Name _______________________________________ Period _______

Secondary 2 Honors Final Review – Part 1 (* means calculator allowed)

Simplify:

1. (𝑥3 − 5) + (6𝑥3 + 2) 2. (−3𝑥2 + 16) − (𝑥2 − 22𝑥 − 4) 3. Find the perimeter of the shape. Lengths are given in cm.

4. (𝑥 + 10)(𝑥 − 7) 5. (𝑥 + 14)(3𝑥 + 1)

6. (2𝑥 + 1)(𝑥4 − 6𝑥 + 3)

Evaluate:

8. f(x) = 2x + 1, g(x) = 1 – x find: (f+g)(x)

9. g(x) = 2𝑥2 − 𝑥, h(x) = 𝑥2 − 4 find (gh)(x)

Simplify:

11. √−325

12. √36𝑔6

14. √80 15. √90𝑥3𝑦4

Simplify:

18. 43 =

19. 64−1

3 =

20. 811

4 = 21. 644

3 = 22. 625−3

4 =

23. 4−2 =

24. 25 = 25. 8−1

3 = 26. 24 = 27. 4−5

2 =

Simplify – do not evaluate

28. 𝑔3

10𝑔2

7 29. (158

5)9

2 Simplify if possible, then state whether each answer is rational or irrational:

31. 12 + √27

3

32. 71

2(√16)

Simplify:

33. √−162 34. 𝑖15 37. (1 + 𝑖)(−15 + 2𝑖) 38. (8 + 2𝑖)(3 + 𝑖) Find the complex conjugate of the number, and then find the product of the number and its conjugate.

39. −1 + 𝑖

Factor completely:

44. 3𝑛2 + 21𝑛 − 24 45. 𝑦4 − 16

46. 5𝑥𝑦 – 3𝑦 + 10𝑥 – 6 47. 7 – 10𝑥 – 7𝑦 + 10𝑥𝑦

48. 6𝑥2 − 7𝑥 − 3 49. 3𝑥2 + 16𝑥 − 35

50. 24𝑥2 − 76𝑥 + 40

51. Determine the midpoint of the segment with endpoints (–10, –11) and (8, –17).

52. A segment has an endpoint at (-10, 3) and a midpoint at (7

2, 4). Find the other endpoint.

53. Determine whether the following transformation represents a dilation. Justify your answer and determine the scale factor if possible.

54*. Determine the scale factor and whether the dilation is an enlargement, a reduction, or a congruency transformation.

55*. △MNO has the following vertices: 𝑀(–5, 8), 𝑁(7, – 3), and 𝑂(−10, – 4). What are the vertices under a dilation with a center at (0, 0) and a scale factor of 75%?

56*. Find all angles and side lengths given ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹

57. Decide whether the pair of triangles is similar. Explain your answer.

61*. Find x.

Use the Triangle Proportionality Theorem and the Triangle Angle Bisector Theorem to find the

unknown lengths of the given segments.

63*. 𝐶𝐷̅̅ ̅̅ , 𝐵𝐷̅̅ ̅̅

64*. Is 𝐴𝐵̅̅ ̅̅ ∥ 𝐸𝐶̅̅ ̅̅ ?

65. Find the value of x.

66. Find the value of a, b and x.

67*. To measure 𝐵𝐶̅̅ ̅̅ , the distance across a lake, a surveyor stands at point A and locates points B, C, D, and E. What is the distance across the lake?

68*. The height of a ramp at a point 2.5 meters from its bottom edge is 1.2 meters. If the ramp runs for 6.7 meters along the ground, what is its height at its highest point, to the nearest tenth of a meter?

69. List a pair of complementary angles. Write a statement about those angles using the

Complement Theorem.

Complementary Angles:

Statement:

In the diagram that follows, 𝐴𝐶⃑⃑⃑⃑ ⃑ and 𝐵𝐷⃑⃑⃑⃑⃑⃑ intersect.

70*. Find m∠4 if m∠1 = 3𝑥 + 4 and m∠2 = 2𝑥 − 4.

71. Find m∠1 if m∠1 = 13𝑥 + 7 and m∠3= 7𝑥 + 49.

Use the following diagram to solve problems 72, 73, given that 𝐴𝐵⃡⃑⃑⃑ ⃑‖ 𝐶𝐷⃡⃑⃑⃑ ⃑ and line 𝑙 is the

transversal. Justify your answers using angle relationships in parallel lines intersected by a

transversal.

72. Find m∠2 if m∠2 = 4𝑥 + 39 and m∠7= 12𝑥 − 17.

73*. Find m∠1 if m∠1 = 11𝑥 + 35 and m∠4 = 𝑥 + 1.

Find the value of the variables:

74.

75.

10.

76.

77.

78.

79. Identify the quadrilateral in as many ways as possible. Then identify the most precise name.

Find the values of the variables. Then find the length of the sides.

80. Kite

81. Rhombus HKJI

82. Graph and label the quadrilateral with the given vertices. Then determine and justify the most

precise name. 𝐽(2,1), 𝐾(5,4), 𝐿(7,2),𝑀(2,−3)

83. Find a

84. Find the value of the variables and the measure of each angle for the following parallelogram

Find the values of x and y for which ABCD must be a parallelogram. 85. 86.

87. Use the given information to find the lengths of all four sides of parallelogram ABCD

The perimeter is 92 cm. AD is 7 cm more than twice AB

88. Find the measures of the numbered angles in the rhombus.

89. Find the measures of the numbered angles in the rhombus.

90. Find the values of the variables for the parallelogram.

91. LMNP is a rectangle. Find the value of x and the length of each diagonal.

𝐿𝑁 = 3𝑥 + 5 𝑀𝑃 = 9𝑥 − 10

92. A carpenter is building a bookcase. How can they use a tape measure to check that the bookshelf is rectangular? Justify your answer and name any theorems used.

93*. Find the measure of each angle in the isosceles trapezoid.

94. Find the value of the variable for the isosceles trapezoid.

98. Identify the terms, coefficients, and constant of 16𝑥2 − 12𝑥 + 20.

99. Simplify the expression 3𝑥2 + 2(5 − 𝑥2) − 8(𝑥2 + 9) and classify it as a monomial, binomial, or trinomial. 100. Write an algebraic expression, and simplify if possible a. twice the sum of a number and 11

b. The product of 7 and the square of 𝑥, increased by the difference of 5 and 𝑥2 101. Simplify and determine if the expression is quadratic.

(2𝑥2 + 9)(𝑥 − 2)

Without your calculator state whether the graph opens up or down and find the y-intercept. Then use your calculator to list the vertex and state whether it is a maximum or minimum, give the x-intercepts and state where the graph is increasing or decreasing.

103*. 𝑦 = −𝑥2 + 10𝑥 − 9

104*. 𝑦 = 1

2𝑥2 + 2𝑥

Without the calculator state whether the graph has a maximum or minimum and find the y-intercept. Then use your calculator to find the max or min and the x-intercepts. 105*. The path of a snowboarder performing stunts is given by the equation

𝑦 = −16𝑡2 + 24𝑡 + 16, where t is the time in seconds and y is the duration of the stunt.

106*. Find the domain of 𝑦 = −𝑥2 + 7𝑥 + 1

107*. A bird is descending toward a lake to catch a fish. The bird’s flight can be modeled by the

equation ℎ(𝑡) = 𝑡2 − 14𝑡 + 40, where h(t) is the bird’s height above the water in feet and t is the time in seconds since you saw the bird. What is the vertex? What does the minimum value mean in the context of the problem?

108*. The path of a snowboarder performing stunts is given by the equation 𝑓(𝑡) = −16(𝑡 − 2)(𝑡 + 1) Where t is time in seconds and y is the duration of the stunt. What are the t-intercepts? Explain the meaning of the t-intercepts in the context of the problem. How long does the stunt last?

109*. The senior class is putting on a talent show to raise money for their senior trip. In the past, the profit from the talent show could be modeled by the function

𝑃(𝑥) = −16𝑥2 + 600 𝑥 − 4000, where 𝑥 represents the ticket price in dollars. What is the reasonable domain for this function? For what domain value will the profits be maximized?

Solve by factoring:

110. 4𝑦2 = 25 111. 3𝑥2 + 15𝑥 + 12 = 0

112. 𝑦2 − 5𝑦 = 0 113. 2𝑥2 + 5𝑥 + 3 = 0

114. Write a quadratic equation that has the zeros x = 4

3,−1

6

115. The altitude of a triangle is 3 inches longer than its base. The area of the triangle is 20 square inches. Find the length of the base of the triangle. Solve:

116. 𝑥2 = 81 117. 𝑥2 + 5 = −3

118. 8(𝑥 − 5)2 = 56 119. (𝑥 + 3)2 + 7 = −2

120. The surface area of a cube with sides of length 𝑎 is given by 6𝑎2. If the surface area of a cube is 200 square inches, what is the length of one side of the cube?

Find 𝒄 so that the expression is a perfect square trinomial.

121. 𝑥2 + 15𝑥 + 𝑐 Solve by completing the square.

122. 𝑥2 − 8𝑥 + 11 = 0

123. 𝑥2 + 12𝑥 − 13 = 0

124. 2𝑏2 − 5𝑏 − 6 = 0

Solve using the quadratic formula.

125. 5𝑥2 + 7𝑥 = −3

126. 2𝑥2 + 7𝑥 − 11 = 0

127. 3𝑥2 + 5𝑥 − 2 = 0

128. Identify the vertex and state whether the function has a minimum or maximum.

𝑦 = −2(𝑥 + 3)2 − 3

129. Find the equation of a quadratic in vertex form with a maximum at (2, 10) and it passes through the point (1, 8)

Convert each equation to vertex form and list the vertex.

130. 𝑓(𝑥) = 𝑥2 − 2𝑥 − 2

131. 𝑓(𝑥) = −2𝑥2 + 6𝑥 − 10

list the vertex, direction of opening, x-intercepts, y-intercept and draw the graph. On the graph label the vertex, intercepts, and another point symmetrical to the y-intercept. 133. 𝑓(𝑥) = (𝑥 − 3)(𝑥 − 4)

Solve the quadratic inequalities. All problems must include number lines.

134. 𝑥2 − 25 > 0

135. 2𝑥2 − 𝑥 − 1 ≥ 0 CALCULATOR PROBLEMS (Round to two decimals.):

136*. 𝑥2 + 4 > 0

138. Tell what changes are made to the graph of 𝑓(𝑥) = 𝑥2 to obtain the graph ℎ(𝑥) = (𝑥 + 2)3 − 4.

139. Tell what changes are made to the graph of 𝑓(𝑥) = |𝑥| to obtain the graph 𝑦 = 3|−𝑥 + 5| − 2.

140. Let 𝒇(𝒙) = 𝒙𝟐. Write a new function that translates 𝒇(𝒙) as described:

Vertical compression by 1

4 , reflect x-axis and down 2 units

x y

–2 –1

–1.5 –1.75

–1 –4

–0.5 –7.75

0 –13

0.5 –19.75

1 28

1.5 –37.75

143. Calculate the average rate of change for ℎ(𝑥) = 𝑥2 − 4𝑥 + 6 between 𝑥 = −1 𝑎𝑛𝑑 𝑥 = 1

144. Calculate the average rate of change using the table below between 𝑥 = −1 𝑎𝑛𝑑 𝑥 = 1

Find the inverse of each function. Restrict the domain of the function if needed.

147. 𝑓(𝑥) = −𝑥2

𝟏𝟒𝟖. 𝑓(𝑥) = −𝑥 + 3

149. 𝑓(𝑥) =1

4𝑥 − 4

150. 𝑓(𝑥) = −10𝑥 + 40

State a reasonable domain for the function. Then find the inverse function 𝑓−1(𝑥).

151. A rock band is selling tickets to a concert at a theater. The band earns money for each ticket

sold, but has to pay some of the earnings to the theater. The total money earned by the band can be

estimated using the function 𝑓(𝑥) = 7.5𝑥 − 300, where 𝑥 is the number of tickets sold.

Graph each of the following piecewise functions

154. 𝑦 = {𝑥2 + 3, 𝑖𝑓 0 ≤ 𝑥 ≤ 3

20, 𝑖𝑓 𝑥 > 3 155. 𝑦 = {

−𝑥 − 6, 𝑖𝑓 𝑥 ≤ 10𝑥 − 26, 𝑖𝑓 𝑥 > 10

156. Estimate the solution to the system that is represented by the circle shown and the line with the given equation by graphing.

2𝑥 + 𝑦 = 14

158. Solve the system algebraically.

{𝑦 = −𝑥2 + 4𝑥 + 6𝑦 = −2𝑥 + 11

159. Solve the system algebraically.

{𝑦 = −𝑥

𝑥2 + 𝑦2 = 32

162*. A company makes candles in the shape of cones. Their best-selling candle has a height of 6 inches and a circumference of 12 inches. What volume of wax is needed to make 1 candle?

163*. A pyramid in Giza, Egypt has a square base with side lengths of 230 meters. Its height is 146.5 meters. What is its volume?

165*. A car loses value each year. The value of the car 𝑡 years from today can be modeled using the

function 𝑓(𝑡) = 15,000(0.85)𝑡. If Elizabeth wants to sell her car in 21

3 years, what will the car’s

value be when she sells it?

166*. Mia is tracking her savings account balance. She knows the equation 𝑦 = 8000𝑝𝑡 can be used to find her balance 𝑦 in any year 𝑡, but she can’t remember what 𝑝 represents. Her balance

today, 32

3 years after opening her account, is $9,905.54. What is the value of 𝑝?

Problems 167 and 168: Find the percent rate of change of f(t) for each unit of t. State whether the

function shows exponential growth or decay.

167. 𝑓(𝑡) = 1.08 (1.07)𝑡

168*. 𝑓(𝑡) = 63(0.87)11𝑡

171. Which function has a greater y-intercept?

𝑚(𝑥) = 10𝑥 − 6 ; 𝑛(𝑥) = (𝑥 − 4)(𝑥 + 2)

172. You are considering investing $100 into a fund. The first fund 𝑓(𝑥) will pay $1,000 each year. The second fund is modeled by the equation 𝑔(𝑥) = 10𝑥 + 100. The third fund is represented by the table.

x (year) 0 1 2 3

h(x) ($) 100 600 2,100 4,600

Which fund would you choose if you withdraw your money after 3 years?

Solve the following problems. Be sure to show how you set them up!

173*. Ashley has 7 places she wants to visit on her trip, but she only has one day free. If she can visit 1

place in the morning, 1 in the afternoon, and 1 in the evening, how many different ways can she plan

her trip?

174*. A pizzeria is offering a special on a large two-topping pizza. They offer 5 vegetable toppings and

4 meat toppings. How many different ways can the two-topping pizza be ordered, if each topping is

different?

179. The following Venn diagram shows a relationship between favorite sport and gender.

What is the probability that a randomly chosen person will choose soccer as their favorite sport?

181. Consuela is playing a card game with a standard 52-card deck. She wants a king or a diamond on

her first draw. What is the probability that she will get a king or a diamond on the first draw?

185*. Jane rolls a pair of dice. What is the probability that the sum is even if the product is even?

What is the probability that the product is even given that the sum is even? Compare the probabilities

and interpret your answer in the context of the problem.

188. Given the Tree diagram below, what is the probability that a person will test positive for TB

(tuberculosis) if they don’t have it? (They get a false positive.)

189*. For a statistics project, Tamara surveys a well-chosen sample that represents all the students at

her school. She finds that 72% have at least one sibling (brother or sister) and 27% have at least one

sibling and at least one pet in their home. Assume that having a sibling and having a pet are

independent events. Based on the survey, what is the probability that a randomly chosen student at

Tamara’s school has at least one pet at home?

193. Find the values of the six trig functions of 𝜃 for the right triangle:

194. Given the value of 1 trig function, find the values of the other 5 trig functions of 𝜃.

cot 𝜃 = 2

195. Find the complement of angle x = 41°

196. cos 25° ≈ 0.906 Find the sine of the complementary angle.

197. Find a value of 𝜃 for which cos 𝜃 = sin59° is true.

198*. Find the value of x. Round to the nearest degree: tan 𝑥 = 0.5923

202*. One section of a ski run is 650 feet long and falls 260 feet in elevation at a constant slope. To the

nearest degree, what angle (θ) does the ski run form with the horizontal?

205*. A blimp provides aerial footage of a football game at an altitude of 400 meters. The TV crew

estimates the distance of their line of sight to the stadium to be 3282.2 meters. What is the television

crew’s angle of depression from inside the blimp?

Simplify:

209. csc𝑥

cot𝑥

210. 𝑐𝑜𝑠𝜃𝑠𝑖𝑛2𝜃 − 𝑐𝑜𝑠𝜃

211. (sec 𝜃 − 1)(sec𝜃 + 1)

212. Change 135° to radian measure. 213. Convert 5𝜋

6 radians to degrees.

217. A circle has a radius of 11 units. Find the length of an arc intercepted by a central angle measuring 72°

218. A circle has a radius of 2 units. Find the arc length of a sector with an area of 12 square units.

219. Find the values of x and y.

222. Find 𝑚𝐵�̂� and 𝑚𝐶�̂�.

223. Given that all circles are similar, determine the scale factor necessary to map ⊙ 𝐶 →⊙ 𝐷.

⊙ 𝐶 has a diameter of 50 units and ⊙ 𝐷 has a diameter of 12 units.

224. What is the value of w?

228. The circumference of the trunk of a tree to be decorated is 12 inches. You have 7 inches of garland to wrap partially around the tree trunk. What is the arc angle of the trunk that you will decorate?

229. The slope of radius 𝑃𝑄̅̅ ̅̅ in circle Q is −𝟐

𝟑 . A student wants to draw a tangent to circle Q at point P.

What will be the slope of this tangent line?

231. The sides of quadrilateral PQRS are tangent to the circle at the points as pictured below. What is the length of 𝑄𝑅̅̅ ̅̅ ?

233. 𝐴𝐵̅̅ ̅̅ and 𝐴𝐶̅̅ ̅̅ are tangent to ⊙ 𝐿 in the diagram below. What is the value of x?

235. Find the length of 𝐵𝐼̅̅ ̅. Assume that I is the incenter.

236. Construct the circumscribed circle for the triangle.

237. The producers of a cooking show on the Snackers Network are designing a new set. A food

preparation station needs to be located between the refrigerator, sink, and stove. Which point of

concurrency, the circumcenter or the incenter, will result in the preparation station being located in a

place that is equidistant from the refrigerator, sink, and stove?