# Brien McMahon High School by 26hw29

VIEWS: 0 PAGES: 35

• pg 1
```									                    Brien McMahon High School
SUMMER REVIEW
ALGEBRA 2/ HONORS ALGEBRA 2

This packet is designed for you to review your Algebra 1 skills and make
sure you are well prepared for the start of your Algebra 2 school year.
Every topic includes an explanation, examples, and practice problems
designed to ensure your readiness for next year. This packet is expected
to be completed upon your arrival back to school and will be counted as a

*It is strongly recommended that calculators NOT be used to complete all
sections of this packet since the objective of this packet is to verify your
understanding of the concepts. Please show all of your work as this will
be required of you throughout the entire school year.

**It is strongly recommended to have a TI-83 or TI-84
graphing calculator for Algebra 2**

Honors Algebra 2 – Complete sections I – XI

Algebra 2 – Complete sections I - VIII

Unit 10
I : Real Numbers and their Properties
Number Systems:

Natural Number (Counting Numbers) 1, 2, 3, …
Whole Numbers (Introduce Zero) 0, 1, 2, 3, ….
Integers (Introduce Negative Numbers) …, -3, -2, -1, 0, 1, 2, 3, …..

Note: The above number systems do not contain fractions or decimals.

Rational Numbers (Numbers that can be written as the ratio of two integers.) When written as
decimals, Rational Numbers terminate or repeat.

2                 3                   7
Examples:        = 0.666….         = 0.75                7
3                 4                   1

Irrational Numbers - Number that when written as decimals neither terminate nor repeat.

Examples:  = 3.1415...           5  2.236067978

Real Numbers (All numbers on the number line.) This system combines the Natural #s, Whole
#s, Integers, Rational #s, and Irrational #s.

Properties of Real Numbers
Let m,n, and p be positive real numbers.

COMMUTATIVE                      m+n=n+m                        mn = nm
ASSOCIATIVE                      (m + n) + p = m + (n + p)      (mn)p = m(np)
IDENTITY                         m + 0 = m, 0 + m = m           m x (1) = m, 1 x (m) = m
INVERSE                          m + (-m) = 0                       1
m x   =1, m  0
m
DISTRIBUTIVE                                          m(n + p) = mn + mp

Additive inverse: the opposite of any m = - m

Example:         the additive inverse of 4 is -4               the additive inverse of -3 is 3

1
Multiplicative inverse: the reciprocal; the inverse of m is
m

1                                            5 6
Examples:       the multiplicative inverse of       =3         the multiplicative inverse of    
3                                            6 5

Unit 10
Practice Problems for Topic I:

Complete the chart by placing a check in all the boxes that apply.

Natural Whole   Integers Rational Irrational Real
Numbers Numbers          Numbers Numbers Numbers
-4
8
3
2
5
1
3

Identify the property being performed:

1. 5  4  4  5                   2. 3(2  4)  4(2  3)

3.  4(1)  4                     4. 10  10  0

1
5. 4(6  9)  4  6  4  9        6.     5 1
5

7. 0  36  0                       8. (9  7)(5)  2(5)

9. (7  4)  1  7  (4  1)      10. 5(12  4)  5(12)  5(4)

11. (.5)  (2)  1                12. (0.3)(0.3)  0

13. 36 jkm  36mjk                  14.  29c  29c  0

Unit 10
II. Orders of Operations
The rules for Order of Operations are as follows:
FIRST: Perform operations inside grouping symbols. Grouping symbols
include parentheses , brackets , braces , radical
symbols , absolute value symbols and fraction bars. If an
expression contains more than one set of grouping symbols,
simplify the expression inside the innermost set first. Follow the
order of operations within that set of grouping symbols and then
work outward.
SECOND: Simplify exponents.
THIRD: Perform multiplication and division from left to right. (Remember
that a fraction bar also indicates division.)
FOURTH: Perform addition and subtraction from left to right.
Hint: You can use the well-known phrase "Please Excuse My Dear Aunt Sally" to help you remember
the Order of Operations. (Remember, however, that multiplication and division must be done in the
order that they appear if they do not appear in parentheses. This is also true for addition and
subtraction.)

Please Excuse My Dear Aunt Sally
Parentheses Exponents Multiplication Division Addition Subtraction

Example 1 Simplify.
- 4 2  24  3  2            (Note that there are no grouping symbols. Therefore the exponent only
applies to the "4" and not the "-".)
 - 16  24  3  2
                      16  8  2 
                      16  16 
                    

Example 2 Simplify.
4 (25 - (5 - 2) 2 )
4(25 - (3) 2 )
4(25 - 9)
4(16)
64

Example 3 Simplify.

3x – 4(8x + 2) + 5x                           distribute the -4 first
3x – 32x – 8 + 5x                              combine all like terms
-24x – 8

Unit 10
Practice Problems for Topic II:
Simplify each expression using the orders of operations.

3(4  11)
1.) 6  9  3                   2.) 4( 2  3) 2                3.)
10  3  3

4.) 47  36  8  2           5.) 4( x  2)  3x             6.) 3(2 x  5)  (5  3x)

3  2  3  1  14  2
7.)                            8.) 2(x-1) – 5(x-3)             9.) 2(5 – 2)2 + 2(4 – 2)2
63 62

Evaluate each expression if a = 2, b = 3, c = -2, d = 4, f = -5

10.) cf                         11.) c  d                       12.)  ca 2

13.) (b  cd )(ac)              14.) (cf )  (ab)                 15.) dc a

16.) a2b2                       17.) 5c + 3bc                     18.) a – b – c – d

Unit 10
Simplify each of the following.
Hint – remember to use orders of operations and to combine like terms

19.)   3x – 4 + 7x – 8 – 10x – 2                         20.)    4x + 2(x + 5)

21.)   3(x + 5) – 4(x – 6)                               22.)    5x3 + 2x2 -7x – x3 + 5x2 – 18

23.)   (5x2 + x – 4) – (9x2 – 4x – 11)                   24.)    12x3(x4 – 5x2 + 2)

25.)   (x + 2)(x2 – 5x + 1)                              26.) 5xy – 20x + 6y – (10xy + 10x)

27.) 2x(3x2 – 4x + 2) – 5x(2x + 3)                       28.) 5x2y2 + 3xy – 3x2y2 + 2x3y2 + 5xy

Unit 10
III.       Plotting Points on the Coordinate Plane

Suppose you were told to locate "(5, 2)" where would you look?

To understand the meaning of "(5, 2)", you have to know the following rule: The x-coordinate
(the number for the x-axis) always comes first. The first number (the first coordinate) is always
on the horizontal axis.

So, for the point (5, 2), you would start at
the "origin", the spot where the axes cross:

...then count over to "five" on the x-axis:

...then count up to "two", moving parallel to
the y-axis:

...and then draw in the dot:

*Remember: “- x” values moves left and “– y” values move down

Unit 10
Practice Problems for Topic III:
Plot the following points of the grid below and label them accordingly:

1.) A - ( 4, -5 )        2.) B - ( 0, -4 )        3.) C - ( -3, -1)       4.) D - ( -2, -2 )

5.) E - ( 3, 0 )         6.) F - ( 5,1 )          7.) G -( -3, 7 )        8.) H - ( 10, -2 )

9.) I – ( -1,-4)       10.) J – (0,2)             11.) K – (-3,10)        12.) L – ( 4,0)

13.) M – (-2,1)        14.) N – (3,4)            15.) P – (0,0)

Unit 10
IV: Solving Linear Equations
Solving with a variable on only one side of an equations:

Ex)     -2x + 7 = 15             subtract 7 from both sides
-2x = 8                   divide -2 from both sides
x = -4
-2(-4) + 7 = 15           check your solution using substitution

8 + 7 = 15
15 = 15

Solving with a variable on both sides of the equation:
Ex)     8x – 10 = 2x + 14              Move the smaller x value to the other side
6x – 10 = 14                  (subtracted 2x from both sides)
6x = 24                    add 10 to both sides (your non x variable)
x=4                             divide both sides by 6
8(4) – 10 = 2(4) + 14                check by substitution
32 -10 = 8 + 14
22 = 22

Solve using the properties of real numbers:
Ex)    -3( 2x + 5 ) + 6x = 11x + 7          Use the distributive property
-6x - 15 + 6x = 11x + 7      Combine like terms (on the same side of the equation)
-15 = 11x + 7                      subtract 7 from both sides
-22= 11x                               divide both sides by 11
-2 = x
-3( 2(-2) + 5) + 6(-2) = 11(-2) + 7           check by substitution

-3(-4+5) - 12 = -22 + 7
-3(1) - 12 = -15
-15 = 15

Unit 10
Practice Problems for TOPIC IV:
Solve the following equations and check your solutions.

1.) -3x + 4 = 11                                        2.) 1/2x - 8 = 3

3.) 8x – 24 = -6 + 18                                   4.) -6 + x – 4 = 7x + 2

2
5.)      x + 6 = 18                                    6.) 6 – x – 5 = -4x – 3 – x
3

b
7.) -5 +     =7                                          8.) 4.2m +4 = 25
4

9.) 4t + 7 + 6t = -33                                   10.) 6 = -z – 4

11.) 4m + 2.3 = 9.7                                     12.) 2a + 5 = 9a – 16

1 4   2
13.)    + y=                                             14.) 6(y – 2) = 8 – 2y
3 6   3

m
15.) n + 3(n – 2) 10.4                                  16.)      = 2.7
12
Unit 10
17.) -4.7 = 3x + 1.3                                   18.) 3w + 2 – w = -4

19.) Solve 2x -4 = -7. Justify each step.

20.) A taxicab company charges each person a flat fee of \$1.85 plus an additional \$.40 per
quarter mile.

A. Write a formula to find the cost for each fare.

B. Use the formula to find the cost for 1 person to travel 8 mi.

21.) Find the dimensions of the rectangle given the area = 164 sq. ft.

3x+5

4

Unit 10
V: Using Slope and y-Intercept to Graph Linear Equations
and to Write Equations of Lines
There are 3 forms to write an equation of a line:

Slope intercept form: y = mx + b        m = slope, b = y intercept

Point slope form: ( y-y1 )=m( x-x1 )        (x1, y1) is a point on the line

Standard form: Ax + By = C

y intercept: where x = 0                 x intercept: where y = 0

*to find these values plug either x = 0 or y = 0 into the equation and solve for the other value

I.) Graphing in slope intercept form, y = mx + b:

Ex. 1      y = ( 2/3 )x – 4

Step 1: Identify “b”           (0,-4)

Step 2: Plot “b” on the grid

Step 3: Use the slope to plot your next point (rise/run) ( 3, -2) , ( 6,0 )

Step 4: Continue step three until at least 3 points are plotted.

Unit 10
Ex. 2

y = –2x + 3

1.) Plot “b” (0,3)

Down 2 and right 1

3.) Repeat this for 3 points

Ex. 3                                                                        Ex. 4

X=3         This is a vertical line with an undefined slope      y = 4 This is a horizontal line with a slope = 0

II.) Writing equations of lines in slope intercept form:
Ex. 1 – Write and equation of a line that passes through the point (2,1) and has a slope of 3.

y = mx + b          plug 3 in for m, 2 in for x, and 1 in for y
1 = 3(2) + b          simplify
1=6+b                 solve for b by subtracting 6
-5 = b
y = 3x – 5          plug 3 in for m and -5 in for b

Unit 10
Given two points (x1, y1) and (x2, y2), the formula for the slope of the straight line going through these
two points is:

Ex. 2 - Write an equation of a line that passes through the points (2, 4) and ( 5, 6 ).
6–4                   start by finding slope using the formula above

plug the slope and one of the points into the Point Slope Formula
y – 4 = 2/3( x – 2 )     simplify by distributing the 2/3
y – 4 = 2/3 x – 4/3      add 4 to both sides to get y by itself
y = 2/3 x + 8/3

Parallel Lines: two lines that have the same slope and never intersect
Ex.      y = 2x + 3    is parallel to   y = 2x -10

Perpendicular Lines: two lines whose slopes are opposite reciprocals
Ex.    y = 2x + 3     is perpendicular to    y = -1/2 x    *2 and -1/2 are opposite reciprocals

Practice Problems for Topic V.
Sketch each of the lines on the graph below.

1.      y = 2x                                                         2.   y = -3x

Unit 10
3.         4.

5.   y=6        6.   x = -4

7.              8.

Unit 10
9. y  3x  2       10. y = -4x+6

11.         x=2     12.             y = -5

13. 2x + 3y = 12   14. -3x + 27y = 8

Unit 10
15. Complete the table and graph the points on the graph below.

16. An hot air balloon is currently at an altitude of 10,000 feet. The pilot begins to descend
the balloon at a rate of 50 feet per minute.

1.     Write an equation for the altitude (a) of the balloon as a function of the time (t)

2.     Find the altitude of the balloon after:

a.     10 minutes            b.     30 minutes             c.    1 hour

3.     How long will it take for the balloon to be:

a.     7000 feet             b.     2000 feet              c,    on the ground

Unit 10
4.     Complete the table and graph the altitude of the balloon as a function of the time in
minutes that the balloon takes to begin its descent and land on the ground.

17. This summer you and a friend recorded your first CD. In order to produce your CD you need to
hire a recording studio. One studio you called charges \$250 for making a master CD and \$3 to
burn each CD after that.

(a.) Write an equation to model the cost of burning the CDs.      ________________________

(b.) How much will it cost to burn 75 CDs? ________________________

(c.) How many CD’s can be made for \$3000? _______________________

More practice for Topic V.
1.     y = 4x – 2                                  2.      6x – 3y = 15

slope = _______                                    slope = _______

y-intercept = _______                              y-intercept = _______

x-intercept = _______                              x-intercept = _______

Unit 10
3.     y = 2x – 5                                    4.      4x + 3y = 12

slope = _______                                      slope = _______

y-intercept = _______                                y-intercept = _______

x-intercept = _______                                x-intercept = _______

5.     3y = 4x – 9                                   6.      5x – y = 15

slope = _______                                      slope = _______

y-intercept = _______                                y-intercept = _______

x-intercept = _______                                x-intercept = _______

Find the slope and y-intercept for each line given below. Then write the equation in

slope-intercept (y = mx+b) form.

7.     A line that passes through the point (-5,6) with a slope of 4.

8.     A line that passes through the points (4,1) and (7,-11)

9.     A line is perpendicular to the line in #2 above and passes through the x-axis at 3.

Unit 10
Find the slope and y-intercept and x-intercept of each line. Then write the equation of each line
in y = mx + b form.

10.                                                   11.

Slope = _________                                       Slope = ________

y-intercept : ________                                  y-intercept: _________

equation: y = _________________                         equation: y = ________________

x-intercept: _______                                    x-intercept:_______

12.                                                      13.

Slope = _________                                        Slope = ________

y-intercept : ________                                   y-intercept: _________

equation: y = _________________                          equation: y = ________________

x-intercept: _______                                     x-intercept:_______

Unit 10
14.                               15.

Slope = _________                 Slope = ________

y-intercept : ________            y-intercept: _________

equation: y = _________________   equation: y = ________________

x-intercept: _______              x-intercept:_______

16.                               17.

Slope = _________                 Slope = ________

y-intercept : ________            y-intercept: _________

equation: y = _________________   equation: y = ________________

x-intercept: _______              x-intercept:_______

Unit 10
18. A tanker that is filled with oil pulls into an oil refinery. The tanker is going to offload the oil as
shown in the graph below.

1.     How many gallons of oil are in the tanker when it pulls into the refinery?_____________

2.     At what rate, in gallons per hour, is the tanker unloading the oil?___________________

3.     Write an equation for the amount of oil (l) after h number of hours: l = _____________

4.     Find the amount of oil after:

a.      15 hours                                b.      25 hours

5.     How long will it take for the tanker to have:

a.      60,000 gallons left                     b.      25,000 gallons left

6.     How long will it take?

Unit 10
Multiplying Binomial Expressions: F.O.I.L. (First, Outer, Inner, Last)
FOIL is the method by which two binomials can be multiplied using the distributive property.

Example 1: Multiply the binomials using FOIL.
x 3x 2Take the first term of the first binomial and distribute it through the second binomial.
Then, take the second term in the first binomial and distribute it through the second binomial.

 x  x  x  2  3 x  3 2 
 ( x 2 )  (2 x)  (3x)  (6)

 ( x 2  5 x  6)

Example 2: (x – 4)(x + 5)

 ( x  x)  ( x  5)  (4  x)  (4  5)
 ( x 2 )  (5 x)  (4 x)  (20)
 x 2  1x  20

Factoring Quadratic Trinomials:                                 (x 2  bx  c) :
When factoring quadratic trinomials with no coefficient in front of the x 2 , ask yourself this
question:
What two numbers add to give "b" and multiply to give "c"?

Example:           Factor: x 2  7x  12

What two numbers add to give "7" and multiply to give "12"?
Obviously, the answer to this question is 3 and 4.
Therefore, the factors of x 2 7x 12 are x 3x 4.

Example 2:           Factor: x 2 - 10x  16

What two numbers add to give "-10" and multiply to give "16"?
The answer to this question is -2 and -8.
Therefore, the factors of x 2 - 10x  16 are x 2x 8.

PARABOLA: graph of a quadratic function

ROOTS: Value of x when y = 0 (where the graph crosses the x axis)

VERTEX: maximum or minimum point on a parabola

Unit 10
To graph a parabola: Fill in the t-chart (function table) by choosing values of x, then plug
those values into your function to find y, then plot your ordered pair.

Example:

x             x 2  4x  5       y

-2             (-2)2-4(-2)-5      7

-1             (-1)2-4(-1)-5      0

0             (0)2-4(0)-5       -5

2             (2)2-4(2)-5       -9

5             (5)2-4(5)-5        0

The vertex is (-2,9), the lowest point of the graph.

The roots are (-1,0) and (5,0), the two places where the parabola crosses the x axis.

Practice Problems for Topic VI:
Foil:

1.) (x – 2)(x + 6)              2.) (3x + 6)(x – 1)      3.) (x + 9)(-2x – 7)    4.) (5x + 1)( 3x – 2)

Factor:

1.)   x 2  x  56              2.) x 2  14  33        3.) x 2  7 x  10      4.) x 2  11x  12

5.) x  9 x  20                  6.) x 2  10 x  16       7.) x 2  x  56       8.) x 2  19 x  90
2

9.) x  14 x  49                10.) x 2  x  72         11.) x 2  2 x  24     12.) 2 x 2  5 x  3
2

13.) x  4 x  4                  14.) x 2  10 x  25     15.) x 2  9            16.) x 2  36
2

Unit 10
Complete the following chart and graph the function. Identify the vertex and the roots.

1.)
x    x 2  4x  4     y

2.)        x    x 2  6x  2     y

3.)
x    x 2  7x  6     y

Unit 10
VII. Properties of Exponents

Unit 10
VIII. Solving Systems of Equations
Solving systems using: Linear combination

Given the equation:

3x – 4 = 7x + 8

-3x          -3x        subtract 3x from both sides to get the x terms together

-4 = 4x + 8

-8           -8        subtract 8 from both sides to get the term by itself

-12 = 4x

4       4               Divide both sides by 4 (opposite operation) to solve for x.

-3 = x                Solution

Substitution

Given two linear equations: y = 5x -1 and -2x + 3y = 23

y = 5x -1 substitute (replace) y in the equation -2x + 3y = 23 with 5x -1

-2x + 3(5x -1) = 23

-2x + 15x -3 = 23         Distribute (multiply) 3 to both 5x and -1

13x – 3 = 23              Combine like terms (-2x and 15x)

+3      +3           Add three to both sides

13x = 23                 Divide by 13 to solve for x.

x=2                Take 2 and substitute it into the equation y = 5x – 1, for x

y = 5x – 1

y = 5(2) – 1             Solve for y.

y = 10 – 1               so, y = 9

Your solution then is x = 2 and y = 9 written (x,y) as (2,9)

Unit 10
GRAPHING:

To solve the system by graphing, graph both linear equations on the same coordinate plane and their
intersection point is their solution.

Practice problems for Topic VIII.
Solve the following systems by the indicated method.

Linear Combination
1. 4x + 3 = 2x – 8        2. 9x – 5 = -6x + 25         3. -2x – 1 = -3x + 11     4. 10x – 40 = 6x

Substitution
1. y = -x + 7            2. y = 3x – 5                 3. x = y + 5              4. y = -7x
2x + 3y =14             -4x + y = -11                  2x – 4y = 2              4x – 5y = -39

Graph
1. y = -3x – 2               2. y = -2x + 5               3. y = -x                    4. y = 3x + 2
y = 2x + 8                   y = 2x – 13                  y=x+5                         y = -3x – 4

Unit 10
IX. Solving Linear Inequalities

EX 1: Solving linear inequalities is almost exactly like solving linear equations.
 Solve x + 3 < 0.

Then the solution is: x < –3
-3         0

When the inequality sign is just greater than (>) or less than (<), we use an open circle to show the answer
does not include this number.

EX 2: Here is another example, along with the different answer formats:
 Solve x – 4 > 0.

Then the solution is: x > 4
0               4

When the inequality sign is just “greater than or equal to” (  ), or “less than or equal to” (  ), we use a
closed circle to show the answer includes this number.

   Solve –2x < 5.

To solve "–2x < 5", I need to divide through by a negative ("–2"), so I will need to flip the inequality:
* When solving inequalities, if you multiply or divide through by a negative, you must
also flip the inequality sign.

2 x 5                        5
                  x >
2  2                        2

Practice Problems for Topic XI:
1. x – 7 < 1                                          2. x + 9 ≤ -6

3. -4y + 2 > 14                                       4. 16 – 3x ≥ 4

Unit 10
5. Wireless company A has a plan that charges \$20 initial fee and \$0.05 for every call. Wireless company B
has a plan that has NO initial fee but charges \$0.25. How much will it cost in each of the two different plans
if you make 80 calls one month and then 110 calls the next month?

Write two different equations, one for each plan
and then graph both.

Describe when you would use each of the different
plans.

Unit 10
X.Graphing Linear Inequalities

Practice problems for Topic X:

Unit 10
Unit 10
XI. Absolute Values and Their Graphs

The absolute value of a number is its distance from zero. The graph of an absolute equation has a “V” shape because
for any input value of “x” (either positive or negative) the output must be the corresponding positive “y” value. Think
of the linear function y = x. It is a line of positive slope 1, going through the origin. You can calculate that if you input
a positive “x” you’ll get a positive “y” out, a negative “x” input gives a negative “y” out. What happens with the
function y = |x| ? Positive in yields positive out, but what about negative values for “x” ? They turn positive also,
thus creating the “V” shaped graph.
To produce an absolute value graph on your graphing calculator, go to your “y=” section (functions) , clear existing
equations, press the MATH key, scroll over to the NUM pulldown, and choose “abs( “ . Enter “x”, and close
parentheses. Press the ZOOM key and then press the number“5”. On screen you should have a 90degree “V” shape
with its vertex at the origin. This is the most basic standard form of an absolute value equation. Any alteration will
cause a transformation.

Practice Problems for Topic XI:
1) y = abs(x) + 5         2) y = abs(x) – 5           3) y = abs(x – 5)           4) y = abs(x + 5)

What has happened in each case? (how has the figure moved?)

1) ____________________________________________________________________________________________

2) ____________________________________________________________________________________________

3) ____________________________________________________________________________________________

4) ____________________________________________________________________________________________

For number three and four, did the figure move in the direction you expected? _____________________________

Now try the following:      5) y = .5 abs(x)          6) y = .75 abs(x)           7) y = 2 abs(x)             8) y = 5 abs(x)

What has happened in each case? (how has the figure changed?)

5) ____________________________________________________________________________________________

6) ____________________________________________________________________________________________

7) ____________________________________________________________________________________________

8) ____________________________________________________________________________________________

What can you say about the effect of numbers between zero and one on the shape of the figure? ______________

______________________________________________________________________________________________

What can you say about the effect of numbers greater than one on the shape of the figure? __________________

Unit 10
Write the absolute value equation for each graph shown.

1. y = _____________________      2. y = ____________________   3. y = ____________________

4. y = _____________________      5. y = ____________________   6. y = ____________________

7. y = _____________________      8. y = ____________________   9. y = ____________________

Unit 10
10. y = ____________________   11. y =___________________   12. y = ___________________

Unit 10

```
To top