Algebra 2 Summer Packet
***Directions: Complete all work on a separate piece of paper. Make sure to show all work. ****
 Section 1-4 Solving Absolute Value Equations
Absolute Value: The distance between the number and 0 on the number line.
Now try: 2.7
+
6
-
2 x if x=4 ANSWER: 4.7
2.
5.
9
= x
+
12
22 x
+
1
x
=
3 x
-
4
3.
6.
-
2 3 a
=
6
32 x
+
2
-
2 x
= x
+
3
1
For any real numbers a and b, where
5 x
-
6
+
9
=
0 b
³
0
, if a
= b
, then a=b or a=-b
Let’s Try:
ANSWER: The absolute value of a number is always positive or zero, an equation like x
= -
8 is never true.
Therefore, it has no solution. The solution set for this type of equation is the empty set, or  .
Practice: x
+
10
=
4 x
-
8
ANSWER: x
=
6
only, x
= -
2
is an extraneous solution
5
Even if the correct procedure for solving the equation is used, the answers may not be actual solutions to the original equation. Such a number is called an extraneous solution. It is because extraneous solutions exist that we must check our answers in the original problem to ensure that they meet the constraints or conditions of the problem.
Practice 1-4:
Solve each equation.
1.
4.
8
+ y
=
2 y
-
3
4 b
+
1
+
8
=
0
 Section 1-5 Solving Inequalities
Inequalities: statements comparing two quantities.
For any two real numbers, a and b, exactly one of the following statements is true: a<b a=b a>b
The solution to an inequality is the set of numbers that make the inequality true.
The procedures for solving an inequality are the same as those for solving an equation, except any time the
inequality is multiplied or divided by a negative number, the inequality sign must be flipped.
Set Builder Notation:
{ y y
<
3
}
is read the set of all numbers y such that y is less than 3.
Interval Notation: Uses ¥ or -¥ to show that the set is unbounded in the positive or negative direction. For example y<3 would be expressed as (
-¥
,3) , which is read the set of all real numbers from -¥ to 3.
Note: While using interval notation, to indicate the endpoint is not included in the set, a parenthesis, ( or ), is used. Parenthesis are always used with the symbols ¥ and -¥
To indicate the endpoint IS included, a bracket, [ or ], is used. Ex
. y
£
3 would be expressed as -¥
,3
] (
When graphing an inequality on a number line, the starting point is noted with either less than or equal to or greater than or equal to ( £
or
³ ).
· or . The open circle is used when the value is strictly less than or greater than (< or >). The closed circle is used when the value is

1. -3(4w-1) >18
3. 9(2 r
-
5)
-
3
<
7 r
-
4
Algebra 2 Summer Packet
2. ³ -
3.5
2
4. 4n −5(n −3) > 3(n+1)-4
2
Practice 1-5:
Solve each inequality. State your answer using set builder notation AND interval notation. Graph the solution on a number line.
4 x
-
3
 Section 1-6 Solving Compound and Absolute Value Inequalities
Compound inequalities: two inequalities joined by the word and or the word or.
“And” statements are only true at the intersection of the two pieces or where both are true.
Example 1: x
³ -
4 and x < 3
Draw a number line to represent where this compound inequality exists.
State solution set using set builder and interval notation.
ANSWER: Interval Notation:
[
-
4, 3)
Set Builder Notation:
{ x
-
4
£ x
<
3}
“Or” statements are true where one or more of the inequalities are true. This is referred to as the union of the solution sets. x
£
3 or x
>
7 Example 2:
Draw a number line to represent where this compound inequality exists.
State solution set using set builder and interval notation.
ANSWER: Interval Notation:
(
-¥
, 3](7,
¥
)
Set Builder Notation:
{ x x
£
3 x
>
7}
Absolute Value Inequalities
For all real numbers a, b, c and x, c>0, the following statements are true.
Absolute Value Inequality ax
+ b
> c ax
+ b
< c
Compound Inequality ax
+ b
> c
or ax
+ b
< c
c
< ax
+ b
< c
Example
If
2 x
-
2
>
4
, then or
If
2 x
-
2
< -
4
2 x
-
2
<
4 , then
-
4
<
2 x
-
2
<
4
Remember: GO LA! (Greater Than=Or, Less Than=And)
Writing an absolute value inequality given a graph.
To write an absolute value graph given the graph, take the following steps.
1.
Identify the middle of the shaded part of the line. This will tell you the shift
2 x
-
2
>
(
4 x
± shift <,>,
£
,
³
2.
The distance is from the center point to an endpoint.
3.
Remember that unions are OR and intersections are AND
4.
Remember that the open and closed circles represent
) distance different things and it is important to demonstrate those differences.

Algebra 2 Summer Packet
3
Practice 1-6:
Solve the following compound inequalities. State solution set using set builder and interval notation. Graph your solution on a number line.
1. 10
£
3 y
-
2
<
19 2. x
+
3
<
2 or
x
£ -
4
Solve each inequality. State solution set using set builder and interval notation. Graph the solution set on a number line.
3. h
+
1
³
5
Write an absolute value inequality for each graph.
5. 6.
4.
3 n
-
2
-
2
<
1
 Section 2-1 Relations and Functions
Function: a relation in which each element of the domain is paired with exactly one element in the range.
One-to-one function: Each element of the domain pairs to exactly one unique element of the range. Ex.
(1, 2), (2, 3), (3, 4)
Onto function: Each element of the range corresponds to an element of the domain. Ex. (1,2), (2, 2),
(3, 3), (4, 6)
One-to-one and Onto: Each element of the domain is paired to exactly one element of the range, and each element of the range element corresponds to a unique element of the domain.
Example: State the domain and range of the relation {(-4, -2), (-3, 1), (0, -2), (1, 2), (3, 3)}. Then determine whether each relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. ANSWER: Domain: { -
4,
-
3, 0,1, 3
} Range: { -
2,1, 2, 3
} function, one-to-one
Discrete Relation: a relation in which the domain is a set of individual points.
Continuous Relation: a relation in which the domain has an infinite number of elements and the relation can be graphed with a line or a smooth curve.
Vertical line test: if no vertical line intersects a graph in more than one point, the graph represents a function. If a vertical line intersects a graph in two or more points, the graph
Equations that represent functions are often written in function notation. For example, y=2x+3 can be written as f(x) = 2x + 3. Function notation emphasizes the fact that the y values, the dependent variables, depend on the values of x, the independent variable.
Example: Given the function f(x) = x 2 + 2 , find f(5) and f(9). ANSWERS: f (5)
=
27
, f (9)
=
83
Practice 2-1:
State the domain and range of each relation. Then determine whether each relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither.
1. {(-6, -1), (-5, -9), (-3,-7), (-1, 7), (6,-9)}
Find each value, if
3. g(5)
6. f(2a) f ( x )
= -
2 x
+
4, and g ( x )
= x
3
4. f(- 3)
7. g (2)
× g (3)
2. {(2, -2), (-1, -1), (-2, 0), (-1, 0), (2, 2)}
x
.
5. g (-2)
8. g (2
×
3)

Algebra 2 Summer Packet
 Section 2-2 Linear Relations and Functions
Linear Relations: a relation that has a straight line graph.
Nonlinear Relations: relations that are nonlinear.
4
Linear Equation: has no operations other than addition, subtraction, and multiplication of a variable by a constant. The variables may not be multiplied together or appear in a denominator. A linear equation does not contain variables with exponents other than 1. The graph is always a line.
Linear Function: a function with ordered pairs that satisfy a linear equation. Any linear function can be written in the form f ( x )
= mx
+ b
, where m and b are real numbers.
Example 1: State whether each function is a linear function. Write yes or no. Explain. a.
3 x
+
4 y
=
12
ANSWERS: a) yes, can be written in form b.
1
+
3 y
= -
5 x f ( x )
= mx
+ b
. b) no, cannot divide by a variable
Standard form of an equation:
Ax
+
By
=
C
, where A, B, and C are integers with a greatest common factor of 1, and A is positive.
Example 2: Write the equation -
2 x
= -
2 y
+
4
3
ANSWER:
2 x
-
6 y
= -
12
, a
=
2, b
= -
6, c
= -
12 in standard form. Identify A, B, and C.
The x-intercept of a graph is in the point on the coordinate plane at which the graph of intersects the xaxis. Remember on the x-axis, the y-value is always 0.
The y-intercept of a graph is in the point on the coordinate plane at which the graph of intersects the yaxis. Remember on the y-axis, the x-value is always zero.
Example 3 Find the x-intercept and y-intercept of the graph of 3x-2y = 24.
ANSWER: y-intercept (
0,
-
12
) , x-intercept
8, 0
Practice 2-2:
State whether each function is a linear function. Write yes or no. Explain.
1. f ( x )
=
8
-
3 x
2. f ( x )
=
2 x
3. f ( x , y )
=
3 xy
-
4
4
Write each equation in standard form. Identify A, B, and C. Then, find the x-intercept and the y-intercept of the graph of each equation.
4. 5x – y= 10
6. 2y+7x=14
5. y = 3x+7
7. y = -2x
 Section 2-3 Rate of Change and Slope
Rate of Change: ratio that compares how much one quantity changes, on average, relative to the change in another quantity. If x is the independent variable and y is the dependent variable, then the rate of change= change in y
. This is sometimes expressed as . change in x

Algebra 2 Summer Packet
5
Slope: the ratio of the change in y- coordinates to the corresponding change in x- coordinates. The slope of a line is the same as its rate of change. Suppose a line passes through y x
2
2
y
x
1
1
.
Facts about slope:
Vertical lines have Undefined slope or No Slope
( x
1
, y
1
)
and
( x
2
, y
2
)
, then slope=
Horizontal lines have 0 slope.
Parallel lines have the same slope.
Perpendicular lines have opposite reciprocal slopes.
Practice 2-3:
Find the slope of the line that passes through each pair of points
1. (5, 10) and ( -1, -2)
3. (6, 4) and ( 3, 4)
2. (5, -3) and (-4, 3)
4. (1, 9) and ( 0 , 6)
Identify the lines parallel, perpendicular, or neither?
5. The line passing through (4,3) and (1, -3) and the line passing through (1, 2) and (-1, 3)
6. The line with x-intercept 1 and y-intercept 3 and the line with x-intercept 3 and y-intercept 1
Slope Intercept form:
Point- Slope form:
( ) + b y y
y
= mx
+ b
 Section 2-4 Writing Linear Equations
, where m is the slope and b is the y-intercept.
= m ( x
x )
Example:1 Write an equation in slope-intercept form for the line that has slope -3/2 and passes through
(-4, 1) y
= mx
+ b
1 1
Example 2 : Write an equation in slope-intercept form of the line through (-1, 4) and (-4, 5). m
=
5
- -
4
-
4
- -
1 y
= mx
+ b
=
-
9
3
= -
3
1
æ
è
ç
2
3
ö
ø
÷ -
4
5
=
(
-
3)(
-
4)
+ b
5
=
12
+ b
1
=
6
+ b -
7
= b
-
5
= b y
= -
3 x
-
7 y
= -
2
3
ö
ø
÷ x
-
5
Practice 2-4:
Write an equation in slope-intercept form of the equation given the following information.
1. Slope=3, and passes through (0, -6)
3. Slope=-1, passes through (0,0)
2. Passes through (-2, 5) and (3, 1)
4. Passes through (-1, -2) and (-3, 1)
5. x-intercept= 5/2, y-intercept=5

Algebra 2 Summer Packet
 2.6 Special Functions
6
Piecewise-Defined Function – A function that is written using two or more expressions. A dot indicates that the point is included on the graph, a circle indicates that a point is not included on the graph.
Example 1: Graph f ( x )
=
îï x
-
2 if x
< -
1 x
+
3 if x
³ -
1
. Identify the domain and range.
Answer: (See graph)
Domain: All real numbers since any value of x can be input
Range:
( -¥
,
-
3
)
[2,
¥
) or
{ y y
< -
3 or y
³
2
}
since the y values can only equal values that are less than -3 and more than 2.
Example 2: Graph f ( x )
=
ï
î
ì
ï
í
2 x
+
3 if x
<
1
x
+
2 if 1
£ x
£
2
3 if x
>
2
. Identify the domain and range.
Answer: (See graph)
Domain: All real numbers
Range: y
<
5 or
( -¥
, 5
)
since all values of y are below 5
Piecewise-Linear Function – otherwise known as a step function or a greatest integer function. Consists of line segments
Greatest Integer Function: f ( x )
= [ ]
.
This means the greatest integer, less than or equal to x.
Graph shown on side
Absolute Value Function – a function that contains an algebraic expression within absolute value symbols.

Algebra 2 Summer Packet
7
2.6 Practice
1. A recycling company pays $5 for every full box of newspaper. They do not give any money for partial boxes.
Draw a graph that shows the amount of money P(x) for the number of boxes x brought to the recycling center.
2. A certain psychologist charges for counseling sessions at the rate of $85 per hour or any fractional hour thereof. Draw a graph that represents this situation.
3. Graph
4. Graph y
= x
+
1 . Identify the domain and range. f ( x )
= x
-
2 . Identify the domain and range.
 2.7 Families of Functions
What is a parent function? The simplest of graphs in a family of functions. This is the graph that is transformed to created other members in a family of graphs.
Translation: Moves a graph up, down, left or right

Algebra 2 Summer Packet
2.7 Practice
Describe the translation of the function, then graph the function.
1. y
= x
2 -
4
2. y
= x
+
1
3. y
= ( x
-
5
) 2 4. y
= x
+
3
8
 2-8 Graphing Linear and Absolute Value Inequalities
To graph a linear inequality, graph the linear equation associated with the inequality. If < or > draw line with a dotted line. If ≤ 𝑜𝑟 ≥ then draw a solid line. Then shade the appropriate area.
Example: Graph 𝑥 − 2𝑦 < 4
ANSWER. Graph y
>
1 x
-
2
with a dotted line. Shade above (see graph).
2
To graph an absolute value inequality follow the same rules for linear equality.
Graph the absolute value, using the appropriate dotted or solid line. Then shade appropriately.
Example: Graph. 𝑦 ≥ |𝑥 − 2|
ANSWER: Graph y
= x
-
2
, solid line, shade above (see graph).
Remember that the shaded region represents the values that make the statement true. The shaded region is your solution set.
Practice 2-8:
Graph each inequality.
1.
2 x
+
4 y
£
12
2. x
-
3 y
<
6
3. y
+
3
³ x
+
1 4. y
<
2 x
-
2
 Chapter 4 Summer Packet.
1. For help ( https://www.youtube.com/watch?v=mDwN1SqnMRU )
Complete parts A-C for each quadratic function.
A) Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex.
B) Make a table of values that includes the vertex.
C) Use this information to graph the function.
A) f ( x )
=
4 x
2 B) f ( x )
= -
2 x
2
+
3 x
+
9
2. Determine whether the function has a maximum or minimum value, and find that value. State the domain and range of the function.
For help (http://www.khanacademy.org/math/algebra/algebrafunctions/domain_and_range/v/domain-and-range-of-a-function-given-a-formula)
A) f ( x )
=
5 x
2 B) f ( x )
=
2 x
2 -
16 x
-
42

Algebra 2 Summer Packet
Use a calculator to find the maximum and minimum value, and find that value. Then state the
9 domain and range of the function. For help (https://www.youtube.com/watch?v=Hpir10S33kM)
3. f ( x )
= -
9 x
2 -
12 x
+
19
4. . Last year, the Sports Time Athletic Club charged $20 to participate in an aerobics class. Seventy people attended the classes. The club wants to increase the class price this year. They expect to lose on customer for each $1 increase in the price.
A) What price should the club charge to maximize the income from the aerobics classes?
B) What is the maximum income the Sports/Time Athletic Club can expect to make?
5. Solve each equation by finding the roots. For help
(https://www.youtube.com/watch?v=W2oL52t0NVo) and
(https://www.youtube.com/watch?v=saXEJW1Y1Oo)
A) x
2 =
5 x
B)
2 x
2 -
3 x
-
15
=
4
C) x
2 -
18 x
= -
81
Use a quadratic equation to find two real numbers that satisfy the situation or show that no such numbers exists. To help (https://www.youtube.com/watch?v=TlV2XSJure8)
6. Their sum is 4 and their product is -117.
7. Use the formula h(t)=
V
0 t
-
16 t
2 , where h(t) is the height of an object in feet ,
V
0 is the object’s initial velocity in feet per second, and t is the time in seconds. For help
(https://www.youtube.com/watch?v=DZAwJVHrk1E)
A baseball is hit with an initial velocity of 80 feet per second. Ignoring the height of the baseball player, how long does it take for the ball to hit the ground?
8. Factor. For help ( https://www.youtube.com/watch?v=sJu_EykJuMU ),
(https://www.youtube.com/watch?v=pU98xk5TBPM),
( https://www.youtube.com/watch?v=T0ItI7q6Q1U ),
(https://www.youtube.com/watch?v=AYkaCZUT4O4)
A.
C. x
2
9 y
2
-
7 x
+
12
B. x
=
6 y
-
1
D. x
2
2
+
5 x
=
0
-
25
=
0
E.
G.
6
4 x x
2
2
-
15 x
-
8 x
+
20
F. x 3 + 125
+
37 x
+
9
 3.1 Solving Systems of Equations
Solve Systems Graphically A system of equations is two or more equations with the same variables. You can solve a system of linear equations by graphing the equations on the same coordinate plane. If the lines intersect, the solution is that intersection point. The following chart summarizes the possibilities for graphs of two linear equations in two variables.

Graphs of Equations
Lines intersect
Lines coincide (same line)
Lines are parallel
Algebra 2 Summer Packet
Slopes of Lines
Classification of
System
Different slopes
Same slope, same yintercept
Same slope, different yintercepts
Consistent and independent
Consistent and dependent
Inconsistent
Number of
Solutions
One
Infinitely many
None
Solve each system of equations by graphing. Describe the system of equations as consistent and independent, consistent and dependent, or inconsistent.
Example 1.
2 x
-
5 y
=
11
-
3 x
+
4 y
= -
13
Step 1: Graph both equations
**Tip: Rewriting equation in the form y
= mx
+ b
may help to quickly
10 identify patterns with slope
Step 2: Find intersection (if it exists)
ANSWER: (see graph) (
3,
-
1
)
Solve Systems Algebraically
SUBSTITUTION: To solve a system of linear equations by substitution, first solve for one variable in terms of the other in one of the equations. Then substitute this expression into the other equation and simplify.
Example 1: Use the substitution method to solve the system of equations.
2 x – y = 9 x + 3 y = – 6
Step 1: Isolate a variable. Typically the variable without a leading coefficient that is positive is the easiest.
(In this example, that would be the x in the second equation. i.e. x
= -
3 y
-
6
)
Step 2: Substitute what is equal to that variable into the other equation. (Example in the first equation instead of writing 2x, I will substitute and rewrite the first equation as
2(
-
3 y
-
6)
y
=
9
Step 3: Solve for the variable you have now. (You can now do this since you have one equation with one
-
6 y
-
12
y
=
9 unknown). -
7 y
=
21 y
= -
3
Step 4: Once you have one of the one of the variables, plug that in to either of the equations and solve for
2 x
y
=
9 the other variable.
2 x
-
(
-
3)
=
9
2 x
+
3
=
9
2 x
=
6
. So the solution (
3,
-
3
) x
=
3

Algebra 2 Summer Packet
ELIMINATION: To solve a system of linear equations by elimination, add or subtract the equations to
11 eliminate one of the variables.
Example 2: Use the elimination method to solve the system of equations.
3 x – 2 y = 4
5 x + 3 y = – 25
Step 1: Multiply one or both equations by a number to result in two equations that contain opposite terms.
Step 2: Add the equations, eliminating one variable. Then solve the equation.
Step 3: Substitute to solve for the other variable.
Practice 3.1
Solve by graphing.
1.
4 x
+
3 y
= -
17
-
7 x
-
2 y
=
20
2.
-
2 x
+
5 y
=
10
4 x
-
10 y
= -
20
3.
6 x
-
4 y
=
15
-
3 x
+
2 y
=
9
Use substitution or elimination to solve the system of equations.
4.
2 x
+
3 y
=
5
5.
5 x
+
3 y
=
52
6 x
+
9 y
=
15 15 x
+
9 y
=
54
6 Alex has a computer support business. He estimates that the cost to run his business can be represented by y = 48x + 500, where x is the number of customers. He also estimates that his income can be represented by y = 65x – 145. How many customers will Alex need in order to break even? What will his profit be if he has 60 customers?
7. Lancaster Woodworkers Furniture Store builds two types of wooden outdoor chairs. A rocking chair sells for $265, and an Adirondack chair with footstool sells for $320. The books show that last month, the business earned $13,930 for the 48 outdoor chairs sold. How many rocking chairs were sold?
 3.2 Solving Systems of Inequalities by Graphing
Systems of Inequalities To solve a system of inequalities, graph the inequalities in the same coordinate plane. The solution of the system is the region shaded for all of the inequalities.
Example 1: Solve the system of inequalities. y
³
2 x
-
3 y
< x
+
2
**Graph each inequality and find the intersection.
Find Vertices of an Enclosed Region Sometimes the graph of a system of inequalities produces an enclosed region in the form of polygon. You can find the vertices of the region by a combination of the methods used earlier in this chapter: graphing, substitution, and/or elimination. a

Algebra 2 Summer Packet
Example 2: Find the coordinates of the vertices of the triangle formed by
2 x
y
³ -
1, x
+ y
£
4 , x
Step 2: Find the intersections.
ANSWER: (1,3), (4,0), (0,1)
+
4 y
³
4
.
Step 1: Turn all inequalities into equations.
3.2 Practice
1. Solve the system of inequalities. y
³ -
3
4 y
£ -
3 x
+
1 x
-
2
4
12
2. Medical professionals recommend that patients have a cholesterol level below 200 mg/dl and a triglyceride level below 150 mg/dl. Write and graph a system of inequalities that represents the range of cholesterol levels c and triglyceride t for patients.
3. Find the coordinates of the vertices or the triangle formed by
5 y
£
2 x
+
9 y
£ x
-
6
9 y
³ -
2 x
+
5