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
Yes, cubic binomial
2. No, rational
3. 2 1
Yes, quadratic trinomial
4. √ 3
No, radical
Perform the following polynomial operations. (4 pts; 2 pts if computation error only)
5. 2 3 7 5 3 17
2 3 10
6. 5 4 5 2 7 3 12
3 7 7 17
7. 3 7 2
3 13 14
8. 3 2 1
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 5.1
Unit 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.
3. = + + 3
Yes, cubic trinomial
6. = − !
Yes, quartic monomial
8. = No, rational
9. = −2 − 8
Yes, linear binomial
10. = − 3 + 4
Yes, quadratic trinomial
11. = 4
+ 2
No, exponential
12. = √ + 3
No, radical
13. = − 1
Yes, cubic binomial
14. = 5 Yes, quadratic monomial
15. = −3 + 6
Yes, cubic binomial
17. = −7 Yes, cubic monomial
18. = # $
%
No, rational
1. 1
Yes, quadratic binomial
2. = Yes, linear monomial
4. = √ − 1
No, radical
5. = Yes, 7th degree monomial
7. = 3
No, exponential
"
"
16. = −
No, rational
%
%
!
21. = +
Yes, cubic binomial
19. = √ + 5
No, radical
20. = 2 &
No, radical
22. = 2 '
No, rational
23. = − − (
Yes, linear binomial
""
24. = ' + No, rational
Lesson 5.2
Perform the following polynomial operations.
1. 1 + + 2 + 8
+ + 2 + 7
2. 3 − 4 − + 3 + 5
− − 9
3. 2 − 1 + 4 + 3
2 ! + 7 − 4 + 6 − 3
4. + 3 − 5 − + 4
! + 2 − 4 + 17 − 20
5. 4 + 6 − 2 + 2 − 4
4 + 8 − 4 − 2
6. − + 4 − 4 + 2
− 5 + 2
7. 2 + 24 − 3 − 2
8 − 6 ! − 4 + 8 − 6 − 4
8. + 2 − 52 − 3 − 3
2 ! + − 19 + 9 + 15
9. 6 − 1 + + − + 1
+ + 5
10. ! − + 7 − + 4 ! − − 3 ! − + 7
11. 2 4 − 2 − 1
8 − 4 ! − 2 12. 3 − 4 − 2
12 − 10 + 2
13. 7 + + 2 + + 3 10 + + 3
14. 5 − 4 − 6 − 2 + 4 − 2 − 6
15. 6 − + 3 + 4
− + 3 + 14 + 24
16. + 13 + 2 + 1
3 ! + 2 + 4 + 2 + 1
17. 5 + 3 + + − 6 + 3
+ − 3 + 8
18. 2 + − 4 − − 4 2 + 5 + − 4
19. 3 − 9 + 1 ! + − 2
20. 4 + 3 − 7 − 2
3 ) − 9 + 4 ! − 9 − 5 + 18 − 2
3 − 17 − 34 − 8
22. 4 − − + 3 + 4
−2 − 3
21. 5 + + 2 + − 3 + + 4
23. 2 − 39 + 4 + 2 24. 2 + + 3 + 4
)
!
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 5.3
Graph the following quadratics. When given, fill out the /+ charts with appropriate values.
1. 2 4
2
0
0
8
1
9
2
8
4
0
4. 2 6 2
4
10
5
4
6
2
7
4
0
0
2
8
3
9
9
3
8
0
7
1
6
0
5
3
5. 2 8
8
10
8
0
6
8
5
9
4
8
4
8
6
0
1
6
0
3
1
6
2
3
5
3
4
6
3
7
2
6
1
3
6. 7 3
2
0
8. 3 6 3
7. 6
3. 6 2
2. 6 8
3
0
4
3
5
4
6
3
7
0
9. 3 6
3
6
5
2
4
5
3
6
2
5
1
2
10. 2 8 12
0
12
1
6
2
4
3
6
11. 2 3
4
12
13. 1 3
1
0
0
3
1
4
2
3
3
0
4
0
3
3
2
4
4
7
3
4
2
3
1
4
0
7
1
3
0
0
5
0
6
3
7
4
8
3
9
0
14. 10 24
15. 2 5 1
7
3
6
0
5
1
4
0
3
3
17. 4 7
16. 4
12. 9 5
6
3
5
6
4
7
3
6
7
9
6
3
5
1
4
3
3
9
18. 4 1
2
3
4
0
2
6
1
6
0
4
1
0
19. 3 7
1
3
2
6
3
7
4
6
20. 6 14
5
3
1
9
2
6
3
5
4
6
21. 3 5 2
5
9
7
10
6
1
5
2
4
1
3
10
22. Zeroes: 3, 9
Max: 4
23. Zeroes: 2, 2
Min: 8
24. Zeroes: 5, 1
Max: 4
25. Zeroes: 2, 3
Max: 6
26. Zeroes: 4, 6
Min: 1
27. Zeroes: 0, 5
Min: 9
Impossible, max below -axis
28. A cannon ball is launched from the ground 5
feet behind you. It reaches a maximum height
of 8 feet and lands 3 feet in front of you.
29. An angry bird is launched 7 feet behind you,
reaches a maximum height of 5 feet, and lands
1 foot behind you.
30. Tarzan starts on a tree branch 3 yards behind
where his rope is attached and swings 6 yards
below the tree line to another tree (same height)
3 yards beyond where his rope is attached.
31. You are holding on to a rope on the edge of a cliff
4 feet from a creek. You swing to the other side and
land 4 feet away on the other side. Your toes barely hit
the water 6 feet below the cliffs.
32. You throw a rock that lands 6 feet in front
of you. It barely missed the bottom of a piñata
that is slightly more than 4 feet off the ground
33. You shoot a basketball that reaches a maximum
height of 3 yards. Unfortunately, it was an air ball and
landed 4 yards in front of you.
Lesson 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 + 12 − 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 − 12 + 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 − 34 + 9
3 + 5 − 1
= −0.4, = 4
= 1.5, = −
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
102 − 1 + 2
33 + 4 − 1
22 − 3 − 4
= 0.5, = −2
= −, = 1
!
(
!
!
= −, = 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 5.5
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
5 4
3
20. 2 6
21. 6 8
2 1
1
0
1
2
3
6 9 10 9 6
Vertex: 3, 10
9
6
5
6
9
Vertex: 1, 5
Vertex: 3, 1; Zeros: 4, 2
22. 8 12
23. 2 8
24. 4 2
6 5 4 3 2
0
3 4 3
0
Vertex: 4, 4; Zeros: 6, 2
1
0
1
2
3
5 8 9 8 5
Vertex: 1, 9; Zeros: 2, 4
5 4 3 2 1
3
0
1
0
3
4 3 2 1
2 5 6 5 2
Vertex: 2, 6
0
Lesson 5.6
Describe the transformation applied to the parent function .
"
1. 3 2
2.
Parabola narrows, stretches 3 times
closer to the -axis, translates up 2
Parabola widens, stretches half as
Reflects across -axis, translates
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
closer to -axis, translated down 8
Reflected across -axis and
translated left 6 units
8. 2 5
9. 2 3
Translated left 2 units and down
5 units
Parabola narrows, stretches 2
times farther from -axis,
translates right 3 units
"
7. 3
"
Parabola widens, stretches times
closer to - axis and translated up 3
∗ 2
3. 5
10. 8 16
11. 2 5
12. 2 12 18
Reflected across -axis and
translated left 4
Translates right 1 unit and down
6 units
Parabola narrows, stretches 2
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
Yes, cubic polynomial
2. = + 2 − 1
No, rational
3. = −2 Yes, quadratic monomial
4. = 2√ − 4
No, radical
5. = − − Yes, 5th degree trinomial
6. = 5 − 3
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
+ 5 − 6 − 17
10. 3 + 6 − 5 − 8 − + 3 + 1
−5 + 7 − 3 − 6
11. 2 − 43 − 1
6 − 14 + 4
12. − − 2 + 3
+ 2 − 5 − 6
13. + 7 − 7
− 49
14. + 3 + 2 − 5
− 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