model the backward algorithm
that you came up with above.
𝑔(𝑥) = 3𝑥 − 5 Using the functions you discovered above, fill in the table.
x 𝑦 = 𝑓(𝑥) 𝑧 = 𝑔(𝑦) 𝑓(𝑧)
10 5 10 5
1 2 1 2
4 3 4 3
19 8 19 8 What patterns have you noticed in the columns (outputs)?
The first and the third columns are equivalent. And the second and the fourth columns are also the
same. In this scenario, 𝑓(𝑥) and 𝑔(𝑥) are inverses of each other because 𝑔(𝑥) will undo the actions of
𝑓(𝑥). Thus, we could write 𝑔(𝑥) as 𝑓−1(𝑦) described as “f inverse.”
Day 27 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2017 Page 82
Day 27
Find the inverse of the following functions. Show your work.
1. 𝑦 = 2𝑥 + 1
2.
𝑥 2 -3 0 1 3 -5
𝑓(𝑥) 6 8 1 3 4 -2
Day 27 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2017 Page 83
Answer Key Day 27
1. 𝑓−1(𝑥) =𝑥+1
2
2.
𝑥 6 8 1 3 4 -2
𝑓(𝑥) 2 -3 0 1 3 -5
Day 28 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 84
Day 28
1. Find the inverse of the function 𝑓(𝑥) = 2𝑥 − 5
2. Find the inverse of the function 𝑓(𝑥) = 25 −1
𝑥
3. Find the inverse of the function 𝑓(𝑥) = 3𝑥 + 4
4. Find the inverse of the function 𝑓(𝑥) = −2𝑥 + 1
Day 28 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 85
Answer Key
Day 28
1. f(x) = 1/2x + 5 2. f(x) = 1/25- x 3. f(x) = 1/3 (x -4) 4. f(x) = ½ (1-x)
Day 28 Activity Name ____________________________________
HighSchoolMathTeachers©2017 Page 86
Find the next two terms of each sequence. Then describe the pattern. The equations will be completed
later.
1, 3, 5, 7, 9, _____, _____ Description: ____________ Equation: ___________
2, 7, 12, 17, 22, ______, ______ Description: ____________ Equation: ___________
-416, -323, -230, -137, _____, _____ Description: ____________ Equation: ___________
-2, -5, -8, -11, _____, ______ Description: ____________ Equation: ___________
All of the patterns above are called arithmetic sequences. Hopefully you noticed something about their
pattern that makes them similar. Complete the sentence below by writing a description of the pattern
you noticed above. (If you need help, look in the textbook in section 11.1).
Arithmetic sequences are sequences of numbers where_______________________________________
____________________________________________________________________________________.
Let’s look more closely at the first pattern 1, 3, 5, 7, 9… Suppose the domain is the position of a term (1,
2, 3, 4, etc.) and the range is the term (1, 3, 5, 7, 9, etc.).
Make a graph of the points that are made (position, term)
with the pattern.
What quadrant(s) are these points in? Why?
What kind of graph do you have?
Write an equation for the graph.
Day 28 Activity Name ____________________________________
HighSchoolMathTeachers©2017 Page 87
How does this equation relate to the graph? How does this equation relate to the pattern?
Do you think the graphs of other arithmetic sequences would look similar? ______ Why or why not?
__________________________________________________________________________________
Checkpoint 1: Stop at this point for class comparison. If you are done before others, make equations for
the other three patterns listed at the top.
Now, everyone should have the same equation ___________________ for the pattern 1, 3, 5, 7, 9…
However, we have a problem. This equation makes us use a number that is not on our pattern (-1).
Let’s say we want to use 1 as a starting point instead of -1 (since 1 is our first term in our sequence).
So, suppose our equation is now y = 2x + 1.
See if this works for the pattern.
Try x = 1 (for the first term).
Try x = 2 (for the second term).
Try x = 3 (for the third term).
What do you notice?
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Our new equation y = 2x + 1 makes our pattern shift over one term (one x value). This means we are
adding one too many times!
Let’s alter the equation slightly to y = 2(x – 1) + 1. This will shift all the x values (just like we’ve done
before) and we won’t be adding the extra value of d.
See if this works for the pattern.
Try x = 1 (for the first term).
Try x = 2 (for the second term).
Try x = 3 (for the third term).
What do you notice?
Now, we have an equation y = 2 (x – 1) + 1 that uses the first term and the common difference (slope).
This can be used to make any equation for any arithmetic sequence. Let’s use d = common difference,
a1 = first term, and an = nth term.
So, the nth term of any arithmetic sequence can be found by
an = a1 + (n – 1) d
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Checkpoint 2: Find the rule/equation for the 2nd pattern using the formula above.
Now that you know arithmetic sequences need a common difference (number added or subtracted to
the pattern) and you know how to find the nth term (or equation) for any arithmetic sequence, let’s try
some problems.
Example 1: Is the sequence arithmetic? If so, what’s the common difference? If not, why not?
A) 2, -3, -8, -13,… B) 1, 5/4, 3/2, 7/4,…
C) an = n2 D) an = 4n + 3
Example 2: Write the first 5 terms if and 𝑑 = −7.
Checkpoint 3: Let’s make sure we are on the right track with examples 1 and 2.
Example 3: Write the rule/equation for the given information.
A) a1 = 2, d = 3 B)
21 a
9,2 21 aa
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Example 4: Find the indicated term of each arithmetic sequence. First find the equation, then plug in
your n.
A) a1 = -4, d = 6, n = 9 B) a20 for a1 = 15, d = -8
Checkpoint 4: Let’s make sure we got the answers to examples 3 and 4.
Example 5: Write the equation for the nth term of each arithmetic sequence.
A) 31, 17, 3, ….
Now, the next two are slightly different. I will give you a term and the d – but the term isn’t the first
one. You need to work backwards to find the first term.
B) a7 = 21, d = 5 We know that an = a1 + (n – 1) d
So, using the given information, we have 21 = a1 + (7 – 1) 5
Simplify and solve for a1.
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Now, find the equation.
C) Follow the steps with this information: a6 =12, d = 8.
Checkpoint 5: Did we follow that?
Example 6: Find the arithmetic means (missing terms) in each sequence.
A) 6, ____, ____, ____, 42 B) 24, _____, _____, _____, _____, -1
Challenge: Let’s do this one together.
Use the given information to write an equation that represents the nth term in each arithmetic
sequence.
The 19th term of the sequence is 131. The 61st term is 509.
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Answer Key Find the next two terms of each sequence. Then describe the pattern. The equations will be completed later. 1, 3, 5, 7, 9, 11, 13 Description: Add 2 to the previous term 2, 7, 12, 17, 22, 27, 32 Description: Add 5 to the previous term -416, 323, 230, 137, 44, 49, Description: Add 93 to the previous term -2, 5, 8, 11, 14, 17, All of the patterns above are called arithmetic sequences. Hopefully you noticed something about their pattern that makes them similar. Complete the sentence below by writing a description of the pattern you noticed above. Arithmetic sequences are sequences of numbers where the difference between one term and the next is a constant. Let’s look more closely at the first pattern 1, 3, 5, 7, 9… Suppose the domain is the position of a term (1, 2, 3, 4, etc.) and the range is the term (1, 3, 5, 7, 9, etc.). Make a graph of the points that are made (position, term) with the pattern.
What quadrant(s) are these points in? Why? We see all points are in the first quadrant. That’s because all domain and range values are positive numbers. What kind of graph do you have? We have a linear graph here. Write an equation for the graph 𝑦 = 2𝑥 – 1 How does this equation relate to the graph? How does this equation relate to the pattern? This equation represents the pattern of the sequence. Do you think the graphs of other arithmetic sequences would look similar? Why or why not? Yes, the graphs of other arithmetic sequences would look similar, because all graphs of arithmetic sequences are linear graphs.
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Now, everyone should have the same equation 𝑦 = 2𝑥 − 1 for the pattern 1, 3, 5, 7, 9… However, we have a problem. This equation makes us use a number that is not on our pattern (1). Let’s say we want to use 1 as a starting point instead of 1 (since 1 is our first term in our sequence). So, suppose our equation is now y = 2x + 1. See if this works for the pattern. Try x = 1 (for the first term). Try x = 2 (for the second term). Try x = 3 (for the third term). What do you notice? Our new equation y = 2x + 1 makes our pattern shift over one term (one x value). This means we are adding one too many times! Let’s alter the equation slightly to y = 2(x – 1) + 1. This will shift all the x values (just like we’ve done before) and we won’t be adding the extra value of d. See if this works for the pattern. Try x = 1 (for the first term). Try x = 2 (for the second term). Try x = 3 (for the third term). What do you notice? We notice it works now Now, we have an equation 𝑦 = 2(𝑥 − 1) + 1 that uses the first term and the common difference (slope). This can be used to make any equation for any arithmetic sequence.
Let’s use d = common difference, a1= first term, and an= nth term.
So, the nth term of any arithmetic sequence can be found by 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 Checkpoint 2: Find the rule/equation for the 2nd pattern using the formula above. 𝑎𝑛 = 2 + 5(𝑛 − 1) Now that you know arithmetic sequences need a common difference (number added or subtracted to the pattern) and you know how to find the nth term (or equation) for any arithmetic sequence, let’s try some problems. Example 1: Is the sequence arithmetic? If so, what’s the common difference? If not, why not? A) 2, 3, 8, 13 This sequence is arithmetic. Its common difference is 5. B) 1, 5/4, 3/2, 7/4 The sequence is arithmetic. The common difference is ¼. C) 𝑎𝑛 = 𝑛2 The sequence is not arithmetic, because the common the difference between one term and the next is not a constant. D) 𝑎𝑛 = 4𝑛 + 3 The sequence is arithmetic. The common difference is 4. Example 2: Write the first 5 terms if a and 1 = 2 d =− 7 𝑎1 = 2 𝑎2 = −5 𝑎3 = −12 𝑎4 = −19 𝑎5 = −26
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Checkpoint 3: Let’s make sure we are on the right track with examples 1 and 2. Example 3: Write the rule/equation for the given information. A) 𝑎1 = 2, 𝑑 = 3 𝑎𝑛 = 2 + 3(𝑛 − 1) B) 𝑎1 = 2, 𝑎2 = 9 𝑎𝑛 = 2 + 7(𝑛 − 1) Example 4: Find the indicated term of each arithmetic sequence. First find the equation, then plug in your n. A) 𝑎1 = −4, 𝑑 = 6, 𝑛 = 9 𝑎𝑛 = − 4 + 6(𝑛 − 1) 𝑎9 = − 4 + 6(9 − 1) = 44 B) 𝑎20 𝑓𝑜𝑟 𝑎1 = 15, 𝑑 = −8 𝑎𝑛 = 15 − 8(𝑛 − 1) 𝑎20 = 15 − 8(20 − 1) = −137 Checkpoint 4: Let’s make sure we got the answers to examples 3 and 4. Example 5: Write the equation for the nth term of each arithmetic sequence. A) 31, 17, 3… 𝑎𝑛 = 31 − 14(𝑛 − 1) Now, the next two are slightly different. I will give you a term and the d – but the term isn’t the first one. You need to work backwards to find the first term. B) 𝑎7 = 21, 𝑑 = 5 We know that 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 So, using the given information, we have 21 = 𝑎1 + (7 − 1)5 Simplify and solve for 𝑎1. 𝑎1 = −9 Now, find the equation. 𝑎𝑛 = −9 + 5(𝑛 − 1) C) Follow the steps with this information: 𝑎6 = 12, 𝑑 = 8 12 = 𝑎1 + (6 − 1)8 𝑎1 = −28 𝑎𝑛 = −28 + 8(𝑛 − 1) Checkpoint 5: Did we follow that? Yes, we did! Example 6: Find the arithmetic means (missing terms) in each sequence. A) 6, 15, 24, 33, 42 B) 24, 19, 14, 9, 4, 1 Challenge: Let’s do this one together. Use the given information to write an equation that represents the nth term in each arithmetic sequence. The 19th term of the sequence is 131. The term is th 61st 509. 𝑎𝑛 = −31 + (𝑛 − 1)9
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Day 28
Given the explicit formula for an arithmetic sequence find the first five terms and the term named in the
problem.
1. 𝑎𝑛 = −11 + 7𝑛 Find 𝑎34
2. 𝑎𝑛 = −7.1 − 2.1𝑛 Find 𝑎27
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Answer Key
Day 28
1. -4, 3, 10, 17, 24: 𝑎34=227
2. -9.2, -11.3, -13.4, -15.5, -17.6: 𝑎27=-63.8
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Day 29
Find the x and y intercepts of the equations.
1. 3𝑥 + 4𝑦 = 12
2. 4𝑥 + 6𝑦 = 16
3. 2𝑥 + 4𝑦 = 12
4. −3𝑦 + 5𝑥 = 30
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Answer Key
Day 29
1. x intercept= (4,0), y intercept = (0,3)
2. x intercept = 4, y intercept = 2 2/3
3. x intercept = 6, y intercept = 3
4. x intercept = 6, y intercept = -10
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1. Complete the table of values and
graph the line f xx
( ) 3
2
x -
3
-
1 1 3 5
y 0
2. Interchange (switch) the x and y coordinates from table 1 and graph the line on the same graph.
x 0
y -
3
-
1 1 3 5
3. Find the slope & y-intercept of the line graphed in step 2. Write the equation (y = mx + b) y = ______________________
Rewrite the equation in function form (replace y with g(x)) g(x) = ____________________
4. What do you notice about the two lines on the graph?
5. Fold the graph along the dotted line y = x. ( Fold so the graph shows on the outside) Take the point of your pencil and poke a hole through the points showing on one side of the fold. Open your paper and write a sentence describing your observations.
6. How can you describe the graphs of f(x) and g(x) with respect to the line y = x?
7. Find f(g(x)) (show the steps)
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8. Find g(fx)) (show the steps)
9. What do f(g(x)) and g(fx)) have in common?
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Day 29
Find a formula for the sequence
1. 4, 7, 10, 13, …
2. 1, -1, -3, -5, …
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Answer Key
Day 29
1. 𝑓(𝑥) = 3𝑥 + 1
2. 𝑓(𝑥) = −2 + 3
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Day 30
1. Find expressions for the 20th term of the following sequences: 2, 4, 6, 8,...
2. Find expressions for the 7th term of the following sequences: 7, 14, 21, 28,...
3. Find expressions for the 15th term of the following sequences: 5, 10, 15, 20,...
4. Find expressions for the 11th term of the following sequences: 6, 12, 18, 24,...
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Answer Key
Day 30
1. 40 2. 49 3. 75 4. 66
105
High School Math Teachers
Algebra 1
Weekly Assessment Package
Week 6
HighSchoolMathTeachers©2017
106
Week #6
1. Emma understands that the function, 𝑓(𝑥) = 3.5𝑥 + 10 gives her the price for the bands t-shirts given the $10 set up fee and the price of $3.50 per shirt.
She also knows that there are 88 band members. What is the total cost for the shirts?
2. Lauren keeps records of the distances she
travels in a taxi and what she pays:
Distance, d (in miles)
Fare, F (in dollars)
3 8.25
5 12.75
11 26.25
a. If you graph the ordered pairs (𝑑, 𝐹) from the table, they lie on a line. How can you tell this without graphing them?
b. Show that the linear function in part (a) has equation 𝐹 = 2.25𝑑 + 1.5.
c. What do the 2.25 and the 1.5 in the equation represent in terms of taxi rides?
3. Solve the following equations and justify the steps.
a. 1
3(4𝑥 + 1) = 9 b. 10 =
5𝑥−3
4
107
Week #6 Continued
4. If you have $10, you can buy 4 cookies and
no brownies or you can buy 5 brownies and
no cookies. There are several other options
as well. Graph the situation.
If you have $10 and you buy 1 cookie a day you will run out of money after 5 days. Graph the situation.
Which situation has the cheaper cookie? (Circle one)
1st 2nd Not enough information
5. Let F assign to each student in your math class his/her locker number. Explain why F is a function.
Describe conditions on the class that would have to be true in order for F to have an inverse.
6. Candy bars cost $1.50 each. What is the total bill?
What is the domain? _____________________ What is the range? _____________________
108
Unit 2 - KEYS
Weekly Assessments
109
Week #6 - KEY 1. mma understands that the function,
𝑓(𝑥) = 3.5𝑥 + 10 gives her the price for the bands t-shirts given the $10 set up fee and the price of $3.50 per shirt.
She also knows that there are 88 band members. What is the total cost for the shirts?
𝒇(𝟖𝟖) = 𝟑𝟏𝟖
$318
2. Lauren keeps records of the distances she travels
in a taxi and what she pays:
Distance, d
(in miles)
Fare, F
(in dollars)
3 8.25
5 12.75
11 26.25
a. If you graph the ordered pairs (𝑑, 𝐹) from the table, they lie on a line. How can you tell this without graphing them? Yes, finding the slopes tells us that they are the same for both intervals.
b. Show that the linear function in part (a) has equation 𝐹 = 2.25𝑑 + 1.5. There is only one possible line in part (a) since two points determine a line. The graph of F= -2.25d + 1.5 is a line, so if we show that each ordered pair satisfies it then we will know that it is the same line as in part (a). (3, 8.25)(5, 12.75)(11, 26.25) 2.25(3) + 1.5 = 8.25 2.25(5) + 1.5 = 12.75 2.25(11) + 1.5 = 26.25
c. What do the 2.25 and the 1.5 in the equation represent in terms of taxi rides? The 2.25 represents the cost per mile for the ride. The 1.5 represents a fixed cost for every ride; it does not depend on the distance traveled.
http://illustrativemathematics.org/illustrations/243 3. Solve the following equations and justify the steps.
a. 1
3(4𝑥 + 1) = 9 b. 10 =
5𝑥−3
4
4x + 1 = 27 (Mult prop of equality) 4x = 26 (Add prop of equality) X = 6.5 (Div prop of equality)
40 = 5x – 3 (Mult prop of equality)
43 = 5x (Add prop of equality)
8.6 = x (Division prop of equality)
110
Week #6 Continued
4. If you have $10, you can buy 4 cookies and
no brownies or you can buy 5 brownies and
no cookies. There are several other options
as well. Graph the situation.
If you have $10 and you buy 1 cookie a day you will run out of money after 5 days. Graph the situation.
Which situation has the cheaper cookie? (Circle one)
1st 2nd Not enough information
5. a. Let F assign to each student in your
math class his/her locker number. Explain why F is a function.
F is a function because it assigns to each student in the class exactly one element, his/her locker number. b. Describe conditions on the class that
would have to be true in order for F to have an inverse.
Students would not share lockers.
6. Candy bars cost $1.50 each. What is the total bill?
What is the domain? Number of Candy Bars What is the range? Cost
cookies
br
o w ni
es
cookies
$
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Day 31
1. Graph the equation: −2𝑥 + 𝑦 = −4
x y
2. Graph the equation : −𝑥 + 2𝑦 = −8
x y
3. Graph the equation: −𝑥 + 3𝑦 = −6
x y
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Answer Key
Day 31
1.
2.
3.
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Consider the graph below
a). Write down the end points of lines A and B.
b). Drag the lines horizontally to the left right so that y-coordinates of the end points of A and B
remains to be 2 and 4, and 4 and 7 respectively. Let the new line corresponding to A to H and
that of B be R. Draw the lines on the same axes.
1 2 3 4 5 6 7 8 9 x
1
2
3
4
5
6
y
A
B
7
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c). Determine the equations of lines A, B, H and R.
d). Compare the equations of A and H then B and R in relation to the 5 units moved. Is there
anything common?
e). Come up with a general formula for a function 𝑦 = 𝑓(𝑥) when translated to right hand side.
f). Using the general formula above, what would be equation if A were moved 15 units to the
right.
h). Taking the movement to the left as a negative movement. The new line when A is moved
would have been 𝑦 = 𝑓(𝑥 + 𝑎) where 𝑎 value of units moved. Using this argument, what
would be equation of 𝐻1 , the line found when A is moved 5 units to the left.
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Answer Keys Day 31:
a). A (2,2) and (1,4);
B (3,4) and (7,1)
b).
c). A; 𝑦 = −2𝑥 + 6
H; 𝑦 = −2𝑥 + 16
B; 𝑦 = −3
2𝑥 +
17
2
R; 𝑦 = −3
2𝑥 + 16
d). A: 𝑦 = −2𝑥 + 6, H; 𝑦 = −2(𝑥 − 5) + 6 = −2𝑥 + 16
B; 𝑦 = −3
2𝑥 +
17
2, R; B; 𝑦 = −
3
2(𝑥 − 5) +
17
2= −
3
2𝑥 + 16
In both cases, the variable 𝑥 is replaced with 𝑥 − 5 to the new line after the movement
e). 𝑦 = 𝑓(𝑥 − 𝑎) where 𝑎 is the units moved to the right
f). 𝑦 = 𝑓(𝑥 − 15)
h). 𝑦 = −2(𝑥) + 6
𝐻1 = 𝑓(𝑥 + 5) = −2(𝑥 + 5) + 6 = −2𝑥 − 4
A H
R B
1 2 3 4 5 6 7 8 9 x
1
2
3
4
5
6
7
y
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Find the equation of functions whose initial functions and the motion is described. 1 – 10.
1.𝑓(𝑥) = −𝑥 + 2; Moved downwards by 4 units.
2. 𝑓(𝑥) = −2𝑥 − 1; Moved to the left by 3 units.
3. (𝑥) = −3𝑥2 + 6; Moved to the left by 1 units.
4. (𝑥) = 𝑥2 + 2; Moved to the left by 2 units and one unit upwards.
5. 𝑓(𝑥) = 10𝑥 + 3𝑥2 − 15 ; Moved to the right by 1 units. Simplify completely
6. 𝑓(𝑥) = 2𝑥2 + 2 ; Moved to the downwards by 3 units.
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7. 𝑓(𝑥) = 𝑥2 + 3𝑥 + 1; Moved upwards by 9 units.
8. 𝑓(𝑥) = −𝑥2; Moved to the left by 1 unit and upwards by 2 units
9. 𝑓(𝑥) = −𝑥2; Moved to the left by 6 units and to the right by 8 units.
10. 𝑓(𝑥) = −2𝑥 + 1; Moved to the right by 5 units.
Find the original function if the function given was arrived at using the provided descriptions.
11-15.
11. 𝑓(𝑥) = 4𝑥 + 1 Had been moved to the left by 2 units.
12. 𝑓(𝑥) = 𝑥2 + 2𝑥 + 1 Had been moved to the right by 2 units.
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13. 𝑓(𝑥) = 2𝑥 Had been moved to the left and vertically downwards by 2 and 3 units
respectively.
14. 𝑓(𝑥) = 𝑥 − 3, Had been moved to upwards by 6 units.
15. 𝑓(𝑥) = −𝑥 − 4 Had been moved upwards by 1 unit and downwards by 3 units.
If the figure below, find the equation of (ii) if that of (i) is given. 16 - 20
16. 𝑦 = 𝑓(𝑥)
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17. 𝑦 = 𝑥2
18. 𝑦 = (𝑥 − 3)2
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19. 𝑦 = 𝑔(𝑥)
20. 𝑦 = 𝑓(𝑥 − 2)
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Answer Keys Day 31:
Answers
1. 𝑓(𝑥) = −𝑥 − 2
2. 𝑓(𝑥) = −2𝑥 − 7
3. 𝑓(𝑥) = −3𝑥2 − 6𝑥 + 3
4. 𝑓(𝑥) = 𝑥2 + 4𝑥 + 7
5. 𝑓(𝑥) = 3𝑥2 + 4𝑥 − 22
6. 𝑓(𝑥) = 2𝑥2 − 1
7. 𝑓(𝑥) = 𝑥2 + 3𝑥 + 10
8. 𝑓(𝑥) = −𝑥2 + 2𝑥 + 1
9. 𝑓(𝑥) = −𝑥2 + 4𝑥 − 4
10. 𝑓(𝑥) = −2𝑥 + 11
11. 𝑓(𝑥) = 4𝑥 − 7
12. 𝑓(𝑥) = 𝑥2 + 6𝑥 + 9
13. 𝑓(𝑥) = 2𝑥 − 1
14. 𝑓(𝑥) = 𝑥 − 9
15. 𝑓(𝑥) = −𝑥 − 2
16. 𝑦 = 𝑓(𝑥 − 2) − 3
17. 𝑓(𝑥) = 𝑥2 − 4
18. 𝑓(𝑥) = (𝑥 − 7)2 + 5
19. 𝑦 = 𝑔(𝑥 − 3)
20. 𝑦 = 𝑓(𝑥 − 1) + 2
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Day 31
Translate the graph down 4 and right 2.
Write a rule for this translation.
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Answer Key Day 31 (𝑥 + 2, 𝑦 − 4) Day 32
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Day 32
1. Find the rule for the translation of graphs A, B, and C
A. ______________
B. ______________
C. ______________
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Answer Key
Day 32
A <4, 4>: Reflection
B <4, -2>: Translation
C <-2, 4>: Rotation
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Consider the graph of the function 𝑦 = 𝑥2 below
We would wish to investigate the effect of multiplying 2 by 𝑥2and also that of 1
2 by the same
expression.
(a). From the graph above, draw a table showing the 𝑥 and 𝑦 values of the function for
−3 ≤ 𝑥 ≤ 3
(b). Generate another table where 𝑦 values are multiplied by 2 for all 𝑥 values
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(c). Generate another table where the 𝑦 values are multiplied by1
2.
(d). Draw the graph of tables in (a), (b) and (c) respectively and label them.
(e). Identify a graph that is stretched and one that is compressed with respect to the graph of 𝑦 =
𝑥2.
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Answer Keys Day 32:
(a).
𝑥 -3 -2 -1 0 1 2 3
𝑦 9 4 1 0 1 4 9
(b).
𝑥 -3 -2 -1 0 1 2 3
𝑦 18 8 2 0 2 8 18
(c).
𝑥 -3 -2 -1 0 1 2 3
𝑦 4.5 2 0.5 0 0.5 2 4.5
(d).
e). Graph of (b) is stretched (vertically) while that of (c) is compressed
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Find the equation of functions whose initial functions and the motion is described. 1 – 6.
1.𝑓(𝑥) = −𝑥 + 8; Vertically stretched by a factor of 4.
2.𝑓(𝑥) = −2𝑥3 − 1; Vertically compressed by1
2.
3.𝑓(𝑥) = 2𝑥2 − 4𝑥 + 1; Horizontally stretched by 1
2 .
4. (𝑥) = 𝑥3 ; Horizontally stretched by a factor of 1
4 and vertically stretched by a factor of 4.
5.𝑓(𝑥) = 4𝑥2; Horizontally compressed by a factor of 3 and vertically stretched by a factor of 2.
6. (𝑥) = −𝑥2 ; Horizontally stretched by a factor of 1
2 and vertically compressed by a factor of 2.
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In each of the following functions, identify all the transformations done with respect to the
simplest function of that category. 7 - 13
7. 𝑓(𝑥) = 2𝑥
8. 𝑓(𝑥) = 3(2𝑥)2
9. 𝑓(𝑥) =1
4𝑥3
10.𝑓(𝑥) =1
2(𝑥 − 3)2 + 1
11. 𝑓(𝑥) = 3(𝑥 + 5)3 + 2
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12. 𝑓(𝑥) = 4 (1
2𝑥 − 2)
4
13. 𝑓(𝑥) = 7(3𝑥 + 1)2 − 6
In each of the following functions, identify all the transformations done with respect to the
given original function of that category. 14 – 18
14. From 𝑓(𝑥) = 2(𝑥 − 2)2 + 1 to 𝑓(𝑥) = (𝑥 − 2)2 + 1
15. From 𝑓(𝑥) =1
3|𝑥 − 2| to𝑓(𝑥) =
1
6|3𝑥 − 2|.
16. From 𝑓(𝑥) =1
4 (𝑥 + 1)3 + 3 to
3
4𝑥3 + 3
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17. From 𝑓(𝑥) = 2(𝑥 + 1)2 − 1 to 2 (𝑥
3+ 1)
2
+ 1
18. From 𝑓(𝑥) = 𝑥4 to 𝑓(𝑥) = (2𝑥 + 1)4
Identify the equation showing transformation mentioned. 19-20
19. Vertical compression.
(𝑖). 𝑓(𝑥) = 𝑥 + 2 (𝑖𝑖). 𝑓(𝑥) = 3𝑥2 + 1 (𝑖𝑖𝑖). 𝑓(𝑥) =1
4(𝑥 − 2)2 (𝑖𝑣). 𝑓(𝑥) = (
1
3𝑥 + 1)
2
20. Horizontal compression.
(𝑖). 𝑓(𝑥) = 𝑥 + 2 (𝑖𝑖). 𝑓(𝑥) = 3𝑥2 + 1 (𝑖𝑖𝑖). 𝑓(𝑥) =1
4(𝑥 − 2)2 (𝑖𝑣). 𝑓(𝑥) = (
1
3𝑥 + 1)
2
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Answer Keys Day 32:
Answers
1. 𝑓(𝑥) = −4𝑥 + 32
2. 𝑓(𝑥) = −𝑥3 −1
2
3. 𝑓(𝑥) =𝑥2
2− 2𝑥 + 1
4. 𝑓(𝑥) =1
16𝑥3
5. 𝑓(𝑥) = 72𝑥2
6. 𝑓(𝑥) = −𝑥2
2
7. Vertical stretch of factor 2 or horizontal
compression of factor 2
8. Vertical stretch of factor 3
Horizontal compression of factor 2
9. Vertical compression of factor 1
4 or
horizontal stretch of factor 1
√43
10. Vertical compression of 1
2
Vertical shift of 1 unit upwards Horizontal shift of 3 units to the right
11. Vertical stretch of factor 3
Vertical shift of 2 units upwards Horizontal shift of 5 units to the left
12. Vertical stretch of factor 4
Horizontal stretch of factor 1
2
Horizontal shift of 2 units to the right
13. Vertical stretch of factor 7
Horizontal compression of factor 3 Horizontal shift of 1 units to the left Vertical shift of 6 units upwards
14. Vertical compression of factor 1
2
15. Vertical compression of factor 1
2
Horizontal compression of factor 3
16. Vertical stretch of factor 3
Horizontal shift of a unit to the right
17. Horizontal stretch of factor 1
3
Vertical shift of 2 units upwards
18. Horizontal shift of 1 unit to the left
Horizontal compression of factor 2
19. (iii)
20. (ii)
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1. Given that 𝑦 = 𝑓(𝑥);
a). Write identify the transformation 𝑦 =1
2𝑓(𝑥) and 𝑦 = 𝑓 (
𝑥
2).
b). Given that 𝑦 = 𝑓(𝑥) as shown below, draw the graph of (i). 𝑦 =1
2𝑓(𝑥) and (ii). 𝑦 = 𝑓 (
𝑥
2) on
the same axes.
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Answer Keys Day 32:
Answers
21.
(a). 𝑦 =1
2𝑓(𝑥) Vertical compression of factor
1
2
𝑦 = 𝑓 (𝑥
2) Horizontal stretch of factor
1
2
(b).
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Day 33
1. Translate the functions so that they intersect at (5, 4). Be sure to show your work graphically and algebraically.
{𝑓(𝑥) =
4
5𝑥 + 2
𝑔(𝑥) = −3
5𝑥 + 6
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Answer Key
Day 33
Day 33 Activity Name ________________________________________
HighSchoolMathTeachers©2017 Page 138
We are interested in determining the effect of reflection of a function about the 𝑦 axis. We will
achieve this by reflecting the graph of 𝑦2 = 𝑥 or precisely 𝑦 = ±√𝑥.
a). Generate a table of values for integer values of 𝑥, 1, 4, 9, 16 and 25.
𝑥
𝑦
b). Plot the points on the graph
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c). Using the properties of reflection, draw the image of the graph above on the same axes
d). Come up with a table showing integer values of 𝑥 and the corresponding 𝑦 axis.
𝑥
𝑦
e).Compare the two tables in (a) and (d) above. What is the relationship between the two?
𝑥
𝑦
𝑥
𝑦
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f). Come up with a general formula for the image of the graph 𝑦 = 𝑔(𝑥) under reflection about
the 𝑦 −axis.
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Answer Keys Day 33:
(a).
𝑥 1 4 9 16 25
𝑦 ±1 ±2 ±3 ±4 ±5
b).
c).
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(d).
𝑥 -1 -4 -9 -16 -25
𝑦 ±1 ±2 ±3 ±4 ±5
e).Comparing the two tables, the original one
𝑥 1 4 9 16 25
𝑦 ±1 ±2 ±3 ±4 ±5
And the new one
𝑥 -1 -4 -9 -16 -25
𝑦 ±1 ±2 ±3 ±4 ±5
We find that the y values remain the same while the 𝑥 values becomes negative.
f). If the original graph is 𝑦 = 𝑔(𝑥), the new one is 𝑦 = 𝑔(−𝑥).
Day 33 Practice Name ________________________________________
HighSchoolMathTeachers©2017 Page 143
Find the equation of a given function after the stated transformation. 1- 6.
1. 𝑓(𝑥) = 2𝑥 + 3, reflection about 𝑥 −axis.
2. 𝑓(𝑥) = 2𝑥2 − 3𝑥 + 1, reflection about 𝑥 −axis.
3. 𝑓(𝑥) = −4𝑥3 − 𝑥 + 8, reflection about 𝑦 −axis.
4. 𝑓(𝑥) = 2 − 7𝑥2 + 5𝑥3, reflection about 𝑦 −axis.
5. 𝑓(𝑥) = 𝑥2, reflection about 𝑦 −axis.
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6. 𝑓(𝑥) = 2 − 4𝑥 − 7𝑥2, reflection about 𝑥 −axis.
7. 𝑓(𝑥) = 𝑥4 + 2𝑥3 + 2𝑥 + 1 , reflection about 𝑦 −axis.
Identify the transformation between the two graphs below. 7 – 12.
8.
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9.
10.
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11.
12.
Day 33 Practice Name ________________________________________
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Identify the transformation given the equations of the original and the new graphs. 13 - 18.
13. From 𝑦 = 2𝑥2 to 𝑦 = 2𝑥2.
14. From 𝑓(𝑥) = 𝑥3 + 2𝑥 + 1 to 𝑓(𝑥) = −𝑥3 − 2𝑥 + 1.
15. From 𝑓(𝑥) = 5𝑥3 − 3𝑥2 + 3 to 𝑓(𝑥) = −5𝑥3 − 3𝑥2 + 3.
16. From 𝑓(𝑥) = 6𝑥4 + 10𝑥2 − 2 to 𝑓(𝑥) = 2 − 6𝑥4 − 10𝑥2.
17. From 𝑓(𝑥) = 4𝑥 − 𝑥4 + 2 to 𝑓(𝑥) = −4𝑥 − 𝑥4 + 2.
18. From 𝑓(𝑥) = 6𝑥2 − 2𝑥 + 1 to 𝑓(𝑥) = −6𝑥2 + 2𝑥 − 1.
Day 33 Practice Name ________________________________________
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Find the resultant function when the stated function undergoes the stated number of
transformations.
19. 𝑦 = 2𝑥3 + 2𝑥2 + 5𝑥 − 1; Reflection about 𝑦 −axis followed by a reflection about 𝑥 −axis.
20. 𝑓(𝑥) = −9𝑥2 + 𝑥 − 12; Reflection about the 𝑥 − 𝑎𝑥𝑖𝑠 then 𝑦 − 𝑎𝑥𝑖𝑠.
Day 33 Practice Name ________________________________________
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Answer Keys Day 33:
Answers
1. 𝑓(𝑥) = −2𝑥 − 3
2. 𝑓(𝑥) = −2𝑥2 + 3𝑥 − 1
3. 𝑓(𝑥) = 4𝑥3 + 𝑥 + 8
4. 𝑓(𝑥) = 2 − 7𝑥2 − 5𝑥3
5. 𝑓(𝑥) = 𝑥2
6. 𝑓(𝑥) = −2 + 4𝑥 + 7𝑥2
7. 𝑓(𝑥) = 𝑥4 − 2𝑥3 − 2𝑥 + 1 8. Reflection about 𝑥 −axis
9. Reflection about 𝑦 −axis
10. Reflection about 𝑦 −axis
11. Reflection about 𝑦 −axis
12. Reflection about 𝑥 −axis
13. Reflection about 𝑦 −axis
14. Reflection about 𝑦 −axis
15. Reflection about 𝑦 −axis
16. Reflection about 𝑥 −axis
17. Reflection about 𝑦 −axis
18. Reflection about 𝑥 −axis
19. 𝑦 = 2𝑥3 − 2𝑥2 + 5𝑥 + 1
20. 𝑓(𝑥) = 9𝑥2 + 𝑥 + 12
Day 33 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2017 Page 150
1. Given that 𝑓(𝑥) = 4𝑥3 − 2𝑥2 + 7;
Describe the transformation that leads to 𝑓(𝑥) = −4𝑥3 − 2𝑥2 + 7
2. Identify the transformation relating the two functions shown below.
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Answer Keys Day 33:
Answers
1. Reflection about 𝑦 −axis
2. Reflection about 𝑦 −axis
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Day 34
Draw basic graphs for the parent functions
1. ℎ(𝑥) = 2|𝑥| – 5
2. 𝑔(𝑥) = 3 + 𝑥2
3. ℎ(𝑥) = −𝑥2 + 3
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Answer Key
Day 34
1.
2.
3.
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We are interested in determining the effect of vertical shift on a function. We will achieve this
by considering a square root function (restricted to positive parts of the 𝑦.
a) Given the Square root function, 𝑦 = √𝑥 generate a table of values for integer values of𝑥, that is, 1, 4, 9, 16, 25, 36 and 49.
𝑥
𝑦
b). Plot the points on the graph
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c). Slide the graph 6 units upwards. Draw the new graph and the original graph on the same
axes.
d). Come up with a table showing integer values of 𝑥 and the corresponding 𝑦 axis for the new
graph.
𝑥
𝑦
e).Compare the two tables in (a) and (d) above. What is the relationship between the two?
𝑥
𝑦
𝑥
𝑦
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f). Come up with a general formula for the image of the graph 𝑦 = 𝑔(𝑥) under reflection about
the 𝑦 −axis.
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Answer Keys Day 34:
(a).
𝑥 1 4 9 16 25 36 49
𝑦 1 2 3 4 5 6 7
b).
c).
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(d).
𝑥 1 4 9 16 25 36 49
𝑦 7 8 9 10 11 12 13
e).Comparing the two tables, the original on
𝑥 1 4 9 16 25 36 49
𝑦 1 2 3 4 5 6 7
And the new one
𝑥 1 4 9 16 25 36 49
𝑦 7 8 9 10 11 12 13
We find that the 𝑥 values remain the same while the 𝑦 values increase by 6.
f). If the original graph is 𝑦 = 𝑔(𝑥), the new one is 𝑦 = 𝑔(𝑥) + 6.
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1. Find the new 𝑦 −intercept of the equation when 𝑓(𝑥) = 2𝑥 + 3, is compressed vertically by
a factor of 1
4.
2. Find the new equation when 𝑓(𝑥) = 2𝑥2 − 4 is reflected about the 𝑦 −axis.
3. Find the new equation when 𝑓(𝑥) = −5𝑥3 − 3𝑥2 + 6𝑥 + 8 is reflection about 𝑥 −axis.
4. The graph of 𝑓(𝑥) = 2(𝑥 − 1)2 + 3 is horizontally compressed by a factor of 6, determine
the new equation, express the equation in its standard form.
5. 𝑓(𝑥) = 𝑥2is reflected about the y axis then compressed horizontally by a factor of 2
3. What
would be its new equation?
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Identify the transformation from (i) to (ii).6 – 8
6.
7.
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8.
Identify all the transformations done on the parent functions of the following transformed
equations. 9 – 16
9. 𝑓(𝑥) = |1
2𝑥 − 2| + 1
10. 𝑓(𝑥) = 3√2𝑥 − 1 − 6
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11. 𝑓(𝑥) = 7√1
2𝑥 + 2
12. 𝑓(𝑥) = 6𝑥 + 1
13. 𝑓(𝑥) =1
6(𝑥 + 4)3 − 3
14. 𝑓(𝑥) =1
3(𝑥 − 5)2 +
1
4
15. 𝑓(𝑥) =√3𝑥+4
6
16. 𝑓(𝑥) =1
8+
1
3|𝑥 − 2|
Day 34 Practice Name ________________________________________
HighSchoolMathTeachers©2017 Page 163
Determine the equation of the graph (b) given that the equation its parent function (a) is as
given. 17 – 20.
17. Parent function, (a), 𝑓(𝑥) = √𝑥
18. Parent function, (a),𝑓(𝑥) = 𝑥2
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HighSchoolMathTeachers©2017 Page 164
19. Parent function (a), 𝑓(𝑦) = 𝑦2 𝑓(𝑦) = (𝑦 − 3)2 − 2
20. Parent function, (a),𝑓(𝑥) = 𝑥3
Day 34 Practice Name ________________________________________
HighSchoolMathTeachers©2017 Page 165
Answer Keys Day 34:
Answers
1. 𝑓(𝑥) =1
2𝑥 +
3
4
2. 𝑓(𝑥) = 2𝑥2 − 4
3. 𝑓(𝑥) = 5𝑥3 + 3𝑥2 − 6𝑥 − 8
4. 𝑓(𝑥) = 72𝑥2 − 24𝑥 + 5
5. 𝑓(𝑥) =4
9𝑥2
6. Horizontal shift of 2 units to the
right and a vertical shift of 3 units
downward
7. Horizontal shift of 4 units to the
right and a vertical shift of 2 units
downwards
8. Horizontal shift of one unit to the
right and a reflection about 𝑥- axis
9. Horizontal stretch of a factor of 1
2
Horizontal shift of 2 units to the right Vertical shift of 1 unit upwards
10. Vertical stretch of factor 3
Horizontal compression of factor 2 Horizontal shift of 1 unit to the right Vertical shift of 6 units upwards
11. Vertical stretch of factor 7
Horizontal stretch of factor 1
2
Vertical shift of 2 units upwards
12. Vertical stretch of factor 6 or Horizontal
compression of factor 6
Vertical shift of 1 units upwards
13. Vertical compression of factor 1
6
Horizontal shift of 4 units to the left Vertical shift of 3 units downwards
14. Vertical compression of factor 1
3
Horizontal shift of 5 units to the right
Vertical shift of 1
4 of a unit upwards
15. Vertical compression of factor 1
6
Horizontal compression of factor 3 Horizontal shift of 4 units to the left
16. Vertical compression of factor 1
3
Vertical shift of an 1
8 of a unit
Horizontal shift of 2 units to the right
17. 𝑓(𝑥) = √𝑥 − 3
18. 𝑓(𝑥) = (𝑥 + 2)2 − 2
19. 𝑓(𝑦) = (𝑦 − 3)2 − 2
20. 𝑓(𝑥) = −(𝑥 + 4)3 − 3
Day 34 Exit Slip Name ________________________________________
HighSchoolMathTeachers©2017 Page 166
Day 34
Explain how to change the equation 𝑦 = |𝑥| to translate the graph vertically. Explain how to change the
equation 𝑦 = |𝑥| to translate the graph horizontally.
Day 34 Exit Slip Name ________________________________________
HighSchoolMathTeachers©2017 Page 167
Answer Key Day 34 To move the graph vertically we can add (to move up) or subtract (to move down) outside of the absolute value signs. To move the graph horizontally, we can add (to move left) or subtract (to move right) inside the absolute value signs.
Day 35 Bellringer Name ____________________________________
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Day 35
Match the functions to their respective graphical illustration.
a) 𝑓(𝑥) = 3𝑥 + 4 b) 𝑔(𝑥) = 9 + (𝑥 − 3)
b) c) ℎ(𝑥) = (𝑥2 + 3) − 2 d) 𝑓(𝑥) = (𝑥 − 4)2
1.
2.
3.
4.
Day 35 Bellringer Name ____________________________________
HighSchoolMathTeachers©2017 Page 169
Answer Key
Day 35
A= 1
B = 4
C = 2
D =3
170
High School Math Teachers
Algebra 1
Weekly Assessment Package
Week 7
HighSchoolMathTeachers©2017
171
Week #7
1. A souvenir shop in Niagara Falls sells picture postcards priced as follows:
a. Graph the price of buying postcards as a function of the
number of cards purchased.
b. Is there something wrong with this pricing scheme?
Explain.
2. Suppose P1= (0,5) and P2= (3, −3). Sketch P1 and P2.
a. For which real numbers m and b does the graph of a linear function described by the equation 𝑓(𝑥) = 𝑚𝑥 + 𝑏 contain P1 and P2? Explain.
Do any of these graphs also contain P2? Explain.
b. Suppose P1= (0,5) and P2= (0,7). Sketch P1 and P2. Are there real numbers m and b for which the graph of a linear function described by the equation 𝑓(𝑥) = 𝑚𝑥 + 𝑏 contains P1 and P2? Explain.
c. Suppose P1= (𝑐, 𝑑) and P2= (𝑔, ℎ) and c is not equal to g. Show that there is only one real
number m and only one real number b for which the graph of 𝑓(𝑥) = 𝑚𝑥 + 𝑏 contains the points P1 and P2.
Postcards 15 cents each
Six for $1
172
Week #7 Continued
3. Given 𝑓(𝑥) = 2𝑥 + 1 and 𝑔(𝑥) =𝑥
2−
1
2. Show that the two functions
are inverses.
4. Graph 𝑓(𝑥) = 2𝑥 + 4 and the inverse of 𝑓(𝑥).
Where do they intersect? _____________________
5. Translate the functions so that
they intersect at (3,4). (Feel free
to use the graph if you like.)
𝑓(𝑥) =1
3𝑥 + 1
𝑔(𝑥) = −1
2𝑥 + 7
𝑓(𝑥) =____________________________________ 𝑔(𝑥) =____________________________________
6. The three graphs show the functions
𝑓(𝑥) = 2𝑥 𝑔(𝑥) = 2(𝑥 + 1) ℎ(𝑥) = 2𝑥 + 1
Label the three graphs below.
173
Unit 2 - KEYS
Weekly Assessments
174
Week #7 - KEY
1. A souvenir shop in Niagara Falls sells
picture postcards priced as follows:
a. Graph the price of buying postcards
as a function of the number of cards
purchased.
b. Is there something wrong with this
pricing scheme? Explain.
Six for $1 cost approximately $0.17 each which is higher than the initial $0.15 per postcard.
2. a. Suppose P1= (0,5) and P2= (3, −3). Sketch
P1 and P2 .
For which real numbers m and b does the graph of a linear function described by the equation 𝑓(𝑥) = 𝑚𝑥 + 𝑏 contain P1 and P2? Explain. m = -8/3 b = 5 b. Suppose P1= (0,5) and P2= (0,7).
Sketch P1 and P2.
Are there real numbers m and b for which the graph of a linear function described by the equation 𝑓(𝑥) = 𝑚𝑥 + 𝑏 contains P1 and P2? Explain. No, because this is not a function. c. Extension: Now suppose P1= (𝑐, 𝑑) and P2=
(𝑔, ℎ) and c is not equal to g. Show that there is only one real number m and only one real number b for which the graph of 𝑓(𝑥) =
𝑚𝑥 + 𝑏 contains the points P1 and P2.
See website for full explanation http://illustrativemathematics.org/illustrations/377
Postcards
15 cents each
Six for $1
Number of Postcards
Pri
ce
(Do
ll
ars)
175
Week #7 Continued
3. Given 𝑓(𝑥) = 2𝑥 + 1 and 𝑔(𝑥) =𝑥
2−
1
2. Show
that the two functions are inverses.
𝒇(𝒈(𝒙)) = 𝟐(𝒙
𝟐−
𝟏
𝟐) + 𝟏 = x
𝒈(𝒇(𝒙)) = 𝟐𝒙 + 𝟏
𝟐−
𝟏
𝟐 = 𝒙
4. Graph 𝑓(𝑥) = 2𝑥 + 4 and the inverse of 𝑓(𝑥).
Where do they intersect? (-4, -4)
5. Translate the functions so that they intersect at
(3,4). (Feel free to use the graph if you like.)
𝑓(𝑥) =1
3𝑥 + 1
𝑔(𝑥) = −1
2𝑥 + 7
𝒇(𝒙) =𝟏
𝟑(𝒙 + 𝟒) + 𝟏
𝒈(𝒙) = −𝟏
𝟐(𝒙 + 𝟒) + 𝟕
6. The three graphs show the functions
𝑓(𝑥) = 2𝑥 (Blue)
𝑔(𝑥) = 2(𝑥 + 1) (Red)
ℎ(𝑥) = 2𝑥 + 1 (Green)
Label the three graphs below.
http://map.mathshell.org/materials/tasks.php?taskid=295&subpage=novice
Unit 2 Test Name ____________________________________
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Please remember to read all directions and show all work.
1. The below table provides some U.S. Population data from 2010 to 2014: Year Population (millions) Change in Population (millions)
2010 309,347,057 ---
2011 311,721,632 2,374,575
2012 314,112,078 2,390,446
2013 316,497,531 2,385,453
2014 318,857,056 2,359,525 http://www.census.gov/popest/data/national/totals/2014/index.html
a. If we were to model the relationship between the U.S. population and the year, would a linear function be appropriate? Explain why or why not.
b. Dan decides to use a linear function to model the relationship. He chooses 2,377,500, the average of the values in the 3rd column, for the slope. What meaning does this value have in the context of this model?
c. Use Dan’s model to predict the U.S. population in 2015
2. Hannah keeps records of the distances she travels in a taxi and what she pays: Distance, d, in miles Fare, F, in dollars
5 10.5
9 17.5
15 28
a. If you graph the ordered pairs (d,F) from the table, they lie on a line. How can you tell this without graphing them?
b. Show that the linear function in part (a) has equation 𝑓(𝑥) = 1.75𝑥 + 1.75.
c. What do the 1.75 and the 1.75 in the equation represent in terms of taxi rides?
Unit 2 Test Name ____________________________________
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3. These three graphs show the functions: 𝑓(𝑥) = 3𝑥 𝑔(𝑥) = 3(2𝑥 + 1)
ℎ(𝑥) = 6𝑥 + 1
Label the three graphs at the right.
4. Given 𝑓(𝑥) = 𝑥2 − 4𝑥 + 12, find:
a. 𝑓( 1
2 ) = _____________________
b. 𝑓(−1) = _____________________
c. 𝑓(4) = _____________________
5. Find the slope of the graph between the two points.
a. (2, 4), (−2, −5) b. ( 8
3, −
10
3 ), (
2
3, −
1
3 ) c. (2, 3), (4, 10)
6. Suppose f is a function. If 𝑓(2) = 16, give the coordinates of a point on the graph of f.
7. Solve the following equation and justify the steps. 1
3(9𝑥 + 12) = 19
Unit 2 Test Name ____________________________________
HighSchoolMathTeachers©2017 Page 178
8. A souvenir shop in Las Vegas sells picture postcards priced as follows:
Postcards
$0.12 each 9 for $1.00
a. Graph both pricing schemes showing the price of buying postcards as a function of the number of cards purchased.
b. Does this pricing scheme make sense? Why or why not?
9
a. Graph 𝑓(𝑥) =1
2𝑥 + 2 and 𝑓−1(𝑥).
b. Where do they intersect? _____________________
10. Referring to Table 1, find the rate of reading for each person. Then decide who reads the fastest.
Table 1 Pages Days
Ben 182 17
Brian 202 23
Kirsten 93 9
Laura 57 4
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11. Referring to Graph 1, find the average rate of change for 2 ≤ t ≤ 2.5.
12. Translate the functions so that they intersect at (6,5). (Only use f(x)+k and f(x+k) translations.)
a. 𝑓(𝑥) = 3𝑥 + 2
b. 𝑔(𝑥) =1
3x + 6
f(x)=____________________________________
g(x)=____________________________________
13. Use Linear Translations to describe in words the difference between
a. 𝑓(𝑥) = 𝑥2 b. 𝑔(𝑥) = −(𝑥 + 3)2
14. Use Linear Translations to describe in words the difference between a. 𝑓(𝑥) = |𝑥| b. 𝑔(𝑥) = |𝑥 − 2| − 7
Unit 2 Test Name ____________________________________
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15. The height of a tree was 6 feet tall. After 4 years it was 12 feet tall. If you were to graph this, what would be the slope of the line? What is the meaning of the slope in context with this situation?
Unit 2 Test Name ____________________________________
HighSchoolMathTeachers©2017 Page 181
ANSWER KEY
1. a. A linear function would be appropriate since the change is almost the same each time
b. 2,377,500 means that each year the population increases by 2,377,500 million people.
c. 321,234,556 million people
2. a. Answers will vary b. Answers will vary c. The 1.75 at the end of the equation
represents the initial fee for getting in the taxi. The 1.75 next to the x represents the cost per mile for using the taxi.
3.
4. f(2) = 8.25
f(-1) = 17
f(4) = 12
5. a. 9
4
b. −3
2
c. 7
2
6. (2, 16) 7.
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8.
a. b. Answers will vary
9. 𝑓(𝑥) =1
2𝑥 + 2 𝑓−1(𝑥) = 2𝑥 − 4
Intersection (4,4)
10. Laura read the fastest. 14.3 pages per day.
11. (2, 9) and (2.5, 9) gives an average zero 12. 𝑓(𝑥) = 3𝑥 + 2 − −> 𝑓(𝑥) = 3(𝑥 − 5) + 2
𝑔(𝑥) =1
3𝑥 + 6 − −> 𝑔(𝑥) =
1
3𝑥 + 6 − 3
13. G(x) shifts 3 units to the left and reflects across the x-axis.
14. G(x) shifts 2 units to the right and shifts 7 units down.
15. The slope of 3/2 tells you that the tree grows a foot and a half each year
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