PRECALCULUS SEM. 1 REVIEW (2011–2012) Name: (additional copies available online!) Period: Unit 1: Functions *** No Calculators!!**** Use the given functions to find solutions to problems 1 – 6.

f (x) = x2 + 3x −10 g(x) = x + 3 h(x) =

x + 3x2 − 4

m(x) = x2 + 7 k(x) = x − 2

1. Evaluate

f −3( ) 2. Evaluate k −2.1( ) 3. State the domain of

h x( ) . 4. Find

g g g x( )( )( ) if x0 = 4 .

5a. Find

f g( ) x( ) and g f( ) x( ) . b. Are

f x( ) and g x( ) inverses? How do you know? 6a. Find the inverse of

m x( ) . b. Is the inverse a function? Explain.

7. Given that x is an integer such that −4 < x < 3 . a. State the relation represented by

y = 2 + x by listing a set of ordered pairs:

b. Is the relation a function? Why?

8. Write the ordered pair when

−1,5( ) is reflected over: a. x – axis b. y – axis c. y = x d. y = –x e. origin 9. Find the coordinates of P’ if P

1,−5( ) and P’ are symmetric with respect to M −3,2( ) .

10. Determine algebraically if y = x2 −1 is symmetric with respect to the x–axis, the y–axis, the line y = x, the line y = –x, and/or the origin. Show all work. 11. Complete each statement: a. Odd functions have symmetry. b. Even functions have symmetry. c. If a function is reflected over , the reflection is its inverse.

12. Determine if the following functions are even, odd, or neither: a.

f x( ) = 13x8 −12x2 + 7 b. g x( ) = 4x3 − 6x + 9 c.

l x( ) = 23x5 − 67x3 − x Sketch the following graphs and state the domain and range when indicated: 13a. y = (x − 4)2 + 2 13b. y = − x + 5 14a. y = x + 23 −1 14b.

y = x +1 D: D: D: R: R: R:

15a. y = x + 3 15b. y =

1x

16. y = x −1( )3

D: D: D: R: R: R:

Unit 1 Answers 1. –10 2. –5 3.

except x = ±2 4. 13 5a.

f g( ) x( ) = x2 + 9x + 8, g f( ) x( ) = x2 + 3x − 7 b. No, explain why.

6a. y = ± x − 7 b. No, explain why. 7a. −3,1( ), −2,0( ), −1,1( ), 0,2( ), 1,3( ), 2,4( ){ }

b. Yes, explain why. 8a. −1,−5( ) b.

1,5( ) c. 5,−1( ) d.

−5,1( ) e. 1,−5( )

9. −7,9( ) 10. y – axis, be sure to show work for all 5 symmetries 11a. origin b. y–axis

c. the line y = x 12a. even b. neither c. odd 13a. 13b. 14a.

D :R : 2,∞⎡⎣ )

D :R : −∞,5( ⎤⎦

D :R :

14b. 15a. 15b.

D : 0,∞⎡⎣ )R : 3,∞⎡⎣ )

D :,x ≠ 0R :, y ≠ 0

16.

D :R :

Unit 2: Polynomial Functions *** No Calculators!!**** 1. Determine whether 2 is a zero of f (x) = x3 − 3x2 − 2x + 6 . 2. Write a polynomial equation of least degree with roots –3 and ±2i . 3. Use synthetic division to divide:

x5 − 3x2 − 20 by x − 2( ) . State the depressed polynomial. 4. Use the remainder theorem to find the remainder of

2x3 − 2x − 3( ) ÷ x −1( ) . 5. Determine how many times 1 is a root of x3 − 3x + 2 = 0 .

***Calculators allowed for the next two problems**** 6. Determine the complex roots of 2x4 − x3 + 5x2 − 4x −12 = 0 a. State the number of complex roots for the above equation. b. State all of the possible rational roots for the above equation. c. Determine all of the roots to the equation. 7. Determine the complex roots of x3 − 3x2 − 2x + 8 = 0 a. State the number of complex roots for the above equation. b. State all of the possible rational roots for the above equation. c. Determine all of the roots to the equation.

1000

500

-500

-1000

-1500

-2000

-5 5

8. Describe the end behavior of each of the following. a.

f x( ) = −x4 − 3x2 + 5 b. f x( ) = x3 + 2x2 − x − 4

x →∞, then y → _____x → −∞,then y → _____

x →∞, then y → _____x → −∞,then y → _____

9. Write a potential equation (in factored form) for the graph below. 10. Given the graph describe the number and nature of the roots. a.

f x( ) = x5 + . . . −5 b. g x( ) = x3 + . . . +2

Total number of complex roots: Total number of complex roots:

Total number of real roots: Total number of real roots:

Number of imaginary roots: Number of imaginary roots:

Number of irrational roots: Number of irrational roots:

40

20

-20

- 5 5

20

10

-10

- 5 5

Unit 2 Answers: 1. No, b/c f 2( ) = -2 . 2. y = x3 + 3x2 + 4x +12 3. x4 + 2x3 + 4x2 + 5x +10 4. –3 5. 2 times

6a. 4 b. ± 12,1, 32, 2, 3, 4, 6,12

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪ c. -1,

32, ±2i 7a. 3 b. ± 1, 2, 4, 8{ } c. 2,

1 ± 172

8a. −∞,−∞ b. ∞,−∞ 9. 5, Explain why. 10a. 1 double rational, 1 other rational, 2 irrational b. 1 rational, 2 imaginary Unit 3: Rational Functions *** No Calculators!!****

1. Sketch the graph of y = x2 + 3xx2 + x - 6

.

a. Hole _______________ (point of discontinuity) b. X-int _______________ (even/ odd) c. Y-int _______________ d. V. A. _______________ (even/odd) e. H.A _______________

2. Sketch the graph of y =x -1( )2

x2 x + 3( ) .

a. Hole _______________ (point of discontinuity) b. X-int _______________ (even/ odd) c. Y-int _______________ d. V. A. _______________ (even/odd) e. H.A _______________

3. Sketch the graph of y =x +1( ) x − 5( )2x −1( )2

.

a. Hole _______________ (point of discontinuity) b. X-int _______________ (even/ odd) c. Y-int _______________ d. V. A. _______________ (even/odd) e. H.A _______________

4. Sketch the graph of y =x − 2( ) 3x − 4( ) x + 3( )

x + 2( ) x − 2( )2.

a. Hole _______________ (point of discontinuity) b. X-int _______________ (even/ odd) c. Y-int _______________ d. V. A. _______________ (even/odd) e. H.A _______________

5. Sketch the graph and write the equation of the function with the given information:

even

odd odd

–10 –5 5 10

Unit 3 Answers:

1. a. -3, 35

⎛

⎝⎜

⎞

⎠⎟ b. 0 (odd) c. 0 2. a. none b. 1 (even) c. none

d. x = 2 (odd) e. y = 1 d. x = 0 (even), x = –3 (odd) e. y = 0

3. a. none b. –1 (odd), 5 (odd) c. -5 4. a. none b. 43

(odd), –3 (odd) c. 3

d. x = 12

(even) e. y = 14

d. x = 2 (odd), x = -2 (odd), e. y = 0

5. y = (x - 1)2

x(x + 3)

6

4

2

–2

–4

–6

–8

–10 –5 5 10

–10 –5 5 10

6

4

2

–2

–4

–6

–8

Unit 4: EXPONENTS & LOGARITHMS *** No Calculators!!****

1. Simplify completely

3x2y( )-2

2xy2. 2. Approximate:

log

217 .

3. Solve for x: .1( )x

= 10000 4. Solve for x: 3x • 27x +3 =

181

⎛

⎝⎜

⎞

⎠⎟

-2x

.

5. Sketch the graph of each, State the intercept, another point and the equation of the asymptote: No Calculator!!

a. y =

12

⎛

⎝⎜

⎞

⎠⎟

x-1

+ 3 b. y = ex +1 + 2

pt:________ pt:_______ a.p._______ a.p._______ H.A.: _______ H.A.: _______ c.

y = log

2x + 1( ) d.

y = ln x - 4( )

pt:________ pt:_______ a.p._______ a.p._______ V.A.: _______ V.A.: _______

Solve. Round answers to the nearest thousandth, if necessary ***Calculators allowed!!****

6. 3.6x = 72.4 7. log317 = x

8. 6x-1 = 8x 9. e

3x + 2 = 7 10. ln4.5 = lne.031x 11. 10 = 5e5x

12. ln3 + lnx = ln45 13. 2log

64 -

13

log68 = log

6x

14. log103 = x 15. log

445 = x

16.

log

3x + 2( ) + log

3x - 5( ) = log

318 17.

log

2x - 3( ) + log

2x + 3( ) = 4

18. Tuition for your first year of college will be $13,000 and you have saved $5000. This money is invested in a savings account paying 7.75% interest compounded quarterly. (Round the nearest hundredth please.) a. What will the value of your investment be two years from now? b. How many years will it take you to reach your goal of $13,000? 19. A certain strain of bacteria grow at a rate of .0567 per day. How long will it take the 2 bacteria to grow to 500 bacteria?

Unit 4 Answers:

1.

118x5y4

2. between 4 and 5 3. x = -4 4. x =

94

5a. int.: ( 0 , 5 ) b. int: ( 0 , 4.72 ) a.p.: ( 1 , 4 ) a.p.: ( –1 , 3 ) asymp: y = 3 asymp: y = 2 5c. int.: ( 0 , 0 ) 5d. int: ( 5 , 0 ) a.p.: ( 1 , 1 ) a.p.: ( 6.72 , 1 ) asymp: x = -1 asymp: x = 4 6. ≈ 3.343 7. ≈ 2.579 8. ≈ -6.228 9. ≈ .536 10. ≈ 48.519 11. ≈ .139 12. 15 13. 8 14. 3 15. 5 16. –4 , 7 17. –5 , 5 18a. ≈ $5829.64 b. ≈ 12.45 years 19. 94.38 days

Unit 5: Probability *** Calculators!!**** 1. A six sided die is rolled. a) Find the probability that a three or an even number is rolled.

b) If the die is rolled twice, what is the probability of getting an even number on each attempt?

2. For each of the following situations, a standard deck of cards is used. a) A single card is drawn, find the probability that a queen or a heart is drawn? b) A single card is drawn, find the probability that a ten or an ace is drawn?

c) A single card is drawn, find the probability that the card drawn is a four given that it was not a face card.

3. If two cards are drawn from a standard deck of 52 cards, in succession with replacement, what is the probability of drawing a 7 then a heart? 4. If four cards are drawn from a standard deck of 52 cards, what is the probability that exactly three kings are selected? 5. If two cards are drawn from a standard deck of 52 cards, what is the probability of exactly one face card? 6. In how many ways can 15 baseball players be placed in a 9 person batting order?

7. A committee of 6 people is formed from 5 women and 4 men. How many committees can be formed if the committee consists of half men and half women? 8. In how many ways can the letters in the word problems be arranged? 9. Mark has decided to buy a new suit, either made of wool, silk or rayon. He has narrowed down the choices of colors to gray, olive, blue, black or brown, and the matching tie to a stripe, solid, or geometric pattern. How many different selections of his suit and tie are possible? 10. If a license plate has three letters followed by a 2 digit number. How many license plates can be made if the second letter is an N, and the first digit can be a 2 or a 5, and none of the letters or numbers can repeat? 11. A box of Frango mints contains 5 mint, 4 raspberry, 3 toffee, and 2 peanut butter. a. If a Frango is selected at random, what is the probability that it is a toffee or a mint? b. If a Frango is selected at random, eaten, and a second Frango is selected, what is the probability they were raspberry then toffee?

U D

C F

3

45

6

15 1

10 12

8

12. A box of Frango mints contains 5 mint, 4 raspberry, 3 toffee, and 2 peanut butter. a. If a Frango is selected at random, replaced (not eaten), and a second Frango is chosen, what is the probability that the Frangos were raspberry, then toffee? b. If a Frango is selected at random, replaced (not eaten), and a second Frango is chosen, what is the probability that the Frangos were raspberry and toffee? 13. 100 people were surveyed and asked whether they owned dogs (D), cats (C) and/or fish (F). The Venn diagram at the right shows the responses If one person is randomly chosen from those surveyed, what is the probability that the person ... a. owns a dog? b. owns a cat, given that he/she has fish? c. owns a cat and a dog? d. owns a cat or fish? e. owns a cat or a dog, given that he/she does not own fish? 14. The chart at the right represents the results of a survey of people at Great America who were asked whether they lived in Wisconsin or Illinois. If a person is chosen at random from those surveyed, what is the probability that he or she... a. lives in Illinois? b. is male, given that the person is from Wisconsin? c. Is from Illinois, given that the person is female? 15. Three cards are drawn from a standard deck of 52. What is the probability that at least one heart is drawn?

men

women

WI IL

180 392

176 401

Unit 5 Answers:

1a.

23

b.

14

2a.

413

b.

213

c.

110

3.

152

4.

192270725

5.

80221

6. 1816214400 7. 40

8. 40320 9. 45 10. 10800 11a.

47

b.

691

12 a.

349

b.

649

13a.

71100

b.

931

c.

750

d.

1325

e. 2223

14a. 7931149

b. 4589

c. 401577

15. 9971700

Unit 6: Statistics *** Calculators Okay!!**** 1. The number of trials required by 20 different puppies to learn a certain trick is as follows: 9, 18, 12, 13, 10, 21, 7, 19, 11, 9, 4, 22, 16, 14, 38, 15, 16, 8, 13, 25 Find the following: a. mean = __________ b. median = __________ c. range = __________ d. Q1 = __________ e. Q3 = __________ f. inter-quartile range = __________ h. Calculate the outliers. SHOW ALL WORK. i. Draw a boxplot modeling the data. Include appropriate labels and outliers (if necessary.) j. Describe in words the shape of the graph.

X

X

2. Given the five data values: 72, 61, 82, 75, 91. Determine the sixth data value if the… a. median is 75. b. median is 77. c. mean is 78. 3. The mean of a set of normally distributed data is 88 and the standard deviation is 5. a. What percent of the data lies within the interval 78 - 98?

b. Find the interval about the mean that includes about 68% of the data. 4. Suppose 150 values in a data set are normally distributed. a. How many values are within one standard deviation of the mean? b. How many values are within two standard deviation of the mean? c. How many values fall in the interval between the mean and one standard deviation above the mean? Unit 6 Answers: 1. a. 15 b. 13.5 c. 34 d. 9.5 e. 18.5 f. 9 h. 38 is an outlier j. strongly skewed right 2. a. 75 b. 79 c. 87 3. a. ≈ 95% b. 83 – 93 4. a. ≈ 102 b. ≈ 142 c. ≈ 51

X