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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.


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.


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.


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.


5 – Slope 25 Assessment Assessment Assessment Assessment




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.



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.



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.



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.



30 Assessment Assessment Assessment Assessment




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.



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.



33


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.



34


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.



35 Assessment Assessment Assessment Assessment

Algebra 1 Unit 1 Skills List


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


Accommodations/Special Circumstances: Technology: Function Matching - http://illuminations.nctm.org/Activity.aspx?id=3520

Reflection:

Extra/Additional Resources:

Unit 2 Lesson Plan



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


Reflection:


Unit 2 Lesson Plan



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


Accommodations/Special Circumstances: Technology: NCTM Function Match - http://illuminations.nctm.org/Activity.aspx?id=3520

Reflection:


Unit 2 Lesson Plan



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)


Technology:

Unit 2 Lesson Plan


Reflection:


Unit 2 Lesson Plan



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


Technology:

Unit 2 Lesson Plan


Reflection:


Unit 2 Lesson Plan



Course: Algebra 1


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


Reflection:


Unit 2 Lesson Plan



Course: Algebra 1


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


Unit 2 Lesson Plan


Reflection:


Unit 2 Lesson Plan



Course: Algebra 1


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



Technology:

Reflection:


Unit 2 Lesson Plan



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



Reflection:


Unit 2 Lesson Plan



Course: Algebra 1


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



Reflection:


Unit 2 Lesson Plan



Course: Algebra 1


Day: 33


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



Reflection:


Unit 2 Lesson Plan



Course: Algebra 1


Day: 34


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



Reflection:


Day 21 Bellringer Name ____________________________________


Graph the functions

1. 𝑓(𝑥) = 2𝑥 − 6

2. 𝑓(𝑥) = −1

3𝑥 + 5

3. 𝑓(𝑥) = 5 + 𝑥

4. 𝑓(𝑥) = .5𝑥 + 1



Answer Key

Day 21

1.

2

3.

4.

Day 21 Activity Name ___________________________________


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



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)



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 ________________________


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)



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)



Answer Key

1.

2.

3.

4. Unidentified

5.

6.

7. 0

8.

9.

10.

Day 21 Exit Slip Name ____________________________________


Day 21

Identify the slope and the y intercept from the following graphs.

1. Slope _________________

y-intercept ____________

2. Slope _________________


3. Slope _________________




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


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

1. -2

2. -1/2

3. 1

4. 𝑦 = −2

Day 22 Practice Name ___________________________


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:



Step 3:

Step 4:



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

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?



Answer Key

Day 22

𝑓(𝑥) = 34 − 0.5𝑥

16 days



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?



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 ____________________________________


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



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

Draw a graph representing the amount of time the constant stream of water is running with the height

on the beaker.



Day 23



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

Answer Key

Answer may vary

Day 24 Activity Name ____________________________________


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



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



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 =



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…



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

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?



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.



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

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)

http://illustrativemathematics.org/illustrations/353



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



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 ____________________________________


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 ____________________________________


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

Create a table and graph the data given in the following function.

𝑓(𝑥) = 3.5𝑥 + 1.5



Answer Key Day 26

𝑥 -2 -1 0 1 2

𝑓(𝑥) -5.5 -2 1.5 5 8.5



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


Mass (grams), x 2 3 4 5

Force (newton), y 5 0 -5 24


Speed (m/s), x 2 4 6 8

Distance (meters), y 8 11 14 17


Time (sec), x 0 1 2 3

Acceleration (m/s2), y 12 16 20 24



Answer Key Day 27

1. y=-2x+6 2. y=5x+15 3. y=3/2x+5 4. y=4x+12



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.



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



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) = ___________________________



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.”



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

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



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

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



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)



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.



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?



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



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



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.



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.



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.



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



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



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



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



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



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



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)



8. Find g(fx)) (show the steps)

9. What do f(g(x)) and g(fx)) have in common?



Day 29

Find a formula for the sequence

1. 4, 7, 10, 13, …

2. 1, -1, -3, -5, …



Answer Key

Day 29

1. 𝑓(𝑥) = 3𝑥 + 1

2. 𝑓(𝑥) = −2 + 3



Day 30

1. Find expressions for the 20th term of the following sequences: 2, 4, 6, 8,...






Answer Key

Day 30

1. 40 2. 49 3. 75 4. 66

105


Algebra 1


Week 6


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

$



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



Answer Key

Day 31

1.

2.

3.



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



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.



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



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.



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.



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)



17. 𝑦 = 𝑥2

18. 𝑦 = (𝑥 − 3)2



19. 𝑦 = 𝑔(𝑥)

20. 𝑦 = 𝑓(𝑥 − 2)



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



Day 31

Translate the graph down 4 and right 2.

Write a rule for this translation.



Answer Key Day 31 (𝑥 + 2, 𝑦 − 4) Day 32



Day 32

1. Find the rule for the translation of graphs A, B, and C

A. ______________

B. ______________

C. ______________



Answer Key

Day 32

A <4, 4>: Reflection

B <4, -2>: Translation

C <-2, 4>: Rotation



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



(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.



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



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.



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



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



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



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


Vertical shift of 2 units upwards Horizontal shift of 5 units to the left


Horizontal stretch of factor 1

2

Horizontal shift of 2 units to the right


Horizontal compression of factor 3 Horizontal shift of 1 units to the left Vertical shift of 6 units upwards


2


2



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


19. (iii)

20. (ii)



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.



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).



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



Answer Key

Day 33

Day 33 Activity Name ________________________________________


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



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



f). Come up with a general formula for the image of the graph 𝑦 = 𝑔(𝑥) under reflection about

the 𝑦 −axis.



Answer Keys Day 33:

(a).

𝑥 1 4 9 16 25

𝑦 ±1 ±2 ±3 ±4 ±5

b).

c).



(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 ________________________________________


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.



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.



9.

10.



11.

12.



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.



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.



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



12. Reflection about 𝑥 −axis







19. 𝑦 = 2𝑥3 − 2𝑥2 + 5𝑥 + 1

20. 𝑓(𝑥) = 9𝑥2 + 𝑥 + 12



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.



Answer Keys Day 33:

Answers



Day 34 Belringer Name ________________________________________


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

Day 34 Belringer Name ________________________________________


Answer Key

Day 34

1.

2.

3.



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



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



f). Come up with a general formula for the image of the graph 𝑦 = 𝑔(𝑥) under reflection about

the 𝑦 −axis.



Answer Keys Day 34:

(a).

𝑥 1 4 9 16 25 36 49

𝑦 1 2 3 4 5 6 7

b).

c).



(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.



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?



Identify the transformation from (i) to (ii).6 – 8

6.

7.



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



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|



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



19. Parent function (a), 𝑓(𝑦) = 𝑦2 𝑓(𝑦) = (𝑦 − 3)2 − 2

20. Parent function, (a),𝑓(𝑥) = 𝑥3



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


Horizontal compression of factor 2 Horizontal shift of 1 unit to the right Vertical shift of 6 units upwards


Horizontal stretch of factor 1

2


12. Vertical stretch of factor 6 or Horizontal

compression of factor 6



6

Horizontal shift of 4 units to the left Vertical shift of 3 units downwards


3


Vertical shift of 1

4 of a unit upwards


6

Horizontal compression of factor 3 Horizontal shift of 4 units to the left


3

Vertical shift of an 1

8 of a unit


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 ________________________________________


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 ________________________________________


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

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.



Answer Key

Day 35

A= 1

B = 4

C = 2

D =3

170


Algebra 1


Week 7


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 ____________________________________


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?

http://www.census.gov/popest/data/national/totals/2014/index.html

Unit 2 Test Name ____________________________________


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 ____________________________________


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

Unit 2 Test Name ____________________________________


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 ____________________________________


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 ____________________________________


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.

Unit 2 Test Name ____________________________________


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

complete unit 2 package - highschoolmathteachers.com · algebra 1 unit 2 skills list number unit...

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