19
Pre-Test Unit 5: Polynomials/Quadratics KEY You may use a calculator on the whole test. Identify whether the following are polynomials or not. If it is a polynomial, give its name by both degree and number of terms. (4 pts; 2 pts for polynomial or not, 1 pt for each name) 1. 4 2. Yes, cubic binomial No, rational 3. 2 1 4. √ 3 Yes, quadratic trinomial No, radical Perform the following polynomial operations. (4 pts; 2 pts if computation error only) 5. 2 3 7 5 3 17 6. 5 4 5 2 7 3 12 2 3 10 3 7 7 17 7. 3 7 2 8. 3 2 1 3 13 14 4 5 2 Graph the following quadratics. (4 pts; partial credit at teacher discretion) 9 2 16 24. 10. 5 8

Pre-Test Unit 5: Polynomials/Quadratics KEY · PDF fileSketch a graph using the vertex and zeros. (4 pts.; 2 pts for zeros, 2 pts for vertex) 21. 8 7 4 ,9 1,7 Describe the transformation

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Pre-TestUnit5:Polynomials/QuadraticsKEY

You may use a calculator on the whole test.

Identify whether the following are polynomials or not. If it is a polynomial, give its name by both degree and

number of terms. (4 pts; 2 pts for polynomial or not, 1 pt for each name)

1. ���� � �� � 4� 2. ���� �

Yes, cubic binomial No, rational

3. ���� � �� � 2� 1 4. ���� � √� � 3

Yes, quadratic trinomial No, radical

Perform the following polynomial operations. (4 pts; 2 pts if computation error only)

5. ��� 2�� � 3� � 7� ��� � 5�� 3� 17� 6. �5�� � 4� 5� � �2�� � 7�� 3� � 12� 2�� � 3�� 10 3�� 7�� � 7� 17

7. �3� � 7��� � 2� 8. ��� � 3� 2��� � 1� 3�� � 13� 14 �� � 4�� 5� � 2

Graph the following quadratics. (4 pts; partial credit at teacher discretion)

9���� � 2�� � 16� 24. 10. ���� � �� 5�� � 8

11. Min: � � �7, Zeros: � � 3 and � � �5 12. A cannon is launched from 4�� behind you and lands

6�� in front of you. It reaches a max height of 8��.

Factor the following quadratic functions using the distributive property. Then list the zeros of the quadratic.

(4 pts.; 2 pts for factoring, 2 pts for zeros)

13. ���� � �� 6� 9 14. ���� � �� � 49

�� 3��; � � �3 �� 7��� � 7�; � � �7, 7

15. ���� � �� � 7� 16. ���� � �� � 3� � 28

��� � 7�; � � 0, 7 �� � 7��� 4�; � � �4, 7

Find the vertex using any method without graphing. (4 pts.; 2 pts for method, 2 pts for vertex)

17. ���� � �� 6� 2 18. ���� � �� 6� 9

��3 � 7� ��3, 0�

19. ���� � �� 4� 14 20. ���� � �� � 4� � 21

��2, 10� �2, �25�

Sketch a graph using the vertex and zeros. (4 pts.; 2 pts for zeros, 2 pts for vertex)

21. ���� � �� � 8� 7

�4, �9�

� � 1, 7

Describe the transformation from the parent function ���� � ��. (4 pts.; partial credit at teacher discretion)

22. ���� � �� � 8� 5 23. ���� � �� � 4�� 2

Translate right 4 and down 11 Translate right 4 and up 2

Sketch the transformation from the parent function ���� � ��. (4 pts.; partial credit at teacher discretion)

24. ���� � �2�� 3�� � 4 25. ���� � �� � 4�� 3

Lesson 5.1Lesson 5.1Lesson 5.1Lesson 5.1 Unit 5 Homework Answer KeyUnit 5 Homework Answer KeyUnit 5 Homework Answer KeyUnit 5 Homework Answer Key

Decide whether or not the following functions are polynomials or not. If they are polynomials, name them both

by degree and the number of terms.

1. ���� � �� � 1 2. ���� =�

�� 3. ���� = �� + �� + 3

Yes, quadratic binomial Yes, linear monomial Yes, cubic trinomial

4. ���� = √� − 1 5. ���� =

� 6. ���� = −�!

No, radical Yes, 7th degree monomial Yes, quartic monomial

7. ���� = 3 8. ���� =�

� 9. ���� = −2� − 8

No, exponential No, rational Yes, linear binomial

10. ���� ="

��� − 3� + 4 11. ���� = 4 + 2� 12. ���� = √�� + 3

Yes, quadratic trinomial No, exponential No, radical

13. ���� = �� − 1 14. ���� = 5�� 15. ���� = −3�� + 6

Yes, cubic binomial Yes, quadratic monomial Yes, cubic binomial

16. ���� = �� −"

17. ���� = −7�� 18. ���� =

#$%

No, rational Yes, cubic monomial No, rational

19. ���� = √� + 5% 20. ���� = 2�

%& 21. ���� = �� +

!

No, radical No, radical Yes, cubic binomial

22. ���� = 2�'� 23. ���� = −""

�� −

( 24. ���� = �'� + ��

No, rational Yes, linear binomial No, rational

Lesson 5.2Lesson 5.2Lesson 5.2Lesson 5.2

Perform the following polynomial operations.

1. ��� � 1� + ��� + 2� + 8� 2. �3� − 4� − ��� + 3� + 5� 3. �2� − 1���� + 4�� + 3� �� + �� + 2� + 7 −�� − 9 2�! + 7�� − 4�� + 6� − 3

4. ��� + 3� − 5���� − � + 4� 5. �4�� + 6�� − 2� + �2�� − 4�� 6. �� − �� + 4�� − �4�� + 2�� �! + 2�� − 4�� + 17� − 20 4�� + 8�� − 4� − 2 � − 5�� + 2�

7. �2�� + 2��4�� − 3� − 2� 8. ��� + 2� − 5��2�� − 3� − 3� 9. �6� − 1� + ��� + �� − � + 1� 8� − 6�! − 4�� + 8�� − 6� − 4 2�! + �� − 19�� + 9� + 15 �� + �� + 5�

10. ��! − �� + 7� − �� + 4�!� 11. 2���4�� − 2� − 1� 12. �3�� − ���4�� − 2� −� − 3�! − �� + 7 8� − 4�! − 2�� 12� − 10�� + 2�

13. �7� + �� + �2� + �� + 3�� 14. �5�� − 4�� − �6 − 2� + ��� 15. �6 − ����� + 3� + 4� 10� + �� + 3� 4�� − 2� − 6 −�� + 3�� + 14� + 24

16. ��� + 1��3�� + 2� + 1� 17. �5 + 3� + ��� + ��� − 6� + 3� 18. �2� + ��� − �4 − � − 4��� 3�! + 2�� + 4�� + 2� + 1 �� + �� − 3� + 8 2� + 5�� + � − 4

19. �3�� − 9� + 1���! + �� − 2� 20. �4 + 3����� − 7� − 2� 21. �5� + �� + �2� + �� − �� 3�) − 9� + 4�! − 9�� − 5�� + 18� − 2 3�� − 17�� − 34� − 8 3� + �� + 4�

22. �4 − ��� − ��� + 3� + 4� 23. �2� − 3��9 + 4�� + 2�� 24. �2�� + ��� + 3��� + 4� −2�� − 3� 4�) − 6� + 8�! − 12�� + 18� − 27 �� + 5�� + 12�

25. What degree of polynomial would you get if you added a 5th degree polynomial to a 3rd degree polynomial

and how do you know?

The sum would be a 5th degree polynomial at most. We know it would remain a 5th degree because the term with

the 5th degree could not cancel with any term in the 3rd degree polynomial.

26. If you added a trinomial to a binomial, how many terms could the sum have?

This cannot be determined because some terms could cancel.

27. If you add or subtract two polynomials, why do you always get another polynomial?

The system is closed system under addition and subtraction.

28. If you multiply two polynomials, why do you always get another polynomial?

The system is closed system under multiplication.

Lesson 5.3Lesson 5.3Lesson 5.3Lesson 5.3

Graph the following quadratics. When given, fill out the �/+ charts with appropriate values.

1. ���� � �� 2��� � 4� 2. ���� � �� 6��� 8� 3. ���� � �� 6� 2

� �2 0 1 2 4 � �9 �8 �7 �6 �5 � �5 �4 �3 �2 �1

� 0 �8 �9 �8 0 � 3 0 �1 0 3 � �3 �6 �7 �6 �3

4. ���� � 2�� � 6�� 2 5. ���� � �� 2��� 8� 6. ���� � �� � 7��� � 3�

� 4 5 6 7 8 � �8 �6 �5 �4 �2 � 3 4 5 6 7

� 10 4 2 4 10 � 0 �8 �9 �8 0 � 0 �3 �4 �3 0

7. ���� � ��� � 6� 8. ���� � 3�� � 6� � 3 9. ���� � ��� 3�� 6

� 0 2 3 4 6 � �1 0 1 2 3 � �5 �4 �3 �2 �1

� 0 �8 �9 �8 0 � 6 �3 �6 �3 6 � 2 5 6 5 2

10. ���� � 2�� � 8� 12 11. ���� � �� 2�� 3 12. ���� � �� � 9��� � 5�

� 0 1 2 3 4 � �4 �3 �2 �1 0 � 5 6 7 8 9

� 12 6 4 6 12 � 7 4 3 4 7 � 0 �3 �4 �3 0

13. ���� � �� 1��� � 3� 14. ���� � ��� � 10� � 24 15. ���� � �2�� 5�� � 1

� �1 0 1 2 3 � �7 �6 �5 �4 �3 � �7 �6 �5 �4 �3

� 0 �3 �4 �3 0 � �3 0 1 0 �3 � �9 �3 �1 �3 �9

16. ���� � ��� 4� 17. ���� � �� 4�� � 7 18. ���� � �� 4��� � 1�

� �4 �3 �2 �1 0 � �6 �5 �4 �3 �2 � �4 �2 �1 0 1

� 0 �3 �4 �3 0 � �3 �6 �7 �6 �3 � 0 �6 �6 �4 0

19. ���� � �� � 3�� � 7 20. ���� � ��� 6� � 14 21. ���� � 3�� 5�� � 2

� 1 2 3 4 5 � 1 2 3 4 5 � �7 �6 �5 �4 �3

� �3 �6 �7 �6 �3 � �9 �6 �5 �6 �9 � 10 1 �2 1 10

22. Zeroes: � � 3, � � 9 23. Zeroes: � � �2, � � 2 24. Zeroes: � � �5, � � �1

Max: � � 4 Min: � � �8 Max: � � 4

25. Zeroes: � � �2, � � 3 26. Zeroes: � � 4, � � 6 27. Zeroes: � � 0, � � 5

Max: � � �6 Min: � � �1 Min: � � �9

Impossible, max below �-axis

28. A cannon ball is launched from the ground 5 29. An angry bird is launched 7 feet behind you,

feet behind you. It reaches a maximum height reaches a maximum height of 5 feet, and lands

of 8 feet and lands 3 feet in front of you. 1 foot behind you.

30. Tarzan starts on a tree branch 3 yards behind 31. You are holding on to a rope on the edge of a cliff

where his rope is attached and swings 6 yards 4 feet from a creek. You swing to the other side and

below the tree line to another tree (same height) land 4 feet away on the other side. Your toes barely hit

3 yards beyond where his rope is attached. the water 6 feet below the cliffs.

32. You throw a rock that lands 6 feet in front 33. You shoot a basketball that reaches a maximum

of you. It barely missed the bottom of a piñata height of 3 yards. Unfortunately, it was an air ball and

that is slightly more than 4 feet off the ground landed 4 yards in front of you.

Lesson 5.4Lesson 5.4Lesson 5.4Lesson 5.4

Multiply the following binomials using the distributive property.

1. �� � 1��� + 1� 2. ��� − 2� 3. �� + 2��� + 2�

�� − 1 �� − 2� �� + 4� + 4

4. 2��� + 3� 5. �� + 3��� + 4� 6. �� − 5��� + 2�

2�� + 6� �� + 7� + 12 �� − 3� − 10

7. −��� + 3� 8. �2� + 1��2� − 1� 9. �� − 4��� − 2�

−�� − 3� 4�� − 1 �� − 6� + 8

10. �� − 5��� + 3� 11. �3� + 2��� − 1� 12. �2� + 5��� + 2�

�� − 2� − 15 3�� − � − 2 2�� + 9� + 10

Factor the following quadratic functions using the distributive property. Then list the zeros of the quadratic.

13. ���� � �� � 2� + 1 14. ���� = �� − 16 15. ���� = �� + 5�

�� − 1�� �� − 4��� + 4� ��� + 5�

� = 1 � = 4, � = −4 � = 0, � = −5

16. ���� = 2�� − 4� 17. ���� = −�� − 3� + 10 18. ���� = �� + 5� + 6

2��� − 2� −�� + 5��� − 2� �� + 3��� + 2�

� = 0, � = 2 � = −5, � = 2 � = −3, � = −2

19. ���� = �� − 9 20. ���� = −2�� − 3� − 1 21. ���� = �� − 7�

�� − 3��� + 3� −�2� + 1��� + 1� ��� − 7�

� = 3, � = −3 � = −0.5, � = −1 � = 0, � = 7

22. ���� = −�� + 8� − 12 23. ���� = �� + 4� + 4 24. ���� = �� − 25

−�� − 6��� − 2� �� + 2�� �� + 5��� − 5�

� = 6, � = 2 � = −2 � = −5, � = 5

Factor the following quadratic functions using the distributive property. Then list the zeros of the quadratic.

25. ���� � 2�� + 5� − 3 26 ���� = 3�� − 2� − 8 27. ���� = 4�� + 4� − 3

�2� − 1��� + 3� �3� + 4��� − 2� �2� − 1��2� + 3�

� = 0.5, � = −3 � = −!

�, � = 2 � = 0.5, � = −1.5

28. ���� = 5�� − 18� − 8 29. ���� = 8�� + 6� − 27 30. ���� = 3�� + 2� − 5

�5� + 2��� − 4� �2� − 3��4� + 9� �3� + 5��� − 1�

� = −0.4, � = 4 � = 1.5, � = −(

! � = −

�, � = 1

31. ���� = 5�� − 5� − 30 32. ���� = 4�� − 48� + 144 33. ���� = 7�� − 28

5�� + 2��� − 3� 4�� − 6�� 7�� + 2��� − 2�

� = −2, � = 3 � = 6 � = −2, � = 2

34. ���� = 20�� + 30� − 20 35. ���� = 9�� + 3� − 12 36. ���� = 4�� − 22� + 24

10�2� − 1��� + 2� 3�3� + 4��� − 1� 2�2� − 3��� − 4�

� = 0.5, � = −2 � = −!

�, � = 1 � = 1.5, � = 4

Answer the following questions.

37. Using technology, graph the function ���� � 4�� + 8� − 45. Are there zeros? Approximately at what �

values are they?

Yes, at approximately � = −4.5 and � = 2.5

38. Do you think the quadratic ���� = 4�� + 8� − 45 will factor? Why or why not?

Answers will vary

39. Using technology, graph the function ���� = �� − 5. Are there zeros? Approximately at what � values are

they?

Yes, at approximately � = −2.25 and � = 2.25

40. Do you think the quadratic ���� = �� − 5 will factor? Why or why not?

Answers will vary

41. What conclusions can you draw about the reliability of factoring?

Factoring is not very reliable because it does not work very often.

Lesson 5.5Lesson 5.5Lesson 5.5Lesson 5.5

Find the vertex using any method.

1. ���� � �� � 2� − 8 2. ���� = �� + 4� − 4 3. ���� = �� − 5� + 4

�1, −9� �−2,−8� �2.5,−2.25�

4. ���� = �� + 4� + 10 5. ���� = 2�� − 12� + 6 6. ���� = �� + 3� − 18

�−2, 6� �3, −12� �−1.5,−20.25�

7. ���� = �� − 4� + 8 8. ���� = 2�� − 12� + 16 9. ���� = 3�� + 9� + 3

�2, 4� �3, −2� �−1.5,−3.75�

10. ���� = �� − � + 5 11. ���� = 3�� + 9� + 6 12. ���� = �� − 4� + 4

�0.5, 4.75� �−1.5,−0.75� �2, 0�

13. ���� = �� + 8� − 4 14. ���� = �� + 2� − 1 15. ���� = �� + 6� + 3

�−4,−20� �−1,−2� �−3,−6�

16. ���� = −2�� + 8� − 4 17. ���� = �� − 6� + 7 18. ���� = �� + 2� + 3

�2, 4� �3, −2� �−1, 2�

Do a quick sketch of the graph of each function by finding first finding the vertex and then filling out the �/+

chart or finding the zeros.

19. ���� � �� 6� � 1 20. ���� � �� � 2� 6 21. ���� � �� 6� 8

� �5 �4 �3 �2 �1 � �1 0 1 2 3 � �5 �4 �3 �2 �1

� �6 �9 �10 �9 �6 � 9 6 5 6 9 � 3 0 �1 0 3

Vertex: ��3,�10� Vertex: �1, 5� Vertex: ��3,�1�; Zeros: �4, �2

22. ���� � �� 8� 12 23. ���� � �� � 2� � 8 24. ���� � �� 4� � 2

� �6 �5 �4 �3 �2 � �1 0 1 2 3 � �4 �3 �2 �1 0

� 0 �3 �4 �3 0 � �5 �8 �9 �8 �5 � �2 �5 �6 �5 �2

Vertex: ��4,�4�; Zeros: �6, �2 Vertex: �1, �9�; Zeros: �2, 4 Vertex: ��2,�6�

Lesson 5.6Lesson 5.6Lesson 5.6Lesson 5.6

Describe the transformation applied to the parent function ���� � ��.

1. ��3�� 2 2. "

�∗ ��� � 2� 3. ���� 5�

Parabola narrows, stretches 3 times Parabola widens, stretches half as Reflects across �-axis, translates

closer to the �-axis, translates up 2 far from �-axis, translates 2 units right 5 units left

4. ���� � �� � 1�� 5. ���� � 2�� � 8 6. ���� � ��� 6��

Translated right 1 unit Parabola narrows, stretches 2 times Reflected across �-axis and

closer to �-axis, translated down 8 translated left 6 units

7. ���� �"

��� 3 8. ���� � �� 2�� � 5 9. ���� � 2�� � 3��

Parabola widens, stretches "

� times Translated left 2 units and down Parabola narrows, stretches 2

closer to �- axis and translated up 3 5 units times farther from �-axis,

translates right 3 units

10. ���� � ��� � 8� � 16 11. ���� � �� � 2� � 5 12. ���� � 2�� � 12� 18

Reflected across �-axis and Translates right 1 unit and down Parabola narrows, stretches 2

translated left 4 6 units times farther from �-axis,

translate right 3 units

Given the original function of ���� � �� .�� � /, do a quick sketch of the transformed function 0��� without

using an �/+ chart.

13. ���� � �� � 3 14. ���� � �� 4�� � 8 15. ���� � 2�� 4�� � 3

16. ���� � ��� 1�� � 3 17. ���� � �� � 2�� 1 18. ���� �"

��� 4��

ReviewUnit5:Polynomials/QuadraticsKEY

You may use a calculator.

Identify whether the following are polynomials or not. If it is a polynomial, give its name by both degree and

number of terms.

1. ���� � �� � 4�� + 3� − 1 2. ���� =

+ 2� − 1

Yes, cubic polynomial No, rational

3. ���� = −2�� 4. ���� = 2√�� − 4

Yes, quadratic monomial No, radical

5. ���� = � − �� − � 6. ���� = 5� − 3

Yes, 5th degree trinomial Yes, linear binomial

Perform the following polynomial operations.

7. �5�� + 4�� − 3� − 7� + �−2�� − 5�� + 6� + 12� 8. �7�� − 4� + 1� − �3�� − 7�� − 2� 3�� − �� + 3� + 5 4�� + 7�� − 4� + 3

9. ��� − 3� − 10� + ��� + 4�� − 3� − 7� 10. �3�� + 6�� − 5� − �8�� − �� + 3� + 1� �� + 5�� − 6� − 17 −5�� + 7�� − 3� − 6

11. �2� − 4��3� − 1� 12. ��� − � − 2��� + 3� 6�� − 14� + 4 �� + 2�� − 5� − 6

13. �� + 7��� − 7� 14. ��� + 3� + 2��� − 5� �� − 49 �� − 2�� − 13� − 10

Graph the following quadratics.

15. ���� � 2�� � 16� 24 16. ���� � �� 5�� � 8

17. ���� � �� � 4� � 5 18. Max: � � 4, Zeros: � � 3 and � � 7

19. Min: � � �4, Zeros: � � �3 and � � 9 20. A cannon is launched from 9�� behind you and lands

9�� in front of you. It reaches a max height of 7��.

Factor the following quadratic functions using the distributive property. Then list the zeros of the quadratic

21. ���� � �� 2� + 1 22. ���� = �� − 64

�� + 1�� �� − 8��� + 8�

23. ���� = �� + 3� 24. ���� = �� + 3� − 28

��� + 3� �� − 4��� + 7�

25. ���� = �� + 6� + 8 26. ���� = 2�� + 8� − 10

�� + 4��� + 2� 2�� + 5��� − 1�

27. ���� = �� − 10� + 25 28. ���� = �� − 5� − 24

�� − 5�� �� − 8��� + 3�

Find the vertex using any method without graphing.

29. ���� � �� 2� + 3 30. ���� = �� + 12� + 36

�−1, 2� �−6,0�

31. ���� = �� + 4� − 6 32. ���� = −�� + 6� − 2

�−2,−10� �3, 7�

33. ���� = �� − 9 34. ���� = �� + 4� − 32

�0, −9� �−2,−36�

35. ���� ="

!�� + � − 8 36. ���� = �� + 14� + 41

�−2,−9� �−7,−8�

Sketch a graph using the vertex and zeros.

37. ���� � �� � 4 38. ���� � �� � 10� 21

Describe the transformation from the parent function ���� � ��.

39. ���� � �� 2� 5 40. ���� � �� 5�� � 7

Translated left 1 unit and up 4 units Translated left 5 units and down 7 units

41. ���� � �� 2� � 3 42. ���� � �2�� � 1�� 3

Translated left 1 unit and down 4 units Parabola narrows, Stretched 2 times farther from �-axis

Reflects over �-axis, Translates right 1 unit and up 3 units

Sketch the transformation from the original function ���� � �� /�� � ..

43. ���� � ��� � 3�� � 4 44. ���� � �� 3�� � 7