1  

Immaculate  Heart  Academy  Summer  Math  Assignment  for  Algebra  I/Pre-­‐Calculus  Honors  

           

           LEARN                           EXCEL  

     

   You   are   taking   Algebra   II/Pre-­‐Calculus   Honors   in   the   fall.     A   mastery   of   and  proficiency  in  performing  the  following  Algebra  and  Geometry  skills  will  be  necessary  for   success   in   this   Algebra   II/Pre-­‐Calculus   honors   level   course.   Referencing   class  notes  from  Algebra  I/Algebra  II  Honors  and  Geometry  Honors  will  be  very  helpful  in  doing   the  problems  presented   in   this   review.  Work  on  each  problem   in  order.    Copy  the  problem  onto  loose-­‐leaf  paper  except  where  you  are  directed  to  show  all  work  in  the  space  provided  or  to  present  graphs.    Show  all  work  in  a  neat  and  organized  manner.     Box   in   your   final   answer.     Complete   this   entire   assignment   and  bring   it   to  class  on  the  first  day.      This   assignment   is   mandatory.     We   recommend   that   you   periodically   go   to   this  packet  during  the  summer  rather  than  attempting  to  do  all  of  it  in  your  last  week.    That  will  allow  you  to  really  process  these  important  skills.    You  will  be  given  a  proficiency  test   within   the   first   week   of   school   on   the   topics   in   this   assignment.     If   you  demonstrate  mastery  of  these  topics  (grade  of  90  or  better)  you  will  be  awarded  a  bonus  point  at  the  end  of  the  first  quarter.    The  most  significant  reward  to  you  will  be  your  smooth  transition  into  Algebra  II  /  Pre-­‐calculus  Honors  this  September!!      At  the  end  of  this  assignment  are  several  links  to  websites  that  you  might  find  helpful  should  you  have  any  problems  with  your  assignments.      Name:  ______________________________________________    Date:  ____________________      Math  Class  last  year:___________________________________________________      Teacher:  _______________________________________________________________________  

PRACTICE        

  2  

I. Perform  the  indicated  operation(s).  You  must  find  the  LCD  and  simplify  your  answer.      Do  not  use  a  calculator.  

 

 

     II. Perform  the  indicated  operation(s).      Simplify  the  result.    Do  not  use  a  calculator.  

1. 3x8÷ 1

22. 8x ÷ − 1

4⎛⎝⎜

⎞⎠⎟

3. − 24x ÷ − 23

⎛⎝⎜

⎞⎠⎟

4. −22

− 13

5.

1234

 

6.

235

7. − 39 ÷ −4 13

⎛⎝⎜

⎞⎠⎟

8. 42t−14z

÷ −67t  

9. 18x − 93

10. 22x +102

11. −56+ x−8

12. 45−5x5

13. 22− 4x4

14. 15x − 75

15. 20x + 35

 

 III.      SLOPE     1.    Slope  Formula:       2.    Determine  the  value  of  “y”  so  the  line  passing  through  this  pair  of  points  has                the  given  slope:      

  (−2,−3),(7, y);m = − 1

3           3.    Determine  the  slope  of  the  graph  of  the  linear  function  f,  given:                       f 3( ) = −1, f −5( ) = −1.            

3 5 7 1 2 1 1 8 2 11. 2. 3. 4.8 13 4 12 5 3 6 9 3 2

1 1 1 9 5 5 3 4 15. 6. 7.2 3 4 11 3 6 12 10 5

2 3 1 7 4 2 1 8 58. 9. 10.8 4 2 15 5 3 2 10 4

1 3 3 5 2 1 1 311. 5 2 12. 4 2 13. 9 3 14. 2 38 4 8 6 5 3 20 8

+ − − − + +

+ + − − − + −

− + − + − +

− − + +

  3  

IV. Use  the  points  A  (4,  1)  and  B  (8,  3).      Show  all  work  in  the  space  provided.  a)  Present  the  graph  of   AB .    

b)  Find  the  midpoint  of   AB                                                          Midpoint:  _________      c)  Find  the  slope  of   AB  using  the            Slope  Formula.    Confirm  by  box-­‐            counting.                                                                    Slope:___________    d)  Determine  the  slope  of  the  line  perpendicu-­‐              lar  to   AB                          Ans.:  ____________      e)  Write  an  equation  of  the  line  that  is  the  perpendicular  bisector  of   AB  in  point-­‐slope  form,  slope-­‐intercept  form  and  standard  form.          f)  Write  the  function  whose  graph  is  the  line    AB .    (Use  correct  mathematical  notation)          

       Point-­‐Slope  Form:                                                            __________________________________________          Slope-­‐Intercept  Form:________________________________      Function:________________________________________________  

   V. Rewrite  Formulas  

1.    Solve  the  formula   I = prt  for  r.                      2.    Solve  the  formula  A = 12bh  for  b.  

3.    Solve  the  formula  F = 95c + 32  for  c                    4.    Solve  the  formula  P = 2l + 2w  for  w.  

               5.    Solve  the  formula  A = 12(b1 + b2 )h  for  b1                  6.    Solve  the  formula  C = 2πr  for  r.  

               7.    Solve  the  formula  A = πr2  for  r.    

  4  

VI.    GRAPHING.    Show  all  work  in  the  space  provided.      1.    Using  the  slope-­‐intercept  method,         GRAPH:   −4x − 5 y = 15                                        m  =  ________              b  =  ________              y-­‐intercept:  ________      Any  line  parallel  to  this  line  would  have  a  slope  of  _______.    Any  line  perpendicular  to  this  line  would  have  a  slope  of    ________.    Domain  =  _________________________        Range  =  _____________________________    Define  3  points  that  lie  on  the  graph  of  this  function:    ________,    ________,  and  ________.    2.    Given:    8x  –  3y  =  –8,  graph  using  intercept                  points.    Be  sure  to  demonstrate  your  procedure  algebraically                    in  the  space  below.                                x-­‐intercept    point:________        y-­‐intercept  point:_________          

Find  all  values  of  x  where:    f x( ) > 0_________________f x( ) = 0_________________f x( ) < 0_________________

 

  5  

3.    GRAPH: f x( ) = −2 x +1 + 4   Vertex:____________              

Define  4  additional  points,  2  to  the  right  of  the  vertex  and  2  points  symmetric  to  these:  

 ____________,    ____________,    

____________,  ____________  

 Domain  =  _____________________________________  

 Range  =  _______________________________________  

     

Axis  of  Symmetry:____________________________    Confirm  your  results  by  means  of  your  calculator.      4.    GRAPH:                      

       f x( ) =

− 23x −1, if x < −3

x + 2, if − 3 ≤ x ≤ 14, if x > 1

⎧

⎨

⎪⎪

⎩

⎪⎪

 

                         

  6  

 SOLVE  this  system  graphically.  

5.        

                   Solution:__________    Confirm  your  solution  by  algebraic  substitution.    6.    SOLVE  this  system  of  linear  inequalities  graphically.        

                         

15x −10 y = −806x + 8 y = −80

Algebra 1 87Chapter 7 Resource Book

Copyright © McDougal Littell Inc. All rights reserved.

Practice CFor use with pages 432–438

7.6LESSON

NAME _________________________________________________________ DATE ___________

Lesson

7.6

Write a system of linear inequalities that defines the shaded region.

1.

Graph the system of linear inequalities.

4.

7.

10.

Find the vertices of the graph of the system.

13.

Plot the points and draw the line segments connecting the pointsto create the polygon. Then write a system of linear inequalitiesthat defines the polygonal region.

3x ! y ≤ 13"2x ! y ≤ 3x ! y ≥ 3

x < 4x ! 2y < 3"2x ! 5y < 5

y < 25 x " 2

y ≤ 0x ≥ 0

8x ! y < 04x ! 2y ≥ "6

2"6 "2

2

"6

"10

x

y 2.

5.

8.

11.

x ≥ 0y ≤ 0

34 x " y ≤ 5

"3x ! 4y < "7

x ≤ 6y > "7y ≤ 0x ≥ 0

2x ! 3y > "5 2x ! 3y < 4

3"3 "1

1

"3

"5

x

y 3.

6.

9.

12.

5x " 13 y < 2

5x " 5y ≤ 105x ! y ≥ "65x " y > "4

y ≤ "2x ≤ 44x " y ≥ 2

3x " y ≥ 2 3x " 5y > 2

31"3 "1

1

3

5

x

y

14.

x ≥ 0y ≤ 4 4x " 5y ≤ 5x ! 4y ≤ 17 15.

x ≤ 3y ≤ 4 2x ! 3y ≥ 02x ! y ≤ 6

16. Rectangle:

18. Study Time You need at least 3 hours to doyour English and history homework. You needto spend twice as much time on your historyhomework as your English homework. It is12:00 P.M. on Sunday and your friend wantsyou to go to the movies at 7:00 P.M. Write asystem of linear inequalities that shows thenumber of hours you could spend doing homework for each subject if you go to themovies. Graph your result.

17. Triangle:

19. Ordering Cups You work at a frozen yogurtshop during the summer. You need to order 5-ounce and 8-ounce cups. The storage roomwill only hold 10 more boxes. A box of 5-ounce cups costs $125 and a box of 8-ouncecups costs $150. A maximum of $1300 is budgeted for yogurt cups. Write a system oflinear inequalities that shows the number ofboxes of 5-ounce and 8-ounce cups that couldbe bought. Graph your result.

!"2, 4", !4, 1", !"4, "1"!2, 1", !"1, 4", !"5, 0", !"2, "3"

  7  

VII.    SOLVE  each  of  the  following  absolute  value  equations.    Be  sure  to  check  solutions.  

1.             2.          

 

3.        

     VIII.    Scatter  Plot  and  Linear  Regression  This  table  shows  the  number  of  audiocassette  tapes  (in  millions)  shipped  for  several  years  during  the  period  1994  –  2002.    Present  responses  in  the  spaces  provided  here.          a)  Enter  this  data  on  your  calculator  (let  x  represent  the  number  of  years  since  1994);              present  scatter  plot.    Does  your  scatter  plot  exhibit  positive  or  negative  correlation            and  WHY?        b)  Use  your  calculator  to  write  an  equation  for  the  best  fitting  line  for  this  data.                Round  to  the  nearest  one  thousandth.               Equation:_____________________________________________    c)  Using  this  equation,  write  the  function  that  models  the  number  of  tapes  shipped  (in              millions)  as  a  function  of  the  number  of  years  since  1994.                       Function:_____________________________________________              At  about  what  rate  did  the  number  of  tapes  change  over  time?    _______________________      d)  Use  this  function  to  approximate  the  year  in  which  125  million  tapes  were  shipped.                                                                          Answer:_______________    e)  Find  the  zero  of  this  function.    Explain  what  this  zero  means  in  this  situation.      

3 4x + 2 − 7 = 11 3 2x − 8 + 3 = 2

4x +10 = 6x

Year     1994   1996   1998   2000   2002  Tapes  shipped  (millions)   345   225   159   76   31  

  8  

IX.    Solve  each  of  the  following  absolute  value  inequalities.    Show  all  work  below.    Present                      final  results  in  set-­‐builder  notation  and  interval  notation.  1.    2 3x + 8 −13 < −5                        

   Set-­‐builder  Notation:________________________    Interval  Notation:___________________________          

2.           25x − 8 + 4 ≥ 12  

                 

   

   Set-­‐builder  Notation:________________________    Interval  Notation:____________________________          

Name ——————————————————————— Date ————————————

Copy

right

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75Algebra 1

Chapter 6 Resource Book

Solve the inequality. Graph your solution.

1. ⏐x 2 4⏐< 10 2. ⏐x 1 7⏐> 4.5

3. ⏐x 2 10⏐ ≤ 13 4. ⏐2x 2 5⏐> 17

5. ⏐8 2 3x⏐< 14 6. 7⏐1}2 x 1 5⏐≥ 14

7. 22⏐4x 1 3⏐< 28 8. ⏐5x 2 2⏐2 8 ≥ 23

9. 6⏐2x 1 9⏐2 14 ≤ 16 10. 3}4⏐4x 2 4⏐2 5 > 10

11. 3}5⏐10 2 5x⏐1 7 > 25 12. 21}2⏐5 2 9x⏐1 4 ≤ 210

Write the verbal sentence as an inequality. Then solve the inequality and graph your solution.

13. Seven more than 2 times the distance between x and 4 is less than 15.

LESSON

6.6 Practice CFor use with pages 398 – 403

LESS

ON

6.6

  9  

X.    SIMPLIFY  each  expression  -­‐  no  negative  exponents.    Review  the  following                  Properties  of  Exponents:        

Rule  #1:  Product  of  Powers   m n m na a a +⋅ =     Rule  #2:    Power  of  a  Power    ( )nm m na a ⋅=  

  Rule  #3:    Power  of  a  Product    ( )m m ma b a b⋅ = ⋅  

  Rule  #4:    Zero  Exponent       0 1 0a ,a= ≠  

  Rule  #5:    Negative  Exponent  Rule       1 0nna ,aa

− = ≠  

  Rule  #6:    Quotient  of  Powers     0m

m nna a ,aa

−= ≠  

  Rule  #7:    Power  of  a  Quotient     0m m

ma a ,bb b

⎛ ⎞ = ≠⎜ ⎟⎝ ⎠  

 1.      23 ⋅24     2.       7( )2 7( )3              3.     (12x) 3     4.      − 4x( )2 ⋅ 5x( )3    

5.       7x3y( ) ⋅ 2x4( )   6.         4r2s( )2 −2s2( )3            7.    m−4     8.         yx−2

 

 

9.           33x−4y3

    10.       −3t( )0 ⋅ 2s−2

         11.       4b−1

2a4⎛⎝⎜

⎞⎠⎟

−2

  12.         211

28  

   

13.         3x2z4

2xa⎛⎝⎜

⎞⎠⎟

3

  14.      18b2c

4bc3⋅ 3ab

−2

5a2c3            15.      

4x3y5

3x2y4⋅ 9x

3y2

12xy      XI. FACTOR  completely  –  remember  to  first  present  polynomial  expression  in  standard  

form  and  when  leading  coefficient  is  negative  to  factor  out  a  negative.    

1.   y2 + 3y − 4   2.    n2 +16n − 57   3.     x2 +17x + 66  4.    −45 +14 − z2   5.    12b2 −17b − 99   6.    2t 2 +17x + 66  7.    18d 2 − 54d + 28   8.    4n2 + 4n − 288   9.    a2 − b2  10.    4x2 − 9   11.    169 − x2   12.    25x2 − 49y2  13.     x3 + 5x2 + 8x + 40   14.    2x3 +18x2 − 5x − 45   15.    3x5 + 6x3 − 45x  

 

  10  

XII. SOLVE.      Remember,  if  a  quadratic  equation  cannot  be  solved  by  factoring,  you  may  solve  it  by  using  the  Quadratic  Formula  or  by  completing  the  square.  

                 19. 619x

= −2x2 + 2

             

     XIII.    GRAPH  the  following  quadratic  functions.    You  may  use  your  calculator  to  confirm  results.  

1.    GRAPH:     f x( ) = −2x2 − x + 3 .      Be  sure  to  present  the  5  key  parts  of  this  parabola.            You  must  show  all  work  in  the  space  provided  and  present  values  in  simplest  form  –  NO  DECIMALS!      

                            x-­‐intercept(s):________,  ________          

 Vertex:____________  

    Axis  of  Symmetry:_____________       y-­‐intercept:_________    

                       point  symmetric  to  y-­‐intercept:_________    Maximum  or  Minimum  Value?    Circle                                This  value  is:________________    Using  interval  notation,  write  the  domain  where  the  value  of  this  function  is:                positive:___________________________            negative:___________________________    Using  set-­‐builder  notation,  write  the  domain  where  the  value  of  this  function  is  zero:                         ________________________________  

1.     (2x − 3)(x + 7) = 0   2.    5(x + 3)(2x − 5) = 0   3.     x2 − x − 2 = 0  4.     x2 + 7x +10 = 0   5.     x2 − 9x = −14   6.    2x2 − 9x − 35 = 0  7.     7x2 −10x + 3= 0   8.    2x2 +19x = −24   9.    10x2 + x −10 = −2x + 8  

10.    x3− x5= 3   11.    − 3

8y = 6   12.− 4

9(2x − 4) = 48  

13.    −(8h − 2) = 3+10(1− 3h)   14.    −2x2 + 5x = 3x2 −10x   15.    −9x2 + 35x − 30 = 1− x  16.    6x2 − 8x + 3= 0   17.    8x2 + 4x + 5 = 0   18.    3(x −1)2 = 4x + 2  

 

  11  

2.    Given:     f x( ) = 2x2 − 8x +14      

     a)    Write  this  function  in  vertex  form.    Neatly  present  your  procedural  steps  in  the  space  below.                                              Vertex  Form:____________________________________________    b)    GRAPH:               c)    Define  the  vertex:______________  

      Label  the  vertex  on  your  graph.        d)    Define  the  axis  of  symmetry:___________  

             Label  the  axis  of  symmetry  on  your  graph.         e)  In  the  space  below,  demonstrate  the  use  of  your      vertex  form  above  to  generate  coordinates  of  two  more      points.    Plot  these  points  and  their  reflections  in  the  axis  of    symmetry.                          f)    Define  the  coordinates  of  your  4  additional  points:           _________,      _________,    _________,      and  ____________.  

 

3.     f x( ) = − 12x + 4( ) x − 2( )      

     x  –  intercepts:    ________  and  ________          

         

     Vertex:    __________        Axis  of  Symmetry:___________________                          y-­‐intercept:_________                    point  symmetric  to  y-­‐intercept:_________      

  12  

XIV.    Quadratic  Formula/Discriminant  1.    Demonstrate  the  use  of  the  Quadratic  Formula  to  solve  each  of  the  following  equations.                Present  your  solutions  rounded  to  the  thousandths  for  part  (b).      

a)    15x2 − 8x +1 = 0       b)      4x2 − 3 = 10x                  Solutions:________________________________     Solutions  in  Simplest  Radical  Form:                                                                                                                                                                                                                              ____________________________________________________    

Solutions  in  Decimal  Form  (rounded  to  the                                                                                                                                                                                                                              thousandths):                                                                                                                                                                                                                              ____________________________________________________      2.    Demonstrate  the  use  of  the  discriminant  to  determine  whether  equation  has  two  solutions,  one            solution  or  no  solution.  

a)    x2 + 2x − 3 = 0                                            b)    3x2 + 8x + 7 = 0        Answer:______________________________       Answer:______________________________  

 c)      4x2 + 20x + 25 = 0        Answer:______________________________  

   

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XV.    Real  World  Applications.    All  work  should  be  done  in  the  space  provided  below                    each  problem.  

A. Modeling  with  mathematical  functions:  For  each  write  a  verbal  model  that  describes  the  given  scenario,  label  each  unknown  and  then  write  an  algebraic  function  that  will  model  the  information  in  each  problem.    Use  the  function  and/or  write  an  equation  that  you  will  use  to  answer  each  question.    Answer  each  question  using  a  complete  sentence  in  correct  units.    

1. In  outer  space,  the  distance  an  object  travels  varies  directly  with  the  amount  of                            time  that  it  travels.    An  asteroid  travels  3000  miles  in  6  hours.      

 Verbal  Model:      “Let”  Statement(s):      Algebraic  Model:          

         Demonstrate  the  use  of  this  model  to  determine  the  distance  traveled  by  an  asteroid              after  10  hours.    

   2.    A  climber  is  on  a  hike.    After  2  hours,  he  is  at  an  altitude  of  400  feet.    After  6  hours,              he  is  at  an  altitude  of  700  feet.    What  is  his  average  rate  of  change  in  feet  per  hour?          

Verbal  Model:      “Let”  Statement(s):      Algebraic  Model:  

                   At  this  rate  determine  the  climber’s  altitude  at  the  end  of  2  additional  hours.                                            Ans.:____________________  

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3.    A diver jumps from a cliff that is 48 feet above the ocean with an initial velocity of 8 feet per second. How long will it take the diver to enter the water? a) Using the Vertical Motion Model, present a model specific to this problem. b) Demonstrate the use of your model to determine how long it will take the diver to enter the water. Answer:________________                                        4.    A room’s length is 3 feet less than twice its width. The area of the room is 135 square feet. What are the room’s dimensions? Be sure to include labeled sketch, “let” state- ments, equation, and neatly executed solution.

Length = ________________ Width = _________________

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5. The height (in feet) of a rocket after t seconds can be measured by the function

. You may use your calculator to confirm results. a) Algebraically determine how long it takes b) Algebraically determine what this

this rocket to achieve its maximum height. rocket’s maximum height is. Answer:___________________ Answer:___________________ c) Approximately how long is the rocket in the air?

d) What are the domain and range of this function? e) Draw a quick sketch of the function that models this scenario. Show only the vertex, x-intercepts, and y- intercept. Label the x and y-axes in correct units that pertain to this problem.

                             

 

2( ) 16 720h t t t= − +

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XVI. Write  in  simplest  form.    No  decimal  answers  are  allowed.      

 1.     32    

2.     56   3.     15 ⋅ 3   4.3 27  

 5.     15 ⋅ 3    

6.    1625

     

   

XVII.  Review  of  Right  Triangle  Trigonometry:    Based  on  the  ratios  of     30° − 60° − 90° and 45° − 45° − 90°  triangles  and  definitions  of  sinθ ,  cosθ ,  tanθ ,  csc  θ ,  sec  θ and  cotθ ,  solve  each  of  the  following  triangles.    Simplify  your  answers.    Rationalize  the  denominator.      No  decimal  answers.                            Solve  for  x  and  y:    1.             2.             3.                        

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   4.             5.  a)  Solve  for  x                    b)  Find  the  6  trigonometric  ratios:                                  sinθ,cosθ, tanθ,cscθ ,secθ ,cotθ                6.    a)  Solve  for  x                    b)  Find  the  6  trigonometric  ratios:  sinθ,cosθ, tanθ,cscθ ,secθ ,cotθ