Honors Precalculus - Blessed Trinity Catholic High … TRINITY CATHOLIC HIGH SCHOOL Honors...

54
BLESSED TRINITY CATHOLIC HIGH SCHOOL Honors Precalculus Summer Assignment I really enjoyed teaching you this past year in Honors Algebra II. Together, we developed a strong foundation in Algebra II that will serve you well in Honors Precalculus, A.P. Calculus and higher levels of Mathematics. Many years ago, when I was a student at GT, I remember students struggling with Calculus. Their biggest struggle was not the Calculus concepts, but the Algebra within the Calculus. It is my hope that this summer assignment keeps you sharp on the concepts that we covered this past year. It is a comprehensive packet covering all the major concepts in Honors Algebra II. This assignment will be collected at the first Honors Precalculus class of the semester and will count as five homework grades. Remember that all the notes from Honors Algebra II are very neatly organized on Moodle. I strongly encourage you to access those notes from our classes this past year and to use those as tool to help you on this summer assignment. If you have any questions, you can email me or Mrs. Muller. -Mr. Johnson Warning: DO NOT WAIT TILL THE LAST TWO WEEKS OF SUMMER TO WORK ON THIS ASSIGNMENT. YOU SHOULD DO ONE PAGE PER DAY OVER THE SUMMER. NAME:____________________________

Transcript of Honors Precalculus - Blessed Trinity Catholic High … TRINITY CATHOLIC HIGH SCHOOL Honors...

BLESSED TRINITY CATHOLIC HIGH SCHOOL

Honors Precalculus Summer Assignment

I really enjoyed teaching you this past year in Honors Algebra II. Together, we developed a strong foundation in Algebra II that will serve you well in Honors Precalculus, A.P. Calculus and higher levels of

Mathematics. Many years ago, when I was a student at GT, I remember students struggling with Calculus. Their biggest struggle was not the Calculus concepts, but the Algebra within the Calculus. It is my hope that

this summer assignment keeps you sharp on the concepts that we covered this past year. It is a comprehensive packet covering all the major concepts in Honors Algebra II. This assignment will be

collected at the first Honors Precalculus class of the semester and will count as five homework grades. Remember that all the notes from Honors Algebra II are very neatly organized on Moodle. I strongly

encourage you to access those notes from our classes this past year and to use those as tool to help you on this summer assignment. If you have any questions, you can email me or Mrs. Muller.

-Mr. Johnson

Warning: DO NOT WAIT TILL THE LAST TWO WEEKS OF SUMMER TO WORK ON THIS ASSIGNMENT. YOU SHOULD DO ONE PAGE PER DAY OVER THE SUMMER.

NAME:____________________________

Honors Algebra II Summer Assignment Name _________________________

1. Write the equation of the line in point-slope, slope intercept and standard form using the given information.

a. Slope = −1

2 and passes through the point (2,3)

b. Passes through the point (5, 2) 𝑎𝑛𝑑 (−3,6)

c. Passes through the points (1,4) 𝑎𝑛𝑑 (5 ,4)

d. Passes through the point (2, 3) and is parallel to the line 𝑦 =2

3𝑥 − 5

e. Passes through the point (−3, 5) and is perpendicular to the line 𝑦 = −𝑥 + 7

2. Write the equation of the line that contains the point(4 , −7) and has:

a. An undefined slope b. A zero slope

3. Write the equation of the line that is a) perpendicular and b) parallel to the line 2𝑥 + 5𝑦 = 8 and passes through

the point (7

6, −

2

3)

4. Find the domain algebraically (using interval notation) of the following functions

a. 𝑓(𝑥) = √3 − 2𝑥 b. 𝑓(𝑥) =3

9−𝑥 𝑐. 𝑓(𝑥) =

2𝑥−1

√2𝑥−5

b. 𝑓(𝑥) =3𝑥

√𝑥2−8𝑥+12 d. 𝑓(𝑥) =

2𝑥

|3𝑥−5|+1 𝑒. 𝑘(𝑥) = |3𝑥 + 1| − 2

f. 𝑔(𝑥) =2𝑥−1

𝑥4−9𝑥2+20

5. Find the zeros of the function algebraically:

a. 𝑓(𝑥) = 6𝑥2 − 19𝑥 − 7 b. 𝑔(𝑥) =3𝑥

9−𝑥2

6. Find the inverse function

a. 𝑓(𝑥) = 3 − 2𝑥 b. 𝑓(𝑥) = √2𝑥 + 1

Domain: Domain:

Range: Range:

7. Verify that f and g are inverse functions

a. 𝑓(𝑥) = 3𝑥 − 5 𝑔(𝑥) =1

3𝑥 +

5

3

b. 𝑓(𝑥) = √𝑥 + 2 𝑔(𝑥) = 𝑥2 − 2 , 𝑥 ≥ 0

c. 𝑓(𝑥) =1

3(𝑥 − 1)3 + 7 𝑔(𝑥) = √3𝑥 − 21

3

8. 𝑎) 𝑓(𝑥) = −2|𝑥 + 2| + 3

Vertex:

Domain:

Range:

Increasing:

Decreasing:

Axis of symmetry:

X-int:

y-int:

8 𝑏) 𝑓(𝑥) = −|𝑥| + 5

Vertex:

Domain:

Range:

Increasing:

Decreasing:

Axis of symmetry:

X-int:

Y-int:

8 𝑐) 𝑓(𝑥) =1

3|𝑥 − 1| − 2

Vertex:

Domain:

Range:

Increasing:

Decreasing:

Axis of symmetry:

X-int:

Y-int:

9.) For each of the following, find f(g(x)) and g(f(x)). State the domain of f, g, and of each of the

composition functions.

a) ( ) 3 2f x x ( ) 2 1g x x

b) 2( ) 1f x x 1

( )5

g xx

c) 2( ) 2f x x ( ) 7g x x

d) 2( )f x x 2( ) 1g x x

e) 1

( )2

f xx

( )g x x

10.) Multiple choice: If (2) 5f find a point that must lie on 3 ( 2) 5f x (hint: use transformations)

a. (4, 10) b. (0, 0) c. (6, 0) d. (0,10) e. (4, 0)

11.) Given the table below, find the missing values.

x -3 -2 -1 0 1 2

( )f x -2 -1 0 3 1 -3

( )g x 0 4 -5 2 -1 3

( 1)f x

( ) 1g x

( ( ))f g x

2 ( )f x

( ( ))f f x

( ( ))g g x

( ( ))g f x

1( )f x

1( )g x

12.) Find the domain of the following functions algebraically. Write your answer in interval notation.

A. ( ) 5 3f x x B. 3( ) 9h x x C. ( )5

xg x

x

DOMAIN: ______________ DOMAIN: ________________ DOMAIN: _______________

D. 2( ) 12g x x x E. 3

( )2 4

f xx

F. 3 2( ) 6 5f x x x

DOMAIN: ______________ DOMAIN: ________________ DOMAIN: _______________

G. 2( ) 6 9f x x x H. 2

5( )

25

xg x

x

I*.

1( ) 10

1h x x

x

DOMAIN: ______________ DOMAIN: ________________ DOMAIN: _______________

13.) Graph each of the following. Write down the domain, range, where the function is not continuous, and the type of discontinuity.

a. 3 1

( )2x+1 1

x if xf x

if x

2.

2 0( )

x+2 0

x if xf x

if x

Domain: __________________ Domain: ___________________

Range: ____________________ Range: ___________________

c.

21 +3 1

( ) 1

x if xf x

x if x

d.

ln 0

( ) e 0 1

7 4

x

x if x

f x if x

if x

Domain: __________________ Domain: ___________________

Range: ____________________ Range: ____________________

e. 3 1 if 0

( ) if 0

x xf x

x x

f. 3

1 if 1

( ) x if 1

2 if 1

3

xx

f x x

xx

x

Domain: __________________ Domain: __________________

Range: ____________________ Range: ___________________

14. Johnny solves the following quadratic inequality as shown:

𝑥2 − 9 > 0

(𝑥 + 3)(𝑥 − 3) > 0

𝑥 + 3 > 0 𝑎𝑛𝑑 𝑥 − 3 > 0

𝑥 > −3 𝑎𝑛𝑑 𝑥 > 3

𝑥 > 3

(3, ∞)

What did Johnny do wrong?

How would you recommend Johnny solve this quadratic inequality?

15. Give an example of two functions with even symmetry and two functions with odd symmetry.

16. Determine whether the following functions are even, odd, or neither.

a. 𝑓(𝑥) = 𝑥7 − 𝑥5 + 2 b. 𝑓(𝑥) = |𝑥|(√𝑥2 + 1) c. 𝑓(𝑥) = 𝑥4−9

𝑥√𝑥2+5

d. 𝑓(𝑥) = √𝑥 − 3 e. 𝑓(𝑥) = 3 f. 𝑓(𝑥) = 0

g. 𝑓(𝑥) = 2𝑥3 − 3𝑥 h. 𝑓(𝑥) = |𝑥3 + 2𝑥| i. 𝑓(𝑥) = |𝑥|√𝑥 + 1

j. 𝑓(𝑥) = 𝑥5

4−3𝑥2 k. 𝑓(𝑥) = 𝑥5 − 3𝑥2 + 1 l. 𝑓(𝑥) = |𝑥5|+1

𝑥

m. 𝑓(𝑥) = |𝑥−2|

𝑥2 n. 𝑓(𝑥) = 9−𝑥2

√𝑥4+1

17. Give an example of a function that is one-to-one.

18. Give an example of a function whose inverse is not a function.

19. Give an example of a function whose inverse is itself.

20. Given the graph of a function, how would you graph the function’s inverse?

21. What does it mean to say a function is one-to-one?

22. Given 𝑓(𝑥) = √𝑥 − 1 and 𝑔(𝑥) = 𝑥2 + 1, Graph 𝑓(𝑔(𝑥)).

23. Given 𝑓(𝑥) = 1

𝑥−2 and 𝑔(𝑥) = 𝑥 + 5,

a) Find 𝑓(𝑔(𝑥)). b) Find 𝑔(𝑓(𝑥)).

Domain: Domain:

24. What is the only type of line that does not have an inverse that is a function?

25. What is the only type of line that is not a function?

26. Which of the following functions are one-to-one and thereby have an inverse that is a function?

a) 𝑓(𝑥) = 1

3|𝑥 − 2| + 5 b. 𝑓(𝑥) = 𝑒𝑥 c. 𝑓(𝑥) = (𝑥 + 2)2 − 1 d. 𝑓(𝑥) = 2𝑥 − 1

e) 𝑓(𝑥) = 2 f. 𝑓(𝑥) = 𝑙𝑛𝑥 g. 𝑓(𝑥) = 1

𝑥 h. 𝑓(𝑥) = 𝑥3

i) 𝑓(𝑥) = 𝑥4 j. 𝑓(𝑥) = √𝑥 k. 𝑓(𝑥) = √𝑥3

27. Given the table for an even function, fill in the remaining blanks.

x -4 -3 -2 -1 0 1 2 3 4

y 1 3 2 0 -2

28. Given the table for an odd function, fill in the remaining blanks.

x -4 -3 -2 -1 0 1 2 3 4

y 1 3 0 -2

Name a function that is:

29. Always increasing

30. Always decreasing

31. Always constant

32. Increasing from (−∞, 0) and decreasing from ( 0, ∞)

33. Decreasing from (−∞, 0) and increasing from ( 0, ∞)

34. Increasing from (−∞, −3) and decreasing from (−3, ∞)

35. Decreasing from (−∞, 4) and increasing from ( 4, ∞)

36. Not bounded

37. Bounded from above

38. Bounded from below

39. Continuous everywhere

40. Symmetric with respect to the origin

41. Symmetric with respect to the y-axis

42. Has an absolute max of 2 at x = 3

43. Has an absolute min of -3 at x = 1

44. Is symmetric about the line x = 7

45. A piecewise function with a removable discontinuity at x = 5

46. A piecewise function with a jump discontinuity at x= -3 and x=2

47. A piecewise function with a removable discontinuity at x = 1,3 and 5 and jump discontinuity at x=2 and

x=4

48. An absolute value function that has no x-intercepts

49. An absolute value function with one x-intercept

Solve the following equation:

50. 3 2 1 2x

51. 6 7

23 1 3 1

x

x x

52. 5 7

53 4 12

x x x

53. 3 4

2 1 5 1x x

Simplify:

54. 2 3

1 2 1

2 2

x

x x x x

55.

2 2

2

9 4 4

2 6 9

t t t

t t t

Solve the formula for the indicated variable:

56. 2

0

1;

2h V at for a 57. ;S C RC for C

58. 2 12 ;x ax bx for x

Solve and write your solution using interval notation.

59. 3 1 5x 60. 1 3 5 1 2 1x

61. Classify the following numbers.

Real Imaginary Irrational Rational Integer Whole Natural

√17

2

3i

9

196

-13

62. Given

2 1 0

( ) 1, 2 0

3 1 2

x x

f x x

x x

find: A. ( 2)f B. (2)f C. (0)f

63. Graph f(x) in number 69 above.

Factor the following polynomials:

64. 4 16x 65. 2 34x x 66.

26 11 10x x

67. 2

2 23 16x x 68. 3 27 5 35x x x

Factor the following polynomials:

69. 26 216x 70.

4 27 12x x

Simplify each expression.

71. (1 ) ( 3 2 )i i 72. 2

1 7i 73. 2 5

12 5

i

i

74. 571i 75.

2

4 2i 76. (1 2 )(3 2 )i i

Solve the following using the indicated method. You must show each step of the process.

77. 25 3 6 0x x (complete the square)

Solve the following using the indicated method. You must show each step of the process.

78. 2

4 2 1 26x (square roots) 79. 23 11 5 0x x (complete the square)

80. What is the discriminant and what are the nature of the roots it describes?

81. a. Find the quadratic with x intercepts at x = -1 and x = 2 that passes through the point (5, 18). Write your final answer in standard form.

b. Find the quadratic with a vertex at (2, -3) that passes through the point (-1, -4). Write your answer in vertex form.

82. 23

( ) 1 22

f x x

Vertex: ____________

Aos: ______________

x-int: ______________

yint: _______________

domain:____________ range:______________

inc:________________ dec: _______________

83. 2( ) 2 18 40f x x x

Vertex: ____________

Aos: ______________

x-int: ______________

yint: _______________

domain:____________ range:______________

inc:________________ dec: _______________

84. ( ) ( 5)( 2)f x x x

Vertex: ____________

Aos: ______________

x-int: ______________

yint: _______________

domain:____________ range:______________

inc:________________ dec: _______________

Simplify the following:

85. 2 6 3

3 22

(2 )

12

x y

x y 86.

3

4 3 4 6

4 15

3 8

xy xy

x y x y

87.

23

2 5

1

a b

c

Simplify the following:

88. 3 2 3 2 3 2(10 4 3) [( 3 2 1) (4 5 3)]x x x x x x 89. 2( 2)( 2 4)x x x

90. ( 2)( 3)( 2)x x x 91. (4 3 )(4 3 )x x

92. 2(3 2 )x y 93. 3 2(5 )x

94. 3(5 2)x 95. 3(3 5 )x y

96. 2[ 2 ]x a 97. (a 1)(a 1)b b

Solve the following by factoring. All factors must be prime. List only indicated zeros.

98. 327 8 0x (all) 99. 38 125 0x (all)

100. 3 25 10 2 0x x x (real) 101. 332 108x (real)

102. 3 22 4 3 6 0x x x 103. 3 22 2 1 0x x x

(all) (all)

104. 481 16 0x (all) 105. 4 29 23 12 0x x (real)

106. 5 36 22 8 0x x x (all) 107.

2

6 12 01 1

x x

x x

(real)

108. 10 9 9 0x x 109. 2 4100 4 0x x 110. 5 23 24x x

(real) (real) (all)

111. Divide using long division and write your answer in proper ( )

( )

f x

d x form.

4 2 2( 2 3 8) ( 3 1)x x x x x

112. Simplify the expression using either long division or synthetic. Your answer should be in proper polynomial form. 3 24 8 3

3

x x x

x

113. Given that 2 2( ) ( 5)( 2)( 1)f x x x x ,

a. List all real zeros

b. List all rational zeros

c. List all irrational zeros

d. List all non-real zeros.

114. Use the factor theorem to determine whether 5x is a factor of 3 2( ) 3 5 15p x x x x . Write the proper

factored form of this polynomial.

115. Find (2)f for 5 3( ) 3 4 1f x x x x using the remainder theorem.

116. List all possible zeros of 3 2( ) 7 11 40 48f x x x x

117. Write a polynomial of least degree with a leading coefficient of 1, that has real coefficients and zeros of 5 and 1+ 3i.

Answer should be in standard form.

118.

21( ) ( 2) ( 1)

3f x x x

End behavior:

Zeros, multiplicity of each zero, and behavior based on

multiplicity: relative max or mins: increasing interval: ___________ decreasing interval: ___________

Additional points:

domain: _______________ range: ______________

119. 3 2( ) 2 5 6f x x x x

End behavior:

Zeros, multiplicity of each zero, and behavior based on multiplicity: relative max or mins: increasing interval: ___________ decreasing interval: ___________

Additional points:

domain: _______________ range: ______________

Simplify the expression.

120. (30𝑥3 − 6𝑥2) ÷90𝑥2−15𝑥

3𝑥2+21𝑥+36 121.

(𝑎3+𝑎2𝑏)3

(𝑎3−𝑎2𝑏)3 ÷

(𝑎+𝑏)2

(𝑎−𝑏)2 ∙

𝑎3−𝑏3

𝑎3+𝑏3

122. 3

𝑎−

7

𝑏+

2

𝑐 123.

3𝑦+1

2𝑦−2−

4𝑦−3

5𝑦−5

124. 𝑥−3𝑦−3

𝑥−2−𝑦−1 125.

𝑥−1−𝑥+1

2𝑥+1

𝑥+1

2𝑥+1

126. 𝑎(𝑎−𝑏)3

𝑎3−𝑏3 ÷ 𝑎4+2𝑎3𝑏+𝑎2𝑏2

𝑎2+𝑎𝑏+𝑏2 127. 2−𝑥

9−𝑥2 − 𝑥+3

𝑥2−9+

3

2𝑥+6

128. 𝑥2−9

𝑥3−27 ÷

𝑥3+27

𝑥2+3𝑥+9 129.

𝑥2+7𝑥𝑦+12𝑦2

𝑥3+𝑥2𝑦 ∙

𝑥5

𝑥2+6𝑥𝑦+9𝑦2

Solve the following.

130. 3

𝑥+1=

𝑥

𝑥2−5 131.

𝑦

𝑦−4 −

4

𝑦+7=

20

𝑦2+3𝑦−28

132. 𝑥+2

𝑥2+3𝑥+

𝑥+4

𝑥2−3𝑥 =

𝑥+1

𝑥2−9 133.

2𝑥+6

𝑥+4− 2 =

2𝑥+20

2𝑥+8

134. Graph the following:

4

( )2

f xx

y-int. ___________

x-int: ___________

va: ___________

ha: ___________

add’l pts:

Domain_______________________

Range ___________________________

Discontinuity: ____________________

End behavior (using limits):

135. 2

2

5 4( )

2 3

x xf x

x x

y-int. ___________

x-int: ___________

va: ___________

ha: ___________

add’l pts:

Domain_______________________

Range ___________________________

Discontinuity: _____________________

End behavior (using limits):

136. 2

3( )

4

xf x

x

y-int. ___________

x-int: ___________

va: ___________

ha: ___________

add’l pts:

Domain_______________________

Range ___________________________

Discontinuity: _____________________

End behavior (using limits):

137. Write the following in exponential form: 𝑙𝑜𝑔1

100= −2

Evaluate the following:

138. 𝑙𝑜𝑔51

625 139. 7𝑙𝑜𝑔75𝑧 142. 𝑙𝑛𝑒−

3

5

145. 𝑙𝑜𝑔123451 146. 𝑙𝑜𝑔999 √(999)34 147. 𝑙𝑜𝑔5(−5)

Solve each of the following:

148. 64𝑥−1 = . 1252𝑥 149. 𝑙𝑜𝑔101(𝑙𝑜𝑔3(𝑙𝑜𝑔2𝑥)) = 0

150. 𝑙𝑜𝑔(𝑥2 − 9) − 𝑙𝑜𝑔(𝑥 − 3) = 𝑙𝑜𝑔8 151. 𝑙𝑜𝑔3(𝑥 − 1) = 𝑙𝑜𝑔9(2𝑥 + 1)

152. 𝑙𝑜𝑔3(𝑥 + 1) + 𝑙𝑜𝑔3(𝑥 + 1) = 2 153. 2𝑒5𝑥 − 1 = 6

154. 2𝑙𝑜𝑔4𝑥 + 𝑙𝑜𝑔4(𝑥 − 1) = 𝑙𝑜𝑔4(−5𝑥 + 5) 155. 𝑙𝑜𝑔16𝑥 − 𝑙𝑜𝑔165 = −5

4

Solve each of the following:

156. 3𝑥+1 = 7𝑥−2 157. 1 − 𝑒𝑥+1 = 5

158. 𝑙𝑛(𝑥 − 3) + 𝑙𝑛(𝑥) − 𝑙𝑛(𝑥 + 2) = 0 159. 1

5𝑙𝑜𝑔81(𝑥 − 3) =

1

4

160) Graph the following exponential functions without a calculator. Show x- and y-intercepts, if any, and find domain, range, equation of the horizontal asymptote, and end behavior limits for each function.

(a) 1

( )3

x

f x

(b) 2( ) 3xf x (c) ( ) 2 1xf x

Domain: __________________ Domain: ___________________ Domain: _________________ Range: ___________________ Range: _____________________ Range: ___________________ Eqn of Asymptote: __________ Eqn of Asymptote: ___________ Eqn of Asymptote: _________ x-intercept: ________________ x-intercept: _________________ x-intercept: ______________ y-intercept: ________________ y-intercept: _________________ y-intercept: ______________

lim ( ) ____________

lim ( ) ___________

x

x

f x

f x

lim ( ) ____________

lim ( ) ___________

x

x

f x

f x

lim ( ) ____________

lim ( ) ___________

x

x

f x

f x

(d) ( ) 2 xf x (e) ( ) 3 xf x (f) 2( ) 1xf x e

Domain: __________________ Domain: ___________________ Domain: _________________ Range: ___________________ Range: _____________________ Range: ___________________ Eqn of Asymptote: __________ Eqn of Asymptote: ___________ Eqn of Asymptote: _________ x-intercept: ________________ x-intercept: _________________ x-intercept: ______________ y-intercept: ________________ y-intercept: _________________ y-intercept: ______________

lim ( ) ____________

lim ( ) ___________

x

x

f x

f x

lim ( ) ____________

lim ( ) ___________

x

x

f x

f x

lim ( ) ____________

lim ( ) ___________

x

x

f x

f x

161) Graph the following exponential functions without a calculator. Show x- and y-intercepts, if any, and find domain, range, equation of the vertical asymptote, and end behavior limits for the first three functions. Then, graph and show the vertical asymptote and two points for the last three functions.

(a) 3( ) log ( 1) 2f x x (b)

3( ) log ( 1)f x x (c) 5( ) logf x x

Domain: __________________ Domain: ___________________ Domain: _________________ Range: ___________________ Range: _____________________ Range: ___________________ Eqn of Asymptote: __________ Eqn of Asymptote: ___________ Eqn of Asymptote: _________ x-intercept: ________________ x-intercept: _________________ x-intercept: ______________ y-intercept: ________________ y-intercept: _________________ y-intercept: ______________

lim ( ) ____________

lim ( ) ___________

x

x

f x

f x

lim ( ) ____________

lim ( ) ___________

x

x

f x

f x

lim ( ) ____________

lim ( ) ___________

x

x

f x

f x

(d) 3( ) log ( )f x x (e) ( ) ln 1 2f x x (f) ( ) log( ) 2f x x

lim ( ) ____________

lim ( ) ___________

x

x

f x

f x

lim ( ) ____________

lim ( ) ___________

x

x

f x

f x

lim ( ) ____________

lim ( ) ___________

x

x

f x

f x

Write each expression in radical form.

162. 33

4 163. 2𝑥−2

3 164. (3𝑎)1.75 165. 𝑥 − 3/5

Write each expression in exponential form.

166. (√𝑎𝑥 + 𝑦5 )2 167. 1

( √𝑥+𝑦3 )5

Simplify the following. If you begin in radical form your answer should be in radical form. If

you begin in rational form, your answer may be in rational exponent form or radical form. Your

answers must be in simplest form.

168. (𝑎3

5)−2

5 169. (1024𝑎𝑏12)1

10

170. (3𝑥−5

12) (1

27𝑥

1

3 ) (1

5𝑥

3

7) 171. 9𝑑

− 74

3𝑑−

15

Simplify the following. If you begin in radical form your answer should be in radical form. If

you begin in rational form, your answer may be in rational form or radical form. Your answers

must be in simplest form.

172. (𝑥10)

52

𝑥−1 173. ( √81

12) ( √243

12)

√72912

174. √5

81

7 175. 3√160

4+ 5√10

4

Solve the following equations:

176. 3(𝑥 − 1)34 = 81 177. 2 = √𝑥 − 2 − √10 + 𝑥

178. √𝑥 − 25

− √3 − 𝑥5

= 0 179. √3𝑥 − 2 = 1 + √𝑥

180. −105 = 138 − 3(𝑐 − 5)4

3

Graph the following and state the domain and range.

181. 𝑓(𝑥) = √−𝑥 − 1 − 2

Domain: __________ Range: ___________

Graph the following and state the domain and range.

182. 𝑓(𝑥) = −3√𝑥 + 2 − 1

Domain: __________ Range: ___________

183. 𝑓(𝑥) = (𝑥 + 3)3 + 1

Domain: __________ Range: ___________

Graph the following and state the domain and range.

184. 𝑓(𝑥) = −2√𝑥 − 33

+ 2

Domain: __________ Range: ___________

185. Find the inverse of 𝑓(𝑥) and graph the inverse below.

𝑓(𝑥) = (𝑥 − 2)3 + 1

Domain: __________ Range: ___________

186. Write the standard equation for each Conic Section. Graph and identify the important characteristics.

a. 022 xyy b. 036422 yxyx

Type of Conic: Type of Conic:

Standard Equation: Standard Equation:

Vertex: Center:

Focus: Radius:

Directrix:

Focal Width:

c. 056832 22 yxyx d. 2 22 2 8 4 8y x y x

Type of Conic: Type of Conic:

Standard Equation: Standard Equation:

Center: Center:

Vertices: Radius:

Co-vertices:

Foci:

e. 4844 22 xxyy

Standard form equation:

Center:

Asymptotes:

Vertices:

Co-vertices:

Foci:

f. 3216164 2 yxx g. 0164244 22 yxyx

Type of Conic: Type of Conic:

Standard Equation: Standard Equation:

Vertex: Center: Vertices:

Focus: Co-Vertices:

Directrix: Foci:

Focal Width: Asymptotes:

h. 04484 22 yxyx

Type of Conic:

Standard Equation:

Center:

Vertices:

Co-vertices:

Foci:

187. Write the following absolute value functions as piecewise functions.

a. 𝑓(𝑥) = |𝑥| b. 𝑓(𝑥) = |𝑥2 − 7|

c. 𝑓(𝑥) = |𝑥3 − 𝑥2 − 4𝑥 − 4| d. 𝑓(𝑥) = |3 − √1 − 𝑥3

|

e. 𝑓(𝑥) = |10 + 4√3 + 𝑥| f. 𝑓(𝑥) = |−(𝑥 + 3)(𝑥 − 1)|

g. 𝑓(𝑥) = |−3(𝑥 + 2)2 − 7| h. 𝑓(𝑥) = |𝑙𝑜𝑔𝑥|

i. 𝑓(𝑥) = |𝑒𝑥−3 + 2| j. 𝑓(𝑥) = |𝑥3 + 5𝑥2 − 8𝑥 − 12|

k.𝑓(𝑥) = |3 + 2𝑥| l.𝑓(𝑥) = |2𝑥2 + 3𝑥 + 5|

m. 𝑓(𝑥) = |1

𝑥| n. 𝑓(𝑥) = |

𝑥−3

2−𝑥|

188. Graph a function with the following characteristics:

a. Even Symmetry b. Odd Symmetry

increasing: (−∞, 0) decreasing: (−∞, 0) ∪ (0, ∞)

decreasing: (0, ∞)

c. lim𝑥→−∞ 𝑓(𝑥) = ∞ d. x-intercept: (2,0)

lim𝑥→∞ 𝑓(𝑥) = ∞ y-intercept: (0,1)

Absolute min of -5 at x=2 jump discontinuity at x=1

Removable Discontinuity at x = -3 Domain: (−∞, ∞)

Range: (−∞, 3)

e. Infinite discontinuities at x=-2 and x =2

lim𝑥→−∞ 𝑓(𝑥) = 1

lim𝑥→∞ 𝑓(𝑥) = 1

x-intercepts: (-3,0) and (3,0)

Relative min of 2 at 𝑥 =1

4

Calculator Portion:

1. Go back to each function that you have had to graph in this summer assignment and

verify that your graph is correct with your graphing calculator. Also, verify that all the

key characteristics of your graph are correct: x-intercepts, y-intercept, domain, range,

relative max/min, intervals of increasing/decreasing.

2. 20 gallons of a 60% acidic solution are diluted with water to make a 40% acidic solution.

How much water was used and how many gallons were contained in the final solution?

3. A chemist needs to make a 900mL of a 30% alcohol solution by mixing 28% and 55%

solutions. How much of each should she use?

4. Sarah travels for 100 miles and then decreases her rate of travel by 5 mph and continued

the remaining 300 miles of her trip. If the total trip took 7 hours and 40 minutes, at what two

speeds did she travel?

5. Coach Johnson can run a mile in 4 minutes and 20 seconds and Brandon D’Arienzo can run a

mile in 5 minutes and 9 seconds. Together, how fast can they run a mile?

6. Peter can make a pizza in 15 minutes and John can make a pizza in 25 minutes. How long

will it take them to make 20 pizzas? (Assume working together doesn’t slow down their rates)

7. A gardener works on a landscaping job for 10 days and then is joined by an associate.

Together they finish the job in 5 more days. The associate could have done the job alone in 30

days. How long would it have taken the gardener to finish the job working alone?

8. Graph each function on your calculator and identify the following information:

a. rel max/min b. increasing and decreasing behaviors

3 2( ) 3 2f x x x 4 3 2( ) 4 12 2g x x x x x

9. The attendance A (in millions) at the NCAA women’s college basketball games for the years 1997 through 2003 is

shown in the table , where t represents the year, with t = 7 corresponding to 1997. (source: NCAA)

year, t attendance, A

1997 6.1

1998 7.0

1999 8.0

2000 8.3

2001 8.7

2002 9.1

2003 10.0

Use the regression facility on your calculator to find a cubic model for the data.

According to your model, in which year did attendance reach 7.5 million?

In which year did it reach 10 million?

According to your model, will attendance continue to increase? Explain.

In what year was the attendance 6 million?

10. The revenue for selling x units of a product is R= 1270x. The cost of producing x units is C= 550x + 12000. To

obtain a profit, the revenue must be greater than the cost. Determine the smallest number of units for which

this product returns a profit.

11. You have 1000 feet of fencing to enclose a rectangular garden. You want the length of the garden to be 40 feet greater than the width. Find the length and width of the garden if you use all of the fencing. (hint: draw a picture first). Label your variables, show your equation, and show all your work.

12. The table shows the average sale price p of a house in Suffolk County, Massachusetts, for various years t since 200.

Find a quadratic model for the data.

Years since

2000

0 2 4 6 8 10

Price (in

thousands

of dollars

200 215 240 255 230 210

a. what was the maximum sales price and in what year did it occur?

b. In what year(s) was the average price $235,000?

c. According to your model what was the average house price in the year 2015?

13.) Given the census data for the city of San Diego,California:

Let 1950 be 0t .

t (years) 1950 1960 1970 1980 1990

P (thousands) 234 473 597 776 1011

(a) Find a linear model that best fits the census data for the city of San Diego, California.

(b) Use the model from part (a) to predict the population in the year 2000 for San Diego.

(c) In what year does your model predict the population will be 1 million?

(d) Is your model a good fit for the data? How do you know?

14.) The following table gives the number of cassette tapes (in millions) sold in the United States for the even numbered

years 1988 through 1996. Let 1980 be 0t .

t (years) 1988 1990 1992 1994 1996

S (millions) 440.1 432.2 356.4 335.4 205.3

(a) Find a linear model that best fits the data. Explain the meaning of the slope for this linear model.

(b) Is your model a good model? How do you know?

(c) How many cassette tapes does your linear model predict will be sold in the year 2000? Is this a reasonable

number based on your understanding of the market place?