Secondary 3 Honors Final Review Simplify each expression. 1. 2. 3. 4. 5. Edith has a rectangular picture frame. The length of the frame is inches, and the width of the frame is 5 inches shorter than the length. Write an expression for the perimeter of the picture frame (in simplified form). 6. What is the degree of the polynomial function that generates the data shown in the table? 7. Describe the end behavior of . x y 5 10 6 17.28 7 27.44 8 40.96 9 58.32
Transcript
Secondary 3 Honors Final Review
Simplify each expression. 1. 2. 3. 4. 5. Edith has a rectangular picture frame. The length of the frame is inches, and the width of the frame is 5 inches shorter than the length. Write an expression for the perimeter of the picture frame (in simplified form). 6. What is the degree of the polynomial function that generates the data shown in the table?
7. Describe the end behavior of .
x y
5 10
6 17.28
7 27.44
8 40.96
9 58.32
8. What is the maximum number of turning points for the graph of ?
9. Classify by degree and number of terms. Simplify first if necessary.
10. Write in standard form.
11. Write the function in factored form: 12. Write a polynomial function in standard form with zeros -1, 1, and 0. 13. What are the zeros of the function? State their multiplicities. 14. Write a polynomial function in standard form that has 3 and -5 as zeros of multiplicity 2. 15*. Use your calculator to find the relative maximum, relative minimum, and zeros of the function . Solve each equation. 16.
17. 18. 19*. Graph the function to find the zeros. Rewrite the function with the polynomial in factored form.
20*. The width of a box is 2 m less than the length. The height is 1 m less than the length. The volume is 60 m2. What is the length of the box? (Show your equations) 21. Use synthetic division and the remainder theorem to find .
22. Divide using long division.
23. Divide using synthetic division.
24. Determine whether is a factor of
25. The expression
represents the volume of a square pyramid (the formula for
volume of a pyramid is
). The expression represents the height of the pyramid. What
expression represents the side length of the base? (Hint: where is the side length of the base.)
26. Use the Rational Root Theorem to list all possible rational roots:
27*. Write a polynomial function P(x) with rational coefficients so that P(x) has the given roots.
x =
28. Factor:
29. Give the number of roots:
Find all the zeros for each function:
30*.
31*.
32*.
33. Graph:
Determine which type of model best fits each set of points:
34. (0,3) (3,4) (5,6)
35. (2,3) (4,2) (6,4) (8,5)
36*. Find the cubic and quartic model for the set of values. Explain why one models the data better:
37*. Find the polynomial function whose graph passes through the points
(-3,-50) (-2, -4) (-1, 10) (0,7) (2,-23)
38. Determine the cubic function that is obtained from the parent function y = after
the sequence of transformations:
A horizontal translation 7 units right, a vertical stretch by a factor of 4, and a vertical translation 2 up.
39. Find all the real zeros of the function: + 1
Graph:
40. 41.
42. Write the equation:
43. What is the simplified form of
? State any excluded values.
44. What is the simplified form of
? State any excluded values.
45. Multiply
46. Divide
47. Simplify
48.
49.
50.
51.
52. Is the relationship direct or inverse variation? Write an equation to model the variation.
u v
-3 -15
5 25
6 30
16 80
53. z varies directly with x and inversely with y. When x = 6 and y = 2, z = 15. Write a function to model the
variation and find z when x = 4 and y = 9.
54*. Find the points of intersection by graphing.
55. Find any points of discontinuity for
56. Find the vertical asymptotes of
57. List the asymptotes and intercepts and sketch the graph of
58. List the asymptotes and intercepts and sketch the graph of
59. Solve
for .
60.
61.
62.
63. Solve the system:
64. Solve the system:
65*. Solve the inequality using your calculator:
66*. Solve the inequality using your calculator:
Simplify each radical expression. Use absolute value symbols when needed.
67. 68.
69. Find the two real solutions of the equation: Simplify each expression. Rationalize all denominators.
70.
71.
72.
73.
Simplify:
74. 75.
Rationalize the denominator and simplify:
76.
Simplify each expression:
77.
78.
79.
80.
Solve. Check for extraneous solutions.
81. 82.
83.
84.
85.
Graph each function. Find the domain and range.
86.
87.
88. Rewrite to make it easy to graph using transformations. Describe the graph.
Solve the following radical equations.
88*.
89*. You can model the time , in seconds, an object takes to reach the ground falling from height , in
meters, by
. The value of is 9.81 m/s2. If an object takes 7 seconds to fall to the ground, what
was its initial height?
Let and . Perform each function operation. 90. 91. 92.
Let and
.
93. Find .
94. Find
95. Let and Find and
96. Find the inverse. Determine if the inverse is a function.
Determine if the inverse is a function. If it is, find the inverse and give the domain and range of the function and the inverse.
97.
98.
99. Use compositions to verify that and
are inverses.
100. Determine whether the function represents exponential growth or decay and find the y-intercept. a. b. 101. The population of a certain animal species decreases at a rate of 3.5% per year. You have counted 80 of the animals in the habitat you are studying. Write a function that models the change in the
animal population. 102. For each annual rate of change, find the corresponding growth or decay factor. a. 500% b. -55% c. -.1% 103*. The value of an industrial machine has a decay factor of .75 per year. After six years, the machine is worth $7500. What was the original value of the machine? Round to the nearest dollar. For each function, identify the transformation and draw the graph. 104. 105. 106*. How much money will you have if you invest $400 at 7.6% compounded continuously, for 1.5 years. Round to the nearest cent.
107*. A student wants to save $8000 for college in five years. How much should be put into an account that pays 5.2% annual interest compounded continuously? 108*. At a constant temperature, the atmospheric pressure p, in pascals, is given by the formula
where h is the altitude in meters. What is p at an altitude of 500 m?
109. Write in logarithmic form:
110. Write in exponential form:
111. Evaluate:
a. b.
112. Find the inverse: 113. Graph the function: 114. Find the domain and range of:
115. Evaluate:
116. Write as a single logarithm:
117. Expand:
118. Expand:
119. Write as a quotient of 2 common logs: Solve. No Calculator 120. 121.
122.
Solve: Use your calculator at the end and round to 4 decimals 123*. 124*.
125. Simplify:
126. Write as a single natural log: May use Calculator at end of problem. Round to 2 decimals 127*. Solve: 128*. Solve:
129*. How long would it take to double your principal in an account that pays 6.5% annual interest compounded continuously. Solve each inequality, round to 3 decimals 130*. 4 131*. 132*. 133*.
134. Sketch the graph of a sound wave with the given period, amplitude, and midline: period 0.02, amplitude
4, midline 6.
135. Find the maximum, minimum, and period of the periodic function.
136. Suppose g is a periodic function. The period of g is 24, g(3)=67, and g(8)=70. Find the following function
values.
a) g(27) b) g(80)
c) g(-16) d) g(51)
137. Find the measure of the angle in standard position.
138. Determine the quadrant or axis where the terminal side of an angle measuring lies.
139*. Using your calculator, find the values of cosine and sine for the angle measuring . Round your
answers to the nearest hundredth.
Find a positive and a negative coterminal angle for the given angles.
140. 141.
142. Given the angle measuring , sketch the angle in standard position, then find the exact values of
the cosine and sine of that angle.
143. Find the radian measure of an angle of
144. Find the degree measure of an angle of
radians.
Determine the quadrant or axis where the terminal side of each angle lies, then find one positive and one
negative coterminal angle.
145.
radians 146.
radians
147. Draw the angle measuring
radians in standard position, then find the exact values of the cosine
and the sine of the angle.
148*. Suppose the radius of a bicycle wheel is 13 in. (measured to the outside of the tire). Find the number of
radians through which a point on the tire turns when the bicycle has moved forward a distance of 12
feet.
Determine the number of cycles each sine function has in the interval from 0 to . Find the amplitude and
period of each function.
149.
150.
151. Find the period of the function:
Find the period and amplitude of each sine function. Then sketch 1 period of the function. (label correctly) 152.
153. y = 0.5
154. Write a cosine function if the amplitude is 3, and the period is . Assume a>0.
Identify the period, range and amplitude of the function.
155.
156.
157. Give the amplitude and period and sketch 1 period of
158*. Solve. Round to the nearest hundredth:
a. –
b.
159. Find the
160. Identify the period and graph 2 periods of the function:
161. Write an equation of a tangent for the graph:
162*. Solve in the interval 0 to and round your answer to 2 decimals
163. Graph
164. What is an equation that models
Draw the graph
165. Write an equation for the translation:
166. Describe any phase shift or vertical shift in the graph
167. Write Find the amplitude, midline, and period. Describe any translations.
168. Find the value for
169. Graph 1 period of
170. Graph 2 periods of
171. Graph at least 1 period of
Find the exact value:
172.
173.
174.
175.
176. arcos(tan(
) 177.
178. Find an algebraic expression for:
Simplify: 179.
180.
Verify each identity: 181.
182.
183. Use the identities: Given that = -.6 and is in the third quadrant, what are (use fractions)
184. Find the exact values of , , and using the double angle formulas.
Rewrite the expression using double angles.
185. 186.
187.
188. Verify the identity.
Solve the equations and give all solutions.
189.
190.
Solve the equations on the interval .
191.
192.
193.
194*. The sides of a triangle are 15 in., 17 in., and 16 in. The smallest angle has a measure of 54 .
Find the measure of the largest angle. Round to the nearest degree.
195*. Find the remaining sides and angles of Round to 1 decimal.
m = 54 , e = 8 and d = 10
196*. Find the area of . Round to 1 decimal. m
197*. In Round to 1 decimal.
198*. In Round to 1 decimal.
Determine whether each formula is explicit or recursive and find the first five terms of the sequence. 199.
200.
Find the next two terms in each sequence. Write an explicit formula for the nth term of the
sequence.
201.
202.
203. Find the 17th term of the sequence:
204. Find the arithmetic mean of the given terms:
205. Write an explicit formula for the sequence:
206. Write the series in summation notation:
207. Evaluate the related series for the following sequence:
208. Write an explicit formula for the sequence and give the first five terms:
Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms.
209.
210.
211*. Find the missing terms of the geometric sequence:
212*. Determine whether the series is arithmetic or geometric. Then evaluate the series for the specified number of terms:
Evaluate each infinite series that has a sum.
213.
214.
215. Find the common ratio, r, for the specified infinite geometric series:
216. Find the third term of
Expand each binomial.
217.
218*.
219.
220. A magazine publisher mails a survey to every tenth person on a subscriber list. The survey asks how
much time you spend online each week. What sampling method is the survey using? Identify any bias in the
sampling method.
221. Student government members survey every tenth student who enters the school building and asks
whether the students favor the school’s new dress code. The sample statistics show that 54% favor the new
dress code, 42% oppose the dress code, and 4% have no opinion.
a) What is the population?
b) What is the sample?
c) What general conclusion can be made about the population?
222. For a class project, a student studies the likelihood that students turn in their homework each day. For
each of her classes, she observes the teacher collect homework. She records the number of students who turn
in homework, and the number who do not. The resulting data show that 86% of students turned in homework
on time, and 5% of students did not turn in any homework at all during the week.
a) What type of sampling method was used?
b) What type of study was performed?
c) Can the student use these statistics to make a general conclusion about all students in her
school? Explain.
223*. Identify the outlier in the set, then find the mean, median, and mode when the outlier is included and
a) Find the mean and the standard deviation of the data. (Label them correctly.)
b) How many values in the data set fall within one standard deviation from the mean? Within two
standard deviations? Within three standard deviation? Show how you determined the answer.
227. Draw a curve to represent a normally distributed experiment that has a mean of 180 and a standard
deviation of 15. Label the x-axis and indicate the probabilities.
228. The scores on an exam are normally distributed with a mean of 85 and a standard deviation of 5. What
percent of the scores are between 85 and 95?
229. To win a prize at the county fair, the diameter of a tomato must be greater than 4 in. The diameters of a
crop of tomatoes grown in a special soil are normally distributed, with a mean of 3.2 in. and a standard deviation
of 0.4 in. What is the probability that a tomato grown in the special soil will be a winner?
230. A normal distribution has a mean of 100 and a standard deviation of 10. Find the probability that a
value selected at random is in the interval is at most 110.
231. The numbers of paper clips per box in a truckload of boxes are normally distributed with a mean of 100
and a standard deviation of 5. Find the probability that a box will not contain between 95 and 105 paper clips.
232*. In a simple random sample of 550 people, 326 reported using social networking sites on the internet.
Find the sample proportion and the margin of error (assuming 95% confidence).
233*. Find the 95% confidence interval of the population mean if .
234*. Tonya wants to estimate what proportion of her school’s seniors plan to attend the prom. She
interviews an SRS of 50 of the 750 seniors in her school and finds that 36 plan to go to prom. Calculate and
interpret a 95% confidence interval for the school proportion p that will attend prom.
235*. In 2012, the mean number of wins for teams in Major League Baseball was 81 with a standard deviation
of 11.9 wins. Find and interpret the z-score for the New York Mets with 74 wins.
236*. George has an average bowling score of 180, and bowls in a league where the average for all bowlers is
150 and the standard deviation is 20. Bill has an average bowling score of 190 and bowls in a league where the
average is 160, and the standard deviation is 15. Who ranks higher in his own league, George or Bill? Justify
your answer.
237*. The state of Connecticut has an area of 4,845 square miles. In 2011, Connecticut had a population of
3,580,709. What was the population density of Connecticut in 2011?
238*. A designer wants to manufacture storage bins in the shape of rectangular prisms. Each bin will have a
volume of 9 cubic feet. The designer wants to choose the dimensions of a bin so that the bases of the prism are
squares and the bin’s surface area is minimized. What dimensions should the designer choose for the bins?
Show all work.
239*. The table shows the mass and volume of different metals. Find the density of each.
a) Aluminum
b) Cesium
c) Gold
d) Lead
240*. A florist recommends that you should grow 4 Flower A plants per square foot, and 9 Flower B plants per
square foot. If you want to grow 50 plants each of Flower A and Flower B, how many more square feet of
garden space do you need for the Flower A plants than for the Flower B plants?
Substance Mass (g) Volume (cm3)
Aluminum 135 50
Cesium 308.8 160
Gold 482.5 25
Lead 339 30
241*. The areas of two similar rhombuses are and . What is the ratio of their perimeters?
The scale factor of two similar polygons is given. Find the ratio of their perimeters and the ratio of their areas.
242*.
243*.
244. Find the values of x and y when the smaller triangle has an area of .
245*. Meteorites (mostly dust-particle size) are continually bombarding Earth. The surface area of Earth is about 65.7 million . The area of the United States is about 3.7 million . What is the probability that a meteorite landing on Earth will land in the United States?
246*. Find the probability that a coordinate of a point chosen at random on satisfies the inequality:
247*. Assume that a dart you throw will land on the 12 in.-by-12 in. square dartboard and is equally likely
to land at any point on the board. The diameter of the center circle is 2 in., and the width of each
ring is 1 in.
248*. For the polyhedron, use Euler’s Formula to find the missing number:
edges = 15, vertices = 9, faces = _______
249*. A cube has a net with area How long is an edge of the cube?
250*. The volumes of two similar containers are The surface area of the smaller container
is . What is the surface area of the larger container?
251. Two similar prisms have heights 4 cm and 10 cm.
a. What is their scale factor
b. What is the ratio of the surface areas?
c. What is the ratio of their volumes?
252*. The volume of a spherical balloon with radius 3.1 cm is about 125 . Estimate the volume of a similar
balloon with radius 6.2 cm.
253*. The volumes of two similar pyramids are 1331 and
a. Find the ratio of their heights.
b. Find the ratio of their surface areas.
254*. Square pyramids A and B are similar. In pyramid A, each base edge is 12 cm. In pyramid B, each base
edge is 3 cm, and the volume is 6
a. Find the volume of pyramid A.
b. Find the ratio of the surface area of A to the surface are of B
255*. A clown’s face on a balloon is 4 in. tall when the balloon holds 108 of air. How much air must the
