Quadratics Unit: Classwork & Homework

Characteristics of Quadratics

Class Work

If the following equation is a quadratic, write the equation in standard form.

  1. y= 2 + 3x2 – 5
  1. 4x – 5 = x + y
  1. 5x +4y= x2 – 2
  1. 4x2 – 2 = 4x
  1. 3x2 + 2x = 3x2 – 6

For each of the following graphs, find the direction the parabola opens, the vertex, state whether the vertex is a maximum or minimum, the domain, the range, the axis of symmetry, and the x-intercepts, if any exist.

  1. 7.

x scale is 1x scale is 1

y scale is 2y scale is 2

Homework

If the following equation is a quadratic, write the equation in standard form.

  1. y= 7 + 3x – 5
  1. 4x – 6 = x2 + y
  1. 10x +2y= 8x2 – 2
  1. 4x2 – 2 +10x = 4x
  1. 3x2 - 2x = 2x2 – 7

For each of the following graphs, find the direction the parabola opens, the vertex, state whether the vertex is a maximum or minimum, the domain, the range, the axis of symmetry, and the x-intercepts, if any exist.

  1. 14.

x scale is 1x scale is 1

y scale is 2y scale is 2

Identifying Parts of a Parabola

Class Work

Find the axis of symmetry, the vertex, and the y-intercept of each parabola. Graph the quadratic.

  1. y= x2 + 2x + 1
  2. y= x2 - 6x + 8
  3. y= x2 -4x -2
  4. y= 2x2 +6x +3
  5. y= 3x2 -4x -2

Without graphing, does the graph of the given equation open up or down? Is the graph wider or narrower than the parent equation of y=x2? What is the y-intercept?

  1. f(x)=2x2 +3x -4
  2. y = -.7x2 -4x +3
  3. y = -1.2x2 +6
  4. g(x)= 3x2 +3x
  5. y = -4x2

Homework

Find the axis of symmetry, the vertex, and the y-intercept of each parabola. Graph the quadratic.

  1. y= x2 +2x -8
  2. y= x2 -4x +3
  3. y= .5x2 + 3x -8
  4. y= x2 + 7x + 6
  5. y= 3x2 -4x

Without graphing, does the graph of the given equation open up or down? Is the graph wider or narrower than the parent equation of y=x2? What is the y-intercept?

  1. f(x)=-.6x2 +3x -6
  2. y = 1.7x2 -4x +5
  3. y = -1.02x2 +8
  4. g(x)= 1.3x2 +4x
  5. y = 5x2

Transformations

Class Work

In each exercise the function is given. Describe the transformation of the parent function. Graph the function.

Home Work

In each exercise the function is given. Describe the transformation of the parent function. Graph the function.

Graphing to Find Zeros

Class Work

Find the zeros of the following quadratics:

Find the zeros of the following quadratics by graphing.

  1. y = x2 -4x +3
  2. h(x)= -x2 +3x -8
  3. y = -x2 -8x -15
  4. y= -x2 -8x -16
  5. f(x)= x2 +3x -10
  6. g(x)= 2x2 +4x +2
  7. y= -3x2 +4x +4

Homework

Find the zeros of the following quadratics:

Find the zeros of the following quadratics by graphing.

  1. y = x2 -6x +5
  2. y = -x2 +3x +10
  3. y = x2 +6x +9
  4. f(x)= x2 +x -12
  5. y = x2 +2x +4
  6. g(x)= 2x2 +5x +2
  7. y= -3x2 +11x +4

Solving by Factoring

Class Work

Solve the following quadratics by factoring.

Algebra II - Quadratics ~1~NJCTL.org

  1. a2 +4a +3= 0
  1. b2 -4b -5= 0
  1. -c2 -6c = -7
  1. d2 +8d = -12
  1. -e2 +9 = 0
  1. f2 +4f +4 = 0
  1. –g2 +5g = 6
  1. 2h2 +7h +6= 0
  1. 3j2 -4j = -1

Algebra II - Quadratics ~1~NJCTL.org

Home Work

Solve the following quadratics by factoring.

Algebra II - Quadratics ~1~NJCTL.org

  1. a2 +6a +5= 0
  1. b2 -b -6= 0
  1. -c2 -6c = 8
  1. d2 +7d = -10
  1. -e2 +16 = 0
  1. f2 +6f +9 = 0
  1. –g2 +7g = 6
  1. 2h2 +8h +6= 0
  1. 3j2 -7j = -4

Algebra II - Quadratics ~1~NJCTL.org

Solving Using the Square Roots Method

Class Work Solve the following quadratics using the square roots method.

Algebra II - Quadratics ~1~NJCTL.org

  1. m2 = 16
  1. n2 = 25
  1. 3p2 = 12
  1. 5q2 = 80
  1. r2 -3 =6
  1. s2 +8 =17
  1. 2t2 -6 = -4
  1. 3u2 +5 = 17
  1. (v -7)2 -5 = 11
  1. 2(w -3)2 +6 = 56

Algebra II - Quadratics ~1~NJCTL.org

  1. The square of six less than a number is twenty-five. Write an equation that models this situation. Solve the equation.

Homework Solve the following quadratics using the square roots method.

Algebra II - Quadratics ~1~NJCTL.org

  1. m2 = 36
  1. n2 = 64
  1. 3p2 = 27
  1. 5q2 = 20
  1. r2 -3 = 13
  1. s2 +8 = 24
  1. 2t2 -6 = 12
  1. 3u2 +5 = 8
  1. (v -2)2 +4 = 13
  1. 3(w +4)2 -4 = 44

Algebra II - Quadratics ~1~NJCTL.org

  1. Two times the square of five more than a number is seventy-two. Write an equation that models this situation. Solve the equation.

Solving by Completing the Square

Class Work

Fill in the blank to complete the square.

  1. a2 +8a +__
  2. b2 +10b +__
  3. c2 -4c +__
  4. d2 -6d +__
  5. e2 +7e +__
  6. f2 -5f +__

Solve the following quadratics by completing the square.

  1. h2 +6h =16
  2. j2 -8j = -7
  3. k2 +9 = -10k
  4. m2 -13 = 12m
  5. 14n + 20= -n2
  6. 8p +p2 = 0
  7. 2q2 -8q = 40
  8. 3r2 +27r = 12
  9. A toy rocket launched into the air has a height (h feet) at any given time (t seconds) as h = -16t2 + 96t until it hits the ground. At what time(s) is it at a height of 7 feet above the ground?

Homework

Fill in the blank to complete the square.

  1. a2 +12a +__
  2. b2 +b +__
  3. c2 -14c +__
  4. d2 - 16d +__
  5. e2 +9e +__
  6. f2 -1f +__

Solve the following quadratics by completing the square.

  1. h2 +4h =12
  2. j2 -10j = -9
  3. k2 +15 = -14k
  4. m2 -21 = 20m
  5. 2n - 80= -n2
  6. 6p +p2 = 0
  7. 2q2 -12q = -22
  8. 3r2 +15r = 18
  9. A toy rocket launched into the air has a height (h feet) at any given time (t seconds) as

h = -16t2 + 160t until it hits the ground. At what time(s) is it at a height of 9 feet above the ground?

Complex Numbers

Classwork

Complete each of the following operations with complex numbers. Simplify as much as possible.

Homework

Solving Using the Quadratic Formula

Class Work

Solve the following using the quadratic formula. Leave irrational answers in radical form.

  1. x2 +8x -6 =0
  1. g2 -4g +2 =0
  1. 3d2 + 4d -3 =0
  1. -2m2 + 3m = 1
  1. 4w2 -8 = 5w
  1. 7z – 9z2 = -4
  1. An employee makes (2x + 3) dollars an hour for x hours. If the employer wants to pay no more than $120 a day, what is the maximum number of hours the employee can work? (Round to the nearest quarter hour)

Homework

Solve the following using the quadratic formula. Leave irrational answers in radical form.

  1. x2 +7x -5 =0
  1. g2 -5g +3 =0
  1. 2d2 + 5d -3 =0
  1. -3m2 + 4m = -5
  1. 5w2 -2 = 5w
  1. 3z – 6z2 = -8
  1. An employee makes (3x - 5) dollars an hour for x hours. If the employer wants to pay no more than $200 a day, what is the maximum number of hours the employee can work? (Round to the nearest quarter hour)

Discriminant

Class Work

Find the discriminant for each quadratic equation. State the number of real roots and then find the real solution(s), if any exist.

  1. x2 - 6x + 5 = 0
  1. 2x2 - 4x - 6 = 0
  1. 3x + 4x2 + 5 = 0
  1. 6x – 9 = x2
  1. 3x2 = 6x – 8
  1. 4x2 - 9x – 2= 0
  1. x (2x – 5) = 10
  1. A rock is thrown with a height equation of h = -16t2 + 20t + 5 (where h is the height of the rock in feet at any given time of t in seconds). Will it reach a height of 30 feet? Explain your answer.

Homework

Find the discriminant for each quadratic equation. State the number of real roots and then find the real solution(s), if any exist.

  1. 3x2 - 9x + 7 = 0
  1. 2x2 - 4x + 2 = 0
  1. 5x + 7x2 + 8 = 0
  1. 7x – 6 = 2x2
  1. 5x2 = 7x – 6
  1. 3x2 - 7x – 8= 0
  1. (x + 3)(2x + 6) = 11
  1. A rock is thrown with a height equation of h = -16t2 + 64t + 5 (where h is the height of the rock in feet at any given time of t in seconds). Will it reach a height of 50 feet? Explain your answer.

Vertex Form of a Parabola

Class Work

Find the vertex, direction of openness, and axis of symmetry for each of the following.

Convert the following equations from standard form to vertex form.

Homework

Find the vertex, direction of openness, and axis of symmetry for each of the following.

Convert the following equations from standard form to vertex form.

Mixed Application Problems

Class Work

Solve the following problems using any method.

  1. The product of two consecutive positive integers is 272, find the integers.
  2. The product of two consecutive positive even integers is 528, find the integers.
  3. The product of two consecutive odd integers is 255, find the integers.
  4. Two planes leave airport at the same time (from different runways), one traveling south and one starveling west. If three hours later they are 500 miles apart and the plane flying south has traveled 200 miles farther, how far did the one flying west travel?
  5. Two cars leave a gas station at the same time, one traveling north and one traveling east. One hour later they are 80 miles apart and the one traveling east went 10 miles farther, how far is it from the gas station?
  6. A square has its length increased by 4 feet and its width by 5 feet. If the resulting rectangle has an area of 132 square feet what was the perimeter of the original square?
  7. A rectangular parking lot has a width 30 feet more than its length. The owners are able to increase the width by 20 feet and the length by 40. The new lot has an area of 27,200 square feet, what is the area of the original lot?
  8. A square’s sides are tripled producing an area of 81 square feet. What is the ratio of the area of the original square to the transformed square?
  9. A garden has a length of (x + 2)feet and a width of (2x - 1)feet. The garden’s total area is 88 square feet. Find the length.

Homework

Solve the following problems using any method.

  1. The product of two consecutive positive integers is 342, find the integers.
  2. The product of two consecutive positive even integers is 168, find the integers.
  3. The product of two consecutive positive odd integers is 483, find the integers.
  4. Two planes leave airport at the same time (from different runways), one traveling south and one traveling west. If three hours later they are 600 miles apart and the plane flying south has traveled 100 miles farther, how far did the one flying west travel?
  5. Two cars leave a gas station at the same time, one traveling north and one traveling east. One hour later they are 90 miles apart and the one traveling east went 15 miles farther, how far is it from the gas station?
  6. A square has its length increased by 6 feet and its width by 8 feet. If the resulting rectangle has an area of 239.25 square feet what was the perimeter of the original square?
  7. A rectangular parking lot has a width 20 feet more than its length. The owners are able to increase the width by 20 feet and the length by 40. The new lot has an area of 7225 square feet, what is the area of the original lot?
  8. A square’s sides are quadrupled producing an area of 64 square feet. What is the ratio of the area of the original square to the transformed square?
  9. A garden has a length of (x - 4)feet and a width of (2x +3)feet. The garden’s total area is 76 square feet. Find the length.

End of Unit Review: Multiple Choice

1. Comparing the graph of y = 2x2 + 6x -3/2 to its parent function, it:

A) opens down and is wider than the parent function graph.

B) opens down and is narrower than the parent function graph.

C) opens up and is wider than the parent function graph.

D) opens up and is narrower than the parent function graph.

2. What is the equation of the axis of symmetry of y = 3x2 + 6x – 7?

A) x = -6

B) x = -1

C) x = 1

D) x = 6

3. What is the coordinates of the vertex of y = 3x2 + 6x – 7?

A) (-6 , 65)

B) (-1 , -10)

C) (1 , -14)

D) (6 , 137)

4. What is the y- intercept of y = 3x2 + 6x – 7

A) (0 , 7)

B) (7 , 0)

C) (0 , -7)

D) (-7, 0)

5. Which graph to the righthas no zeros?

A

B

C

D

6. Which of the following is a step in solving y = x2 + 5x +6 by the Factoring Method?

A) x + 3 = 0 or x + 2 = 0

B) 2x + 2 = 0 or x + 3 = 0

C) x + 1 = 0 or x + 6 = 0

D) 2x + 6 = 0 or x +1 = 0

7. The solution to (x + 16)2 = 25 is

A) -5 and 5

B) -4 and 4

C) -3 and 3

D) -11 and -21

8. What value goes on the blank to complete the square: x2 - 8x + ___

A) -4

B) -16

C) 16

D) 4

9. What is the discriminant of 2x2-8x -6 = 0

A) -112

B) -16

C) 16

D) 112

10. How many real zeros does an equation have if the discriminant is -9?

A) 0

B) 1

C) 2

D) Not enough information

11. Solve 2x2 -4x + 6 =0

A.

B.

C.

D.

12. Simplify

A) -6 and -2

B) -4 and -2

C) -4 and -16

D) -6 and -4

13. Given the height of a ball as h = -16t2 + 48t + 32, the class was asked how long was the rock in the air. Kim solved to find the zeros are t = -.56 and t =3.56, assuming the calculations were correct what should the answer to teacher’s be?

A) -.56 seconds

B) 3.56 seconds

C) 4.12 seconds

D) Not enough information

14. Solve 3x2 + 6x -9 = 0

A) -3 and -1

B) 3 and 1

C) 3 and -1

D) -3 and 1

15. The vertex of

A) (4,8)

B) (-4,8)

C) (4,-8)

D) (-4,-8)

Extended Response

  1. A quadratic’s equation in standard form is
  1. What is the vertex form of the quadratic?
  1. What is the quadratic’s y-intercept?
  1. Why is the constant different for both forms of the same quadratic?
  1. What are the x-intercepts?
  1. Which equation did you use from part c and why?
  1. The height of Carl, The Human Cannonball, is given by h(t) = in feet t seconds after launch.
  1. What was his height at launch?
  1. When does he land in the safety net which is 8’ off the ground?
  1. What is his maximum height?
  1. The vertex of a parabola is (4,1) and it opens down at twice the rate of the parent function.
  1. Write the equation of the parabola in vertex form.
  1. Does the parabola have a maximum or a minimum? Where is it? Explain.
  1. What is the y-intercept?
  1. What is the x-intercept?

Answers

Algebra II - Quadratics ~1~NJCTL.org

1.

2.Not quadratic

3.Yes

4.Yes

5.Not quadratic

6.Up; (0,-4) min; D: reals, R: [-4, infinity); x=0; 2 and -2

7.Down, (1,2) max, D: reals; R: (negative infinity, 2]; x=1; 0 and 2

8.Not quadratic

9.

10.

11.

12.

13.Down; (2,8) max; d:reals; R: (negative infinity, 8]; x=2; 0 and 4

14.Up; (-3,-2) min; D:reals; R:[-2, infinity); x=-3; -4.5 and -1.5

15.X=-1; (-1, 0); (0,1)

16.X=3; (3, -1); (0,8)

17.X=2; (2, -6); (0,-2)

18.X=-3/2; (-3/2,-3/2); (0,3)

19.X=2/3; (2/3, -10/3); (0, -2)

20.Up; narrower; (0,-4)

21.Down; wider; (0,3)

22.Down; narrower (0,6)

23.Up; narrower; (0.0)

24.Down; narrower (0,0)

25.X=-1; (-1,-9) (0,-8)

26.X=2; (2, -1) (0,3)

27.X=-3; (-3, -12.5); (0,-8)

28.X=-7/2; (-7/2, -25/4) (0,6)

29.X=2/3; (2/3, -4/3) (0,0)

30.Down, wider, (0,-6)

31.Up, narrower, (0,5)

32.Down, narrower (0,8)

33.Up; narrower; (0,0)

34.Up; narrower; (0,0)

35.up 4

36.down 2

37.right 2

38.left 3

39.reflects over x-axis

40.no change, symmetric about y-axis

41.vertical stretch of 3

42.horizontal shrink of

43.vertical stretch of 4

44.vertical shrink of .3

45.1 left, reflect over x-axis and vertical stretch of 2

46.3 right, up 4

47.1 and -5

48.1 and 3

49.1 and 3

50.None

51.-5 and -3

52.-4

53.2 and -5

54.-1

55.-2/3 and 2

56.1 and -6

57.-1 and 3

58.5 and 1

59.-2 and 5

60.-3

61.-4 and 3

62.None

63.-1/2 and -2

64.-1/3 and 4

65.(a+3)(a+1)=0; -3 and-1

66.(b-5)(b+1)=0; 5 and -1

67.-1(c+7)(c-1)=0; -7 and 1

68.(d+6)(D+2)=0; -6 and -2

69.–(e-3)(e+3)=0; 3 and -3

70.(f+2)(f+2)=0; -2 and -2

71.-1(g-3)(g-2)=0; 3 and 2

72.(2h+3)(h+2)=0; -3/2 and -2

73.(3j-1)(j-1)=0; 1/3 and 1

74.(a+5)(a+1)=0; -5 and -1

75.(b-3)(b+2)=0; 3 and -2

76.(c+4)(c+2)=0; -4 and -2

77.(d+5)(d+2)=0; -5 and -2

78.–(e-4)(e+4)=0; 4 and -4

79.(f+3)(f+3)=0; -3

80.–(g-6)(g-1)=0; 6 or 1

81.(2h+2)(h+3)=0; -3 or -1

82.(3j-4)(j-1)=0; 1 or 4/3

83.

84.

85.

86.

87.

88.

89.

90.

91.

92.w=8 or -2

93.(x-6)2 = 25; 11 or 1

94.m=+/- 6

95.n=+/- 8

96.p=+/- 3

97.q=+/- 2

98.r=+/- 4

99.s=+/- 4

100.t=+/- 3

101.u=+/- 1

102.v=5 or -1

103.w=0 or -8

104.2(x+5)2 = 72; 1 or -11

105.16

106.25

107.4

108.9

109.12.25

110.6.25

111.2 or -8

112.7 or 1

113.-1 or -9

114.13 or -1

115. -1.61 or -12.39

116.0 or -8

117.

118.0.424 and -9.424

119.5.93 seconds and .07 seconds

120.36

121.0.25

122.49

123.64

124.20.25

125.0.25

126.2 or -6

127.9 or 1

128.

129.21 or -1

130.8 or -10

131.-6 or 0

132.

133.1 or -6

134..06 sec and 9.94 sec

135.

136.

137.

138.

139.

140.

141.

142.

143.

144.

145.

146.

147.

148.

149.

150.

151.

152.

153.

154.

155.

156.

157.

158.

159.

160.

161.

162.

163.

164.-3 or 0.5

165.

166.

167.

168.

169.16; 2 real roots; 5 and 1

170.64; 2 real roots; 1 and -3

171.-71; no real roots

172.0; 1 real root; 3

173.-60; no real roots

174.113; 2 real irrational roots;

175.105; 2 real irrational roots;

176.-16t2+20t+5=30; discriminant = -1200; no real solution; rock will not reach 30 ft high.

177.-3; no real solution

178.0; 1 real solution; 1

179.-199; no real solution

180.1; 2 real solutions; 1.5 and 2

181.-71; no real solution

182.145; 2 real irrational solutions;

183.88; 2 real irrational solutions;

184.-16t2+64t+5=50; discriminant=1216; 2 real solutions; The vertex is (-2, 69) and the parabola is open downward so rock reaches 50 feet twice; once on the way up; once on the way down

185.(2, 10), up, x=2

186.(-5,-4), down, x=-5

187.(6,8), down, x=6

188.(-7,0), up, x=-7

189.(3,4), up, x=3

190.(-2,-5), down, x=-2

191.(4,6),down, x=4

192.(0,-4),up, x=0

193.

194.

195.

196.

197.

198.

199.

200.

201.

202.

203.(3.1); down; x=3

204.(-6,0); up; x=-6

205.(-4,1): down; x=-4

206.(8,11); up; x=8

207.(-2,-3); down; x=-2

208.(-1,-2); down; x=-1

209.(3,8); up; x=3

210.(0,-3); up; x=0

211.

212.

213.

214.

215.

216.

217.

218.

219.

220.

221.16 and 17

222.22 and24

223.15 and 17

224.239.12 miles

225.61.35 miles east

226.S=7ft; P=28ft

227.X=120; A= 18,000ft2

228.1/9

229.(x+2)(2x-1)=88

X=-7.5 or 6

Length = 8 feet

230.18 and 19

231.12 and 14

232.21 and 23

233.371.31 miles

234.55.69 miles

235.P=34 ft

236.A=2925 ft 2

237.1/16

238.(x-4)(2x+3)=76

X=-5.5 or 8

Length= 4 feet

Unit Review Answers:

Multiple Choice:

1. D

2. B

3. B

4. C

5. C

6. A

7. D

8. C

9. D

10. A

11. B

12. A

13. B

14. D

15. A

Extended Response:

1a)

b) (0,7)

c) The equations are in different forms which tell you different information about the quadratic.

d) none

e) answers vary:

standard form – check discriminant which is negativeno zeros

vertex form – realize vertex is at (2,3) and parabola is opening upwards no zeros

2a)40 feet

b) t=4 seconds

c) 89 feet

3a)

b) maximum at (4,1); parabola opens downward, vertex is maximum

c) (0,-31)

d); (3.293,0) and (4.707,0)

Algebra II - Quadratics ~1~NJCTL.org