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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
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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
+ − − − + +
+ + − − − + −
− + − + − +
− − + +
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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.
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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_________________
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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"
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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
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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 ————————————
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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
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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
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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
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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:_________
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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