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File: Algebra2B 2011 Questions 8.12.11
6.
Write the equation for the conic function below.
Updated: August 12, 2011
Section 5.00
1.
What are the four conic sections shapes generated
from the intersection of a plane and a cone?
Unless specific instructions are given below,
perform the following conic operations:
First state type of conic function, orientation and
provide a sketch of the conic function and put
equation in standard form.
If conic is a Parabola find:
a) Vertex (h,k) =
b) p value and direction of opening
c) Focus point (F = )
d) Equation of Directrix
If conic is a Circle find:
a) Center (h,k) =
b) Radius (r = )
If conic is a Ellipse find:
a) Center (h,k) =
b) Vertices (V = ); as well as (U = )
c) Length of Major
d) Length of Minor axis
e) Foci ordered pairs (F = )
If conic is a Hyperbola find:
a) Center (h,k) =
b) Vertices (V = ); as well as (U = )
c) Length of Transverse axes
d) Length of Conjugate axis
e) Foci ordered pairs (F = )
f) Asmptotes:
7.
2.
Explain how the formula of a circle is derived. Use
an example to help illustrate your explanation.
11.
3.
(x + 5)2 + (y – 3 )2 = 49
i)
ii)
iii)
iv)
8.
9.
10.
12.
4. Complete the following table for circles with the
given center and radius
Center Radius
(h,k)
(r)
Circle Equation
a) (-5, 2)
6
b) (6, -3)
5
c) (4, -1)
8
d) (-5, 6)
4
13.
14.
15.
5. Complete the following table for circles with the
given equations.
a)
b)
c)
d)
2
(x + 5)
(x – 7)2
(x – 2)2
(x + 8)2
Equation
+ (y + 3)2 = 49
+ (y – 2)2 = 81
+ (y + 1)2 = 100
+ (y – 4)2 = 4
Center
(h, k)
16.
Radius
(r)
Below are the equations of four conic sections.
Indicate the conic type for each equation and sketch
the parameters that are easily obtained.
a)
b)
c)
d)
17.
Write the equation of a circle with center (0, 3) and
a radius of 3.
18.
Write the equation of a circle with center (-1, 2) and
a radius of 4.
19.
(x – 4)2 + y2 = 16
20.
(x – 1)2 + (y + 2)2 = 25
21.
Determine which of the four conic equation is/are
functions.
22.
Write the equation for the conic function below.
26.
Write the equation for the conic function below.
Section 5.07
\
23.
27.
(x – 5)2 + (y + 3)2 = 16
28.
Write the equation for the conic function below.
(Each square is 1 x 1)
Write the equation for the conic function below.
29.
30.
24.
25.
Write the equation of the ellipse with foci (0, +3)
and y-intercepts +5.
Write an equation of a circle with a diameter of 20
inches and a center at the point (4, -5)
Section 5.10
Write the equation for the conic function below.
31.
32.
33.
(x – 3)2 + (y + 2)2 = 25
y = 2x2
34.
Write the equation for the conic function below.
39.
Given the conic equation: y = -2(x – 3)2 – 4
Determine:
a) Conic type:
b) If any horizontal or vertical shift
c) If any reflection
d) If any vertical shrink or stretch
e) The vertex
40.
x2 = - 16y
41.
y = ( 1/12) x2
42.
43.
35.
Write the equation for the conic function below.
44.
Below are the equations of four conic sections.
Indicate the conic type for each equation and sketch
the parameters.
a)
b)
c)
d)
45.
36.
37.
38.
46.
y = 4(x + 3)2 + 5
a) Conic type:
b) If any horizontal or vertical shift
c) If any reflection
d) If any vertical shrink or stretch
e) The vertex
f) The focus and directrix (sketch)
47.
Where do the asymptotes of a hyperbola intersect
the hyperbola? How is the hyperbola intersections
defined?
48.
Sketch and label: (x + 2)2 + (y – 3)2 = 16
49.
Sketch and label: (x + 4)2 + (y – 1) = 25
50.
Write the equation for the conic function below.
Write the equation for the conic function below.
Write the equation for the conic function below.
51.
Sketch and label: (x – 5)2 + y2 = 16
52.
Write the equation for the conic function below.
53.
Write the equation of the ellipse with foci at ( +2, 0)
and x-intercepts at +5
54.
Write the equation of the parabola with focus at
(0 , 6) and directrix y = -6
55.
Write the equation for the conic function below.
56.
57.
58.
59.
Write the equation for the conic function below.
60.
Write an equation of a circle with a diameter of 10
inches and a center at the point (3, -6)
Write an equation of a circle with a diameter of 16
inches and a center at the point (-2, 1)
61.
62.
63.
64.
(x – 3)2 + (y + 1)2 = 5
Write the equation for the conic function below.
65.
Write the equation for the conic function below.
Write the equation for the conic function below.
66.
67.
(Harder Problem)
Algebra 2B: Unit 6
68. Find the least common denominator for the
following rational expressions:
80.
Simplify
69.
Find the least common denominator for the
following rational expressions:
81.
Simplify
70.
Find the least common denominator for the
following rational expressions:
82.
Simplify
83.
Simplify
84.
Simplify
85.
Simplify
86.
Simplify
87.
Simplify
88.
Simplify
89.
Simplify
90.
Simplify
71.
72.
73.
Simplify
Simplify
Simplify
74.
Simplify
75.
Simplify
76.
Simplify
77.
Simplify
78.
Simplify
79.
Simplify
91.
Simplify
92.
Simplify
93.
Simplify
103. Simplify
104. Simplify
105. Simplify
94.
Simplify
106. Simplify
95.
Simplify
107. Simplify
96.
Simplify
108. Simplify
97.
Simplify
109. Simplify
98.
Simplify
110. Simplify
99.
Simplify
100. Simplify
101. Simplify
102. Simplify
111. Simplify
112. How many solutions does each of the following
equations have?
2x3 + 3x2 – 18x – 27 = 0
3x2 + 4x – 6 = 0
6x4 + 3x3 + 2x + 4 = 0
____
____
____
113. Look at the factored form of P(x) and determine the
solutions for x?
P(x) = (x + 7) (x + 3) (x – 4)
114. Look at the factored form of P(x) and determine
the solutions for x?
126. Solve if possible and list any restrictions to the
domain of the variable.
P(x) = (x + 3) (x – 5) (x + 1) (x – 2)
115. Look at the factored form of P(x) and determine
the solutions for x?
P(x) = x(x + 1) (x – 7)
127. Solve if possible and list any restrictions to the
domain of the variable.
116. Look at the factored form of P(x) and determine
the solutions for x?
P(x) = (x – 2) (x – 3) (x – 4)
117. Look at the factored form of P(x) and determine
the solutions for x?
128. Solve if possible and list any restrictions to the
domain of the variable.
P(x) = (x + 2) (x + 3) (x + 4)
118. Look at the factored form of P(x) and determine
the solutions for x?
P(x) = -2(x – 2) (x – 3) (x – 4)
119. Look at the factored form of P(x) and determine
the solutions for x?
f(x) = (x – 4) (x + 2) (x – 1)
129. Solve if possible and list any restrictions to the
domain of the variable.
130. Solve if possible and list any restrictions to the
domain of the variable.
120. Solve if possible and list any restrictions to the
variable.
121. Solve if possible and list any restrictions to the
variable.
122. Solve if possible and list any restrictions to the
variable.
131. Solve if possible and list any restrictions to the
domain of the variable.
132. Solve if possible and list any restrictions to the
domain of the variable.
123. Solve if possible and list any restrictions to the
variable.
133. Solve if possible and list any restrictions to the
domain of the variable.
124. Solve if possible and list any restrictions to the
variable.
134. State whether the graph is of odd degree or even
degree; positive or negative; then state the number
of relative minima and relative maxima.
125. Solve if possible and list any restrictions to the
variable. (If necessary use quadratic equation to
find the exact solution)
135. Draw a graph representing all positive even degree
polynomials. Give two examples.
136. Draw a graph representing all negative even degree
polynomials. Give two examples.
137. Draw a graph representing all positive odd degree
polynomials. Give two examples.
138. Draw a graph representing all negative odd degree
polynomials. Give two equation examples.
139. Factor, then list all the solutions.
P(x) = x2 + x – 6
140. Factor, then list all the solutions.
P(x) = -x2 – x – 6
141. Factor, then list all the solutions.
P(x) = 2x2 + 2x – 12
142. Factor by grouping, then list all the solutions.
x3 – 2x2 – x + 2 = 0
143. Factor by grouping, then list all the solutions.
x3 – 6x2 – x + 6 = 0
144. Factor by grouping, then list all the solutions.
-x + 4x + x – 4 = 0
3
2
145. Find the remainder when dividing. Check your
answer by substitution.
-x3 + 2x – 6 = 0 ÷ by (x – 3)
146. Find the remainder when dividing. Check your
answer by substitution.
x3 – 4x + 1 = 0 ÷ by (x – 2)
147. Find the remainder when dividing. Check your
answer by substitution.
x3 + 2x + 1 = 0 by (x – 5)
148. Use Descartes’ Rule to analyze the possible SIGNS
of the zeros of the following function. Next use the
Rational Zeros Theorem to identify the possible
RATIONAL zeros.
P(x) = 2x3 + 5x2 – 3x – 4
149. Use Descartes’ Rule to analyze the possible SIGNS
of the zeros of the following function. Next use the
Rational Zeros Theorem to identify the possible
RATIONAL zeros.
P(x) = x + 3x – 5x – 35
3
2
150. Use Descartes’ Rule to analyze the possible SIGNS
of the zeros of the following function. Next use the
Rational Zeros Theorem to identify the possible
RATIONAL zeros.
P(x) = 2x3 - 4x2 + 5x – 14
151. Use Descartes’ Rule to analyze the possible SIGNS
of the zeros of the following function. Next use the
Rational Zeros Theorem to identify the possible
RATIONAL zeros.
f(x) = 2x3 + 3x2 – 5x – 15
152. Use Descartes’ Rule to analyze the possible SIGNS
of the zeros of the following function. Next use the
Rational Zeros Theorem to identify the possible
RATIONAL zeros.
P(x) = 3x2 + 4x – 6
153. Use Descartes’ Rule to analyze the possible SIGNS
of the zeros of the following function. Next use the
Rational Zeros Theorem to identify the possible
RATIONAL zeros.
f(x) = 3x4 – 11x3 + 10x – 4
154. Use Descartes’ Rule to analyze the possible SIGNS
of the zeros of the following function. Next use the
Rational Zeros Theorem to identify the possible
RATIONAL zeros.
P(x) = 6x4 + 3x3 + 2x + 4
155. Use Descartes’ Rule to analyze the possible SIGNS
of the zeros of the following function. Next use the
Rational Zeros Theorem to identify the possible
RATIONAL zeros.
P(x) = x5 – 3x2 + 1
156. Use Descartes’ Rule to analyze the possible SIGNS
of the zeros of the following function. Next use the
Rational Zeros Theorem to identify the possible
RATIONAL zeros.
P(x) = x7 + 37x5 – 6x2 + 12
157. Factor by grouping, then list all the solutions.
P(x) = x3 – 2x2 – 9x + 18
158. Factor by grouping, then list all the solutions. “i"
f(x) = x4 + 4x2 – 45
159. Factor by grouping, then list all the solutions.
f(x) = x3 + 3x2 – 2x – 6
160. Factor by grouping, then list all the solutions.
f(x) = x3 – x2 – 3x + 3
161. What are all the asymptotes in this expression?
162. What are all the asymptotes in this expression?
174. Factor the polynomial given one of its factors.
List all zeros and their multiplicities.
P(x) = x3 – 6x2 + 11x – 6; x – 3
163. What are all the asymptotes in this expression?
175. Factor the polynomial given one of its factors.
List all zeros and their multiplicities.
P(x) = 6x3 + 19x2 + 2x – 3; x + 3
164. Use the Remainder Theorem and synthetic
division to find: (a) f(-2) and (b) f(2) in the
following function. Check your answer using
substitution.
f(x) = 3x4 – x2 + 2x – 6
165. Use the Remainder Theorem and synthetic
division to find (a) f(-3) and (b) f(2) in the
following function. Check your answer using
substitution.
f(x) = 2x + x – 10x – 5
3
2
166. Use the Remainder Theorem and synthetic
division to find f(-4) and f(3) in the following
function. Check your answer using substitution.
176. Factor the polynomial given one of its factors.
List all zeros and their multiplicities.
x4 + 2x3 – 7x2 – 20x – 12; (x + 2) 2
177. List all zeros. (Use the Quadratic formula to
find other zeros)
x3 – 4x2 + 21x – 34 = 0; x – 2
178. Factor the polynomial given one of its factors.
List all zeros. ****
P(x) = x3 + x2 – 4x – 24; -2 + 2i is a zero
179. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
f(x) = -2x3 – 8x2 + 7
167. Use the Remainder Theorem and synthetic
division to find f(-6) and f(3) in the following
function. Check your answer using substitution.
180. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
f(x) = -x3 – 5x2 + 8
168. Factor the polynomial given one of its factors.
List all zeros.
x3 – 7x + 6 ; x – 2
169. Factor the polynomial given one of its factors.
List all zeros.
x + 6x – x – 30; x + 5
3
2
181. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
182. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
170. Factor the polynomial given two of its factors.
List all zeros.
x3 + 2x2 – x – 2; (x – 1),(x + 1)
171. Factor the polynomial given two of its factors. List
all zeros. (Use the Quadratic formula to find other
zeros)
x4 + 2x3 + 2x2 – 2x – 3; (x + 1), (x – 1)
183. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
184. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
172. One zero is given for the following polynomial.
Including the given list all zeros. (Use the Quadratic
formula to find other zeros)
P(x) = x3 + 2x2 – 3x + 20; x + 4
173. Factor the polynomial given one of its factors. List
all zeros and their multiplicities.
P(x) = x3 – 3x – 2; x + 1
185. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
186. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”). Then graph.
197. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
187. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
198. Solve if possible.
199. Solve if possible.
188. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
200. Solve if possible.
189. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”). ****
201. Given the graph of the following function
determine all asymptotes.
190. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
191. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
192. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
193. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
194. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
195. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
196. Simplify, if possible and determine any asymptotes
or points of discontinuity (“holes”).
202. Given the graph of the following function
determine all asymptotes.
203. Given the graph of the following function
determine all asymptotes.
204. Given the graph of the following function
determine all asymptotes.
205. Given the graph of the following function
determine all asymptotes.
206. Given the graph of the following function
determine all asymptotes.
Algebra 2B Unit 7 NO CALCULATOR SECTION
=================================
Rewrite with negative exponents:
207.
2y3
/x5
Use logarithmic properties to write the following
expressions in logarithmic form.
229.
25 = 32
230.
4-2 = 1/16
231.
34 = 81
53 = 125
208.
1
209.
3
232.
210.
2x3/y2
233.
23 = 8
=================================
Sketch a graph of the following exponential functions.
Indicate the y-intercept. Determine whether any of
these equations are exponential growth functions or
decay functions?
234.
y = 3x
/x4
211.
2ab3c5/d2
=================================
Rewrite with positive exponents.
212.
5/3x -3
213.
2x -2/y -3
214.
215.
235.
y = -3x
2a-4b-3
236.
y = (2/3)x
5x2 y -3
237.
y = -(2/3)x
238.
y = (1/2)x
239.
y = 35x
240.
y = (1/4)x
241.
y = (1/2)2x
216.
3x2y -3 /7z -5
=================================
Write the following expressions in reduced
simplified, radical form.
217.
x -3/4
2/3
218.
16
219.
5 2/3
220.
x2/5
221.
72/3
222.
x 3/4
223.
x 2/3
=================================
Write the following expressions in exponential form.
224.
225.
242.
y = 2x
=================================
First state the inverse function of each of the following.
Next sketch and label a graph of both the function and
inverse.
243.
y = log 4 x
244.
y = log2 x
=================================
What is the domain and range of the following?
245.
g(x) = 3x
246.
g(x) = 5x
=================================
247.
Make a table of the following functions then
explain how they relate to each other.
y = 5x and y = log5 x
=================================
226.
227.
228.
=================================
Which of the following is NOT an exponential function?
What type of function is it?
248.
f(x) = x3
f(x) = 3x
2x
f(x) = 5
f(x) = (3/4)(x+1)
249.
f(x) = 4x
f(x) = x4
f(x) = 34x
f(x) = (1/2)(x+2)
250.
f(x) = 5x
f(x) = 54x
f(x) = x5
f(x) = (1/2)(x+2)
=================================
State the base of the following:
251.
y = log 3 x
252.
y = log x
253.
y = Ln x
=================================
Solve for x or simplify.
254.
Solve: 2 x = 8
255.
Solve: 2 x = 16
256.
Solve: 3 x = 27
257.
Solve: 2 2x = 4 2x + 4
258.
Solve: 5 2x = 25 3x – 4
Write the following expressions in exponential form.
287.
log 8 4 = 2/3
288.
log 5 125 = 3
=================================
Solve for x in each of the following using
logarithmic properties.
289.
Solve: log x 9 = 2
290.
Solve: log2 32 = x
291.
Solve: log4 x = 3
292.
Solve: log x 216 = 3
293.
294.
Solve: log5 x = 2
Solve: log6 x = 2
295.
296.
Solve: log10 (x2 – 3) = log10 6
Simplify: log3 (1/81)
297.
Simplify: log2 128
298.
Solve: log x 125 = 3
299.
Solve: log 4 x = 3
300.
Solve: log 7 (1/49) = x
259.
260.
261.
262.
263.
Solve: 3 2x = 9 2x + 4
Solve: 3 4x = 3 5 – x
Solve: 5 x - 3 = 1/25
Solve: 9 x = 81x + 4
Solve: 8 4 – 2x = 4x + 2
264.
265.
Solve: 3x – 2 =
Solve: 2 x = 4 5/2
266.
267.
268.
Solve: 3 3x = 813x – 4
Solve: 8 = 2 3x+1
Solve: 5 x = 1/125
301.
Solve: log x (1/64) = -3
302.
Solve: log 4 x = -2
303.
Solve: log 9 27 = x
269.
270.
Solve: 16 x = 8 x + 1
Solve: 2 2x = 1/16
304.
Solve: logx 9 = -2
305.
Solve: logx 8 = -3
271.
Simplify: 16 -3/4
306.
Solve: logx 625 = -4
272.
Solve: 16 = 2 3x + 1
307.
Solve: log 9 x = 3/2
273.
Simplify: 3 /33
308.
Solve: log 7 (x2 – 6) = log 7 x
274.
Simplify: (8 -1/ 3)2
275.
Simplify: (4 )
276.
Simplify: (22 ) (23 )
277.
Simplify: 32 4/5
278.
Simplify: 27 4/3
311.
Solve: 3log5 x – log5 4 = log5 16
279.
Simplify: 16 -1/4
312.
Solve: 2log3 x – log3 2 = log3 x
280.
Solve: x 2 = 25
313.
Solve: 2log2 x + log21 = log2 4
281.
Solve: 5x = 25
314.
Solve: 2log2 x – log21 = log2 9
315.
Solve: log3 (x+7) – log3 (x-1) = 2
316.
Solve: log2 (3x + 1) – log2 x = 2
-1
5x
(1 – x)
=4
309.
Solve: log 3 (3x – 5) = log3 (x + 7)
=================================
Solve for x for each of the following using
logarithmic properties.
310.
Solve: log3 2 + log3 7 = log3 x
x–3
282.
Solve: (2 ) 16
283.
Solve:
284.
Solve: 4x =
317.
Solve: log5 x + 3 = log5 (x – 20) + 4
285.
Solve: 64 = 23x + 1
318.
Solve: log2 x + 5 = 8 – log2 (x + 7)
286.
Simplify: (2-1) (43) (8-1)
319.
Solve: Ln x – Ln (x – 4) = Ln 2
320.
Simplify: log2 8 + log2 2
= 8x + 2
=================================
321.
Solve: x = 4(5/2)
322.
Solve: log2 (3x + 5) – log2 x = 3
323.
Solve: 2log3 6 – (1/4)log3 16 = log3 x
324.
Solve: log x + log (x + 3) = 1
325.
Simplify: log2 4 + log2 2
=================================
Use logarithmic properties to express the following as
a single logarithm whose coefficient is 1:
326.
log2 x + log2 5
347.
A bank account starts with $100 and has an
annual interest rate of 4%.
(a) Start a data table for this problem for the first
3 years.
(b) Determine the type of equation this is and then
write the equation (given any time limit).
(c) How much money will be in the account in 12
years?
327.
3log(x 2 – 4) + (1/3)log(x + 2) – 4log(x + 7)
328.
2log a – 3log b – 5log c
329.
2log a + log b - 3log c - log d
330.
3log5 a – (1/2)log5 b + log5 c
331.
(1/2)log3 4 + log3 5 – log3 x
348. In 1985, there were 285 cell phone subscribers in
the small town of Centerville. The number of
subscribers increased by 75% per year after 1985.
(a) Start a data table for this problem for the first
3 years.
(b) Determine the type of equation this is and then
write the equation (given any time limit).
(c) How many cell phone subscribers were in
Centerville in 1994?
332.
log3 x + 2log3 b – (1/2)log3 c
349.
333.
3x = 64
334.
5x = 12
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
A CALCULATOR CAN BE USED FOR
THE FOLLOWING SECTIONS
(a) Convert the following to: x = logarithmic form
(Do Not use the calculator up to this point).
(b) Using your calculator solve for x
(Round your answer to the thousandths place)
335.
2x = 3
336.
4x = 12
337.
5x = 15
338.
3x – 2 = 4
339.
6 x + 2 = 17.2
For the following use your calculator solve for x
(Round your answer to the thousandths place)
340.
53x = 8x – 1
341.
4x = 13x – 3
342.
2x – 7 x + 2 = 0
343.
3 2x = 13 x– 5
344.
7 2x + 1 = 3 3x – 2
345.
12x = 5 2x + 4
=================================
346.
Evaluate each of the following expressions:
Round answer to the thousandths place.
A. log3 54
B. log 49
C. ln 23
D. log5 22
E. log4 5 - log4 3
=================================
Each year the local country club sponsors a
tennis tournament. Play starts with 128
participants. During each round, half of the
players are eliminated. How many players
remain after 5 rounds?
(a) Complete a data table for this problem for the
first 3 rounds
(b) Determine the type of equation this is and then
write the equation (given any time limit).
(c) How many players will there be after 5 rounds?
350.
A teacher has a class of 100 students
simulate a half life of a radioactive
substance by having each student roll a die.
If they roll a 1 they must sit down. Each
time the teacher records the number of
students standing and records this.
(a) Complete a data table for this problem for the
first 2 rounds for those students still standing
(simulating that they are radioactive)
(b) Determine the type of equation this is and then
write the equation (given any time limit).
(c) How many students will be standing after 10
rounds?
351.
A species of bacteria doubles every ten
minutes.
(a) Starting out with only one bacterium, how many
bacteria would be present after one hour?
(b) Determine the type of equation this is and then
write the equation (given any time limit).
352.
The half-life of carbon 14 is 5,700 years.
After 5,700 years, the amount of carbon 14
left in the body is half of the original
amount.
(a) Complete a data table for this problem for the
first 3 half lives.
(b) Determine the type of equation this is and then
write the equation (given any time limit).
(c) How much C-14 will there be after 30,000 years?
Algebra 2B Unit 8A
C. Looking at the original question above, one
can see that:
Sn = a1 + a1 ( r ) + a1 ( r )2 + a1 ( r )3 + a1 ( r )4
Arithmetic Series and Sums
353.
Make up a sequence starting at some number
greater than two (a1). Add some number (d) to
your original number, and continue to add the
same number. Record your beginning number and
the six subsequent numbers.
Lastly total: a1 + a2 + a3… an = Sn
n = 7 ; d = ____
a1
a2 =
a3 =
a4 =
a5 =
a6 =
a7 =
= ___
a1 + d
= ___
a1 + d + d
= ___
a1 + d + d + d
= ___
a1 + d + d + d + d
= ___
a1 + d + d + d + d + d
= ___
a1 + d + d + d + d + d + d = ___ = an
Sn = ____
A. How does the total number of d’s in an relate to
what n is?
B. Using only your numbers for a1, d, and n
determine a formula to calculate the last number
(an). Check that your formula works for your
values.
C. Now look at the variables: a1, an and n.
Determine a formula to calculate the total (Sn)
using these variables. (Hint: Average). Check
first that your formula works for your values.
D. Modify the formula determined in part C (Sn = ),
by replacing an with what you found in part B.
Do not bother simplifying this new formula.
=================================
Geometric Series and Sums
354.
Make up a sequence starting at some number
greater than two (a1). Multiply your original
number by some number (r). Continue to multiply
your answer by the same number. Record your
beginning number and the 4 subsequent numbers.
Lastly total: a1 + a2 + a3… an = Sn
n = 5 ; r = ____
a1
a2 =
a3 =
a4 =
a5 =
= ___
a1 ( r )
= ___
a1 ( r )( r )
= ___
a1 ( r )( r )( r )
= ___
a1 ( r )( r )( r ) ( r ) = ___ = an
Sn = ____
A. How does the total number of r’s in an relate
to what n is?
B. Using only your numbers for a1, r, and n
determine a formula to calculate the last
number (an). Check that your formula works
for your values.
From this start show how to derive the
formula for finding the sum of a geometric
sequence. Check that your formula works for
your values.
=================================
Determine whether the following sequences are
arithmetic or geometric - next determine d or r:
355.
3, 5, 7, 9, …
356.
2, 6, 18, 54, 162, …
357.
4, 7, 10, 13, …
358.
1, - 1/2 , 1/4 , - 1/8 , …
359.
1, 1, 1, 1, …
360.
1, 3, 6, 10, 15, …
=================================
Determine: (a) The first 4 terms and a10, and a15.
(b) The sequence type including d or r.
(c) The formula for an using a1.
(d) Solve for the exact value of a10, using the
formula for an.
361.
an = 3n
362.
an = n – 2
363.
an = (n – 3)/3
364.
a1 = 3,
365.
a1 = -2,
d = -3
366.
a1 = -3,
r = -2
367.
a1 = -54,
368.
an = 2n + 1
d =5
r = 2/3
369.
an = (n – 1)/ n
=================================
Determine:
(a) The formula for an.
(b) Solve for the exact indicated an value
using the formula for an.
370.
-12, -7, -2, 3, … ; a21
371.
10, 7, 4, 1, … ; a95
372.
2, 2 1/4, 2 1/2, 2 3/4, … ; a43
373.
5, -10, 20, - 40, … ; a7
Find the indicated term of the given sequence
388.
a1 = 4, d = 3 ; a81
389.
374.
/2, 1/20, 1/200, 1/2000… ; a9
a1 = 15, r = - 2/3 ; a6 (fraction)
1
375.
-1, - 4, -16, -64, … ; a6
376.
3, 7, 11, 15, … ; a10
377.
9, 3, 1, 1/3 , … ; a20
378.
8, 4, 2, 1, 1/2 , 1/4 , … ; a15
379.
12, 9, 6, 3, 0, -3, … ; a25
380.
2, 9, 16, 23, … ; a51
381.
4, 2, 1, 1/2 , … ; a7
382.
3, -1, -5, -9, … ; a21
383.
3, -6, 12, -24, … ; a8
=================================
384.
Given that: a3 = 8, and a16 = 47. Find:
a) d
b) a 1
c) The formula for an
d) Write the first five terms of the sequence.
e) Find a61
390.
391.
=================================
Terry accepts a job, starting with an hourly wage
of $14.25, and is promised a raise of $0.25 per
hour every 2 months for 5 years. Find:
a) a 1
b) d
c) n
d) a n
e) Terry’s hourly wage at the end of 5 years.
Given: a3 = - 4 and d = -2 . Find:
f) a 1
g) The formula for an
h) a 56
=================================
Arithmetic Sums
Find and evaluate the sum
392.
393.
394.
385.
386.
387.
Given that: a5 = 19, and a19 = 75. Find:
a) d
b) a 1
c) The formula for an
d) Write the first five terms of the sequence.
e) Find a55
Given that: a3 = 3, and a6 = -81. Find:
a) r
b) a 1
c) The formula for an
d) Write the first five terms of the sequence.
Given that: a4 = 375, and a7 = 46875 Find:
a) r
b) a 1
c) The formula for an
d) Write the first five terms of the sequence.
=================================
395.
396.
=================================
Write in summation (sigma) notation with k = 1
397.
3 + 6 + 9 + 12
398.
12 + 8 + 4 + 0 + (-4)
399.
1+3+5+7+9
=================================
400.
401.
The first term in the arithmetic series is 3, the
last term is 136, and the sum is 1,390. What
are the first four terms?
408.
Find the sum of the first 15 terms in the
arithmetic sequence: 4, 7, 10, 13, …
409.
Find the sum of the first 20 terms in the
arithmetic sequence: 5, 11, 17, 23, …
410.
A stack of telephone poles has 30 poles in the
bottom row. There are 29 poles in the second
row, 28 in the next row, and so on. Use the
formula for Sn to find how many poles are in
the stack if there are 5 poles in the top row?
How many poles will be in a stack of telephone
poles if there are 50 poles in the first layer, 49 in
the second, and so on, with 6 in the top layer?
Find the sum of the first one hundred terms in
the progression: -6, -2, 2, …
=================================
Find and evaluate the sum
402.
411.
403.
412.
=================================
404.
Carl Gauss (1777-1855) was the greatest
mathematician of his time. When he was in
elementary school his teacher wanted to keep
students busy by asking them to add the
numbers 1 to 100. Within seconds, Gauss had
found the answer. Show how to find the sum
of the first 100 natural numbers.
(1 + 2 + 3 + … + 99 + 100)
Theaters are often built with more seats per row
as the rows move toward the back. Suppose that
the first balcony of a theater has 28 seats in the
first row, 32 in the second, 36 in the third, and so
on, for 20 rows. How many seats are in the first
balcony altogether?
=================================
More Geometric Sums
413.
Use the formula for Sn to find the sum of the
first 10 terms in the geometric sequence:
16, 32, 64, 128, …
414.
405.
You are building a staircase out of cubes.
1 step = 1 cube
2 steps = 2 cubes
3 steps = 3 cubes
How many cubes does it take to build a
staircase that is: 25 cubes high?
406.
If your first step is:
1 step = 3 cube
2 steps = 4 cubes
3 steps = 5 cubes
Use the formula for Sn to find how many cubes
does it take to build a staircase that is 7 cubes
high?
407.
If your first step is:
1 step = 50 cube
2 steps = 51 cubes
3 steps = 52 cubes
How many cubes does it take to build a
staircase that has 78 cubes in the last row?
415.
Use the formula for Sn to find the sum of the
first 7 terms of the geometric sequence:
3, 15, 75, 375, …
Quite a while ago, a king wished to reward a
royal mathematician, and asked him what he
desired. The mathematician, who by the way
appeared both modest and humble, replied that
if just one grain of rice was placed on a square
of an ordinary 8 by 8 chess board, and then two
grains of rice in the next square, and so forth,
doubling the previous amount of rice, until the
last square on the chessboard was reached, then
he would be totally content with the total sum of
all the grains of rice. How much rice was this?
=================================
Algebra 2B Unit 8B
416. There are 5 girls and 3 boys in my family.
In how many different ways can my
mother choose a girl and a boy to do the
dishes?
417.
418.
There are 4 sopranos and 7 altos in the
choir. How many different soprano/alto
duets can be formed?
The menu at the cafeteria lists 3 different
sandwiches, 6 different drinks, and 5
varieties of chips. How many different
sandwich / chip / drink meals are possible?
426.
A social security number is a 9-digit
number like 522-77-0823.
(a) How many different social security
numbers can there be?
(b) There are about 275 million people in the
U.S.. Can each person have a unique
social security number? Explain
427. A U.S. postal zip code is a five digit number.
(a) How many zip codes are possible if any of
the digits 0 to 9 can be used?
(b) If each post office has its own zip code, how
many possible post offices can there be?
================================
Permutations – an ordered arrangement of objects
419.
When creating your fall class schedule you
discover that there is a choice of 5 sections
of English, 3 sections of Math, 4 sections
of Science, and 2 sections of History. How
many schedule arrangements are possible?
420.
How many different 3-digit code symbols
can be formed with the letters A, B, C with
repetition (that is, allowing letter to be
repeated)?
421.
How many different 3-digit code symbols
can be formed with the letters A, B, C , D,
and E with repetition (that is, allowing
letter to be repeated)?
422.
How many different 5-digit code symbols
can be formed with the letters A, B, C ,
and D if we allow a letter to occur more
than once?
423.
Using only the odd digits (1, 3, 5, 7, 9)
how many different 3-digit numbers can
you form (repetitions are allowed)?
424.
How would one find the total number of
license plate possibilities given that a
license plate is three letters followed by
three digits?
425.
How many 7-digit phone number can be
formed with the digits 0-9, assuming that
the first number cannot be 0 or 1?
428.
Three finalists: Sue, Tim, Pat are in the
annual bake off contest.
(a) Show all the different orders that they can
finish?
(b) How many total different ways is this?
(c) Show this same answer using factorial
notation.
(d) Show this same answer using permutation
notation. Explain what each number /
symbol stands for.
429. (a) Show all the different orders that the letters
in the word MATH be arranged.
(b) How many total different ways is this?
(c) Show this answer using factorial notation.
(d) Show this same answer using
permutation notation. Explain what each
number / symbol stands for.
430. (a) Using factorial notation to show how
many different ways 8 packages be placed
in 8 mailboxes, one package in a box?
(b) Show this same answer using
permutation notation.
431. (a) Show all the different 3-digit code symbols
that can be formed with the letters A, B, C
without repetition (that is, using each letter
only once)?
(b) How many total different ways is this?
(c) Show this same answer using factorial
notation.
(d) Show this same answer using
permutation notation. Explain what each
number / symbol stands for.
432.
On one Saturday night, all 6 employees
decided to line up and sing “Happy
Birthday” to a customer.
(a) Using factorial notation to show how
many different ways could they have
lined up?
(b) Show this same answer using
permutation notation.
433. (a) Using factorial notation to show how
many different ways can 5 people can
line up for a photograph?
(b) Show this same answer using
permutation notation.
434. (a) Using factorial notation to show how
many different ways 5 starters on the
basketball team be assigned to their
positions?
(b) Show this same answer using
permutation notation.
435.
436.
437.
(a) Using factorial notation to show how
many different ways 7 athletes be
arranged in a straight line?
(b) Show this same answer using
permutation notation.
(a) Using factorial notation to show how
many different ways 6 classes be
scheduled during a 6-period day?
(b) Show this same answer using
permutation notation..
(a) Using factorial notation to show how
many different ways 9 starters of a
baseball team be placed in their
positions?
(b) Show this same answer using
permutation notation.
=============================
438. (a) Show all the different 3-digit code symbols
that can be formed with the letters A, B, C
and D without repetition (that is, using each
letter only once)?
(b) How many total different ways is this?
(c) Show this same answer using
permutation notation. Explain what
each number / symbol stands for.
439. (a) Show all the different ways can you arrange
the letters of MATH taking two at a time?
(b) How many total different ways is this?
(c) Show this same answer using
permutation notation. Explain what each
number / symbol stands for.
440. (a) If six baseball teams must play each other
team twice. Demonstrate with a chart
showing each team’s opponents.
(b) How many total different ways is this?
(c) Show this same answer using
permutation notation. Explain what each
number / symbol stands for.
441.
Using permutation notation show how
many ways can you arrange 3 books on a
book shelf from a group of 7 books?
Explain what each number / symbol stands
for.
442.
Using permutation notation show how many
different ways you can arrange the letters of
the word COMPUTER taking 4 at a time?
Explain what each number / symbol stands
for.
443.
Using permutation notation show how
many different ways you can arrange the
odd digits (1, 3, 5, 7, 9) for a 3-digit
numbers (when repetitions are NOT
allowed)? Explain what each number /
symbol stands for.
444.
Using permutation notation show how
many different 3-digit code symbols can
be formed with the letters A, B, C, D, E
and F without repetition (that is, using
each letter only once)? Explain what each
number / symbol stands for.
445.
The flags of many nations consist of three
vertical stripes. For example, the flag of
Ireland has its first stripe green, second
white, and third orange. Suppose that the
following colors are available: black,
yellow, red, blue, white, gold, orange,
pink, purple. Using permutation notation
show how many different flags of 3 colors
can be made without repetition of colors?
This assumes that the order in which the
stripes appear is considered. Explain what
each number / symbol stands for.
446.
A president and a vice-president are to be
selected from a 7-member student council
committee. Using permutation notation
show how many different ways can the
selection be made? (Each member may be
a president or a vice-president, but not
both).
447.
12 students are in a race. Using
permutation notation show how many
different ways possible that they can win
the 1st, 2nd and 3rd place trophies?
448.
A baseball manager arranges the batting order
as follows: The 4 infielders will bat first. Then
the 3 outfielders, the catcher, and the pitcher
will follow, not necessarily in that order. How
many different batting orders are possible?
452.
Find the total number of distinguishable
permutations to the last three questions:
Permutations of Sets with
Nondistinguishable Objects.
Question 33: How many distinguishable
permutations are possible using this set of
3 marbles - 1 which is blue and 2 of
which are red.
Question 34: How many distinguishable
permutations are possible using this set of
4 marbles - 2 which are blue and 2 of
which are red.
Question 35: How many distinguishable
permutations are possible using this set of
5 marbles - 3 which are blue and 2 of
which are red.
453.
Consider a set of 7 marbles, 4 of which are
blue and 3 of which are red. Although the
marbles are all different, when they are
lined up, one red marble will look just like
any other red marble and are nondistinguishable, similarly, the blue marbles
are also non-distinguishable from each
other. How many distinguishable
permutations are possible using this set of
7 marbles?
454.
How many distinguishable code symbols
can be formed from the letters of the word
MATHEMATICS?
455.
How many distinguishable code symbols
can be formed from the letters of the word
BUSINESS?
=================================
Permutations of Sets with Nondistinguishable Objects
449.
Consider a set of 3 marbles, 1 of which is blue
and 2 of which are red. Although the marbles
are all different, when they are lined up, one
red marble will look just like any other red
marble and are non-distinguishable.
(a) First show all the distinguishable permutations
possible using this set of 3 marbles, than
(b) state the total number of distinguishable
permutations.
450.
Consider a set of 4 marbles, 2 of which are
blue and 2 of which are red. Although the
marbles are all different, when they are lined
up, one red marble will look just like any other
red marble and are non-distinguishable,
similarly, the blue marbles are also nondistinguishable from each other.
(a) First show all the distinguishable permutations
possible using this set of 4 marbles, than
(b) state the total number of distinguishable
permutations.
451.
Consider a set of 5 marbles, 3 of which are
blue and 2 of which are red. Although the
marbles are all different, when they are lined
up, one red marble will look just like any other
red marble and are non-distinguishable,
similarly, the blue marbles are also nondistinguishable from each other.
(a) First show all the distinguishable permutations
possible using this set of 3 marbles, than
(b) state the total number of distinguishable
permutations.
Suppose the expression a2b3c4 is rewritten
without exponents. In how many ways can
this be done?
================================
456.
Combination – a selection without regard to the order
457. (a) If seven soccer teams must play each
other one time only. Demonstrate with a
chart showing each team’s opponents
(b) How many total different ways is this?
(c) Show this same answer using
combination notation. Explain what each
number / symbol stands for.
458. (a) Show all the combinations of three letters
taken from the set of 5 letters:{A, B, C, D, E}
(b) How many total different ways is this?
(c) Show this same answer using
combination notation.
459.
In how many ways can three numbers be
chosen from 10 numbers? .
460.
There are 8 finalists in a trivia contest. If
each finalist plays a round against each of
the other finalists, how many rounds will
have been played in all? (Use combination
notation).
461.
Yummy Ice Cream shop serves 12
flavors of ice cream. How many
different flavored double scoop cones
can they make? (No two scoops of the
same flavor).
462.
Two of the 6 players on a basketball team
are to be selected as co-captains. In how
many ways can the selection be made?
463.
Students are asked to choose 4 numbers
from 1-10 for a school game of chance.
How many different combinations are
possible?
464.
A principal wants to start a peer
counseling group to work with students in
need. He needs to narrow down his choice
to six students from a group of nine
students. How many ways can a group of
six be selected?
Use Combination notation to find the total
number of diagonals (connecting two
vertices – not on the perimeter) that can be
drawn in each of the following figures:
Check your answers by drawing the total
number of diagonals for each figure.
a) Triangle
b) Quadrilateral
c) Pentagon
d) Hexagon
e) Heptagon
f) Octagon
g) Determine a quadratic equation for
determining this total number of ydiagonals given the total number of xsides.
===============================
466.
How many committees can be formed
from a group of 5 governors and 7
senators if each committee consists of 3
governors and 4 senators?
467.
A bag of marbles contains 4 red
marbles, 5 green marbles, and 8 blue
marbles. How many ways can 2 red
marbles, 1 green marble, and 2 blue
marbles be chosen?
You can do this long-hand or use your
calculator to get the answer but show
work.
468.
A bucket of flowers contains 6 red
carnations, 5 white daisies, and 7 yellow
tulips. How many bouquets could be
created so that each bouquet has 3 red
carnations, 1 white daisy, and 2 yellow
tulips?
===============================
469. Suppose that 3 people are selected at random
from a group that consists of 6 men and 4
women. What is the probability that 1 man
and 2 women are selected?
470.
471.
465.
Suppose that 2 cards are drawn from deck
of 52 cards. What is the probability that
both of them are spades?
==============================
Eight cards are drawn from a standard
deck of 52 cards. How many 8-card hands
having 5 cards of one suit and 3 cards of
another suit can be formed? You may use
a calculator but show all work.
472.
To win the jackpot in the Colorado lottery
you must pick 6 correct numbers out of 49
total numbers. How many 6-number
combinations are possible? (For this
problem work does not have to be shown
if you put down what you entered into
your calculator and the result.)
473.
How many 5-card poker hands are
possible with a 52-card deck? (For this
problem work does not have to be shown
if you put down what you entered into
your calculator and the result.)