ALGEBRA 2 MIDTERM REVIEW 1 Name: ______________
Show all work for all problems.
TOPIC 1: LINES, LINEAR EQUATIONS & INEQUALITIES
Forms of Equations of Lines:
General ( Standard ) Form:
Ax  By  C
Point – Slope Form:
y  y1  m( x  x1 )
Vertical Line:
xa
Slope-Intercept Form:
y  mx  b
Horizontal Line:
yb
Parallel lines have same slopes
Perpendicular lines have slopes that are opposite signed reciprocals
Slope = change in y/change is x
1. Line l contains the points (1, 5) and (4,-1).
(a)
Determine the slope of the line. _____________
(b)
Write an equation for the line in point-slope form. _____________________
(c)
Rewrite the equation for the line in slope-intercept form. _____________________
2. Write an equation of the horizontal line that goes through the point (-7, 10).
_____________________
3. Line k passes through the point (-3,1) and is parallel to the line y  3x  1 . Write an equation for line k.
_____________________
1
4. Find the slope of the following line:
 12 x  2 y  34
_____________________
5. Line m is perpendicular to y  2 x  1 and passes through the (4,9). What is the equation of line m?
_____________________
6. Graph the following:
a) y  2 x  1
b) 2 x  3 y  6
2
c) 2  y 
1
x
3
(recall, you flip the sign when you multiply or divide by a negative!) Don’t forget to shade!
7. Solve the following equations. Show your work and check your answers.
(a) (2 x  1)  (4 x  6)  8
(d) Solve for x:
2ax  b
d
c
(b) 4  7 x  2 x  5(4  9 x)
c)
(e) Solve A = P + Prt
f).
for P
5
6
x  32  114
2x + 3
- 6 = -3
5
__________________
3
g.
2x  4
 7  3
5
h.
2
3
5
x 1  x 
3
4
6
8. Evaluate each expression if a = -5, b = - 2, c = 3, and d = 4
a. ab2 - 9(bc) 2
c.
9.
______________
b.
3a + 4c
2c
_____________
a 2  b  c2
a) (3, 3), (-3, 2), (4, 1), (0, 1), (3, 4)
Is this relation a function?
Domain:_____________________________
Range: _____________________________
10. Use the graph to determine the domain and range:
A) domain: [0, ¥ ) range: (- ¥, ¥ )
B) domain: (- ¥, ¥) range: [-1, ¥ )
C) domain: [0, ¥) range: [0, ¥ )
D) domain: [0, ¥) range: [-1, ¥)
E). None of these
4
11.
Number of Individual Income Tax
Returns Filed Electronically*
Number of returns
Year
(in thousands)
1995 11,807
1996 14,968
1997 19,136
1998 24,580
1999 29,349
2000 35,394
2001 40,245
2002 46,890
*Source: http://www.infoplease.com/ipa/A0004902.html
a) Use your calculator to calculate the regression line that best fits the data. Write it exactly as it is shown in
the calculator.
c) Use the regression equation to predict the approximate number of electronic returns that will be filed in the
year 2010.
d) Use regression equation to estimate the approximate number of electronic returns that were filed in the year
1990.
e) Explain what your answer to part e means? What really happened in 1990?
f) Use regression equation to determine the year in which, the number of electronic returns that should filed is
approximately 121,507.
5
12. Solve each inequality and graph the solution set on a number line. Express solution in interval notation:
(a) 4( t + 2 )  3  7 ( t + 5 )
interval notation: _______________________
(b)  1  3(n  4)  7  14
interval notation: _____________________________
(c) 9x  7  20 or 20  9x  16
Interval notation: ___________________________
13. Solve the inequality 2  3 y  2(4  x) for y and graph the solutions on the Cartesian plane.
14. Is (0, -5) a solution to the inequality above? Show work.
6
TOPIC 2: SYSTEMS
1. Write a system of inequalities that describes each graph pictured below. (Write the “equations” of both lines for
each problem. Include the greater than/less than where appropriate).
a)
b)
2. Solve each system of equations for x and y. Check your work
(a)
 x  3y  0

 2 x  6 y  12
(b)
 y  3x  12

 x  2 y  14
3. Graph the following system of inequalities:
2x  5y  10
y  3x – 2
7
4. Is (-2, 1) a solution to the system above? Show work.
5. Write a system of equations to solve each of the following problems. Then, solve using any method.
a) A boy has 14 coins in his pocket, all of which are dimes and quarters. If the total value of his change is
$2.75, how many dimes and quarters does he have?
b) A boat travels about 20 miles downstream from Springfield, MA to Hartford, CT in one hour. The return
trip, against the current takes 2 21 hours. What is the boat’s speed in still water? (D = rt)
c) A woman invests her savings in two accounts, one paying 6% interest and the other paying 10% simple
interest per year. She puts twice as much in the lower yielding account because it is less risky. Her annual
interest is $3520. How much did she invest at each rate?
8
d) A certain brand of razor blades comes in packages of 6, 12, and 24, blades, costing $2, $3, and $4 per
package, respectively. A store sold 12 packages containing a total of 162 razor blades and took in $35.
How many packages of each type were sold? ( feel free to use POLYSIMULT TO SOLVE)
e. You are in charge of allocating funds for the UConn basketball teams and need to order new sneakers. The
sneakers that they players and coaches can choose from are $100 and $125. If 50 pairs are ordered and the total
cost is $5,850, how many of each type of sneaker are there?
f. A vending machine has $41.25 in it. There are 255 coins total and the machine only accepts nickels, dimes and
quarters. There are twice as many dimes as nickels. How many of each coin are in the machine.
g)
Solve
a. (2, 2, 1)
b. (–2, 2, –1)
c. (2, –2, –1)
d. (2, 2, –1)
9
h .Ed Carter paid $5.80 for some $0.20, $0.05, and some $0.03 stamps. He bought 93 stamps in all. The number
of $0.20 stamps was 7 less than one-half of the $0.05 stamps. How many of each type did he buy?
Equations:
Polysimult Entry Make sure your entry reflects eqs in std form)
i. Scott is in charge of purchasing a variety of tickets to a baseball game to be given to employees who met or
exceeded their sales quotas for the past month. There are 3 types of tickets available for the game at a cost of
$25 apiece, $35 apiece, or $50 apiece. Scott needs to buy 40 tickets in all and can spend $1670. He wants to buy
twice as many $35 tickets as $25 tickets. How many of each type of ticket should he buy?
10
TOPIC 3 LINEAR PROGRAMMING
1. Write the system of inequalities whose solution forms the triangular region shown in the graph.
2) A housing contractor has subdivided a farm into 90 building lots. He has designed two types of
homes for these lots: colonial and ranch styles. A colonial requires $30,000 of capital* (see below)
and produces a profit of $27,000 when sold. A ranch style house requires up to $40,000 of capital
and provide an $30,000 profit. If he has $2 million of capital on hand and wants to maximize his
profit, how many houses of each type should he build?
Objective (Profit) Function:
Constraints:
S
o
l
u
t
i
o
n
:
Solution in a sentence!: ______________________________________________________________
11
3. A furniture manufacturer makes wooden tables and chairs, on which the profits are $65 per table and $35 per
chair. The production process involves two basic types of labor carpentry and finishing. A table requires 2 hours of
carpentry and 2 hours of finishing, whereas a chair requires 3 hours of carpentry and 2 hour of finishing. They
company must produce a minimum 6 tables and 6 chairs per day. The manufacturer’s employees can supply a
maximum of 108 hours of carpentry work and 96 hours of finishing work per day.
How many tables and chairs should be made each day to maximize profit?
a) x =
y =
b) Objective function:
c) List all constraints below
d) Vertices of feasible region
SOLUTION IN A COMPLETE SENTENCE
12
TOPIC 4 ABSOLUTE VALUE
1. GRAPHING ABSOLUTE VALUE RECALL y = a x - h + k where (h, k) is the vertex and “a” is the
“Slope” of the right side of the “V”.
a) f ( x)   | x  3 |
b) g ( x)  2 | x  3 | 4
vertex: _______________
vertex: ___________________
“slope” of right side: _______
“slope” of right side: ___________________
c) y 
3
x 14
2
vertex: _______________
slope of right side: _______________
slope of left side: ________________
olid or dashed? __________________
(Be sure you shade the solution set!)
13
3. . Write an equation or inequality for the graphs pictured below.
__________________________
SOLVING ABSOLUTE VALUE EQUATIONS AND INEQUALITIES
EQUATIONS
Step 1: Isolate the absolute value
Step 2: Set up 2 possible equations
Step 3: Solve each, check
INEQUALITIES
Step 1: Isolate the absolute variable.
Step 2: < AND
> OR so write your
inequality (ies)
Step 3: Solve and graph!
SOLVE
4. (a)
20  3 | x  2 | 5
4. (b)
3 | x  4 | 5  1
Solution in interval notation: ___________________
14
c).
4 1 - 2x + 2 = 10
d)
x +1 -6 <2
X = ___________
solution in interval notation:
X = ___________
___________________________
TOPIC 5: PIECEWISE
1. For the piecewise below, evaluate at each value of x.
ìx 2 - 3
if x ³ 8
ï
ïï 2x - 11
if 3 < x < 8
f x =í
if - 6 < x £ 3
ï 2x + 3
ï
if x £ -6
ïî7
()
a.
f(10) =__________
b.
f(2) =__________
c.
f(-8.25) =__________
d.
f(5) =__________
15
GRAPH
2.
1
 x 5
f x   2
3x  2

if x  0
if x  0
3.

3x  5

f  x   4
 1
 x  1
 2
if x  2
if  2  x  2
if x  2
TOPIC 6 OPERATIONS WITH POLYNOMIALS
. Simply the expressions:
(4x - 3)(x + 5)
(a) ( x3  3x 2  2)  (5 x3  x  8)  (9 x3  x 2  4)
(b)
(c) (4 x  3)(3x 2  2 x  1)
(d) (5 x  2) 2
16
(e) 2( x3  5 x 2  6 x)  ( x 2  3x)
f). (3x+4)(3x – 4)
______________________
g. Simplify (7x3 – 2x2 + 3) + (x2 – x – 5).
A.) 7x3 – 2x2 – x – 2
B.) 7x3 – 3x2 – 2
C.) 8x5 – 3x3 – 2
D.) 7x3 – x2 – x – 2
SIMPLIFY
2
4
3
3
4
2
h). 9 x  9 x  4  6 x   5  2 x 5x 3x 
i). .
4
3
2
A). 4 x  8x  12 x  9
B) 14 x 4  8x 3  12 x 2  9
C) 4 x 4  4 x 3  6x 2  1
D) 14 x 4  8x 3  12 x 2  1
(-5x + 3y) (2x – 12y + 1)
2
2
A). 10x  60xy  5x  36 y
B) 10 x 2  66 xy  66 y 2
C) 10x 2  66xy  5x  36 y 2  3y
D) 10x 2  6xy  5x  36 y 2  3y
17
TOPIC 7: Factoring: a 3  b3   a  b   a 2  ab  b 2 
Difference of two perfect squares:
Perfect Trinomial Squares:
a 3  b3   a  b   a 2  ab  b 2 
a2 - b2 = (a + b)(a - b)
a 2 + 2ab + b2
a 2 - 2ab + b2
1. Factor completely:
(a) 4 z 2 m5  2 z 6 m  16 z 3m3
(b) x 2  x  30
(c) 16x 2 - 40x + 25
(d) 9 x 2  4
(e) x 3  4 x 2  3 x
(f) 2 x 2  5 x  3
g).
8x 3 - 4x 2 - 2x +1
_________________________________________
18
TOPIC 8 INTRO TO IMAGINARY NUMBERS
Imaginary Numbers:
i  1
i 2  1
Complex Number written in Standard Form:
a + bi
Simplify each expression.
1.
100
3. 10 - -8
5. Write in simplest radical form:
2.
-24
4.
10 ± -32
2
8  72
=
4
TOPIC 9 SOLVING QUADRATICS ALL METHODS
To solve by factoring: Set equation = 0, factor, use zero product property
Quadratic Formula: Given ax 2  bx  c  0 , then x 
Solving by +/- square root
 b  b 2  4ac
.
2a
solve by completing the square
solve using PTS
1. A. What must be added to x 2  10 x to complete the square?
2.
Determine the nature of the roots of the equation 2 y 2  7 y  10  0
A) No real roots
B) One real rational solution
C) Two real solutions (2 rational roots)
D) Two real solutions (2 irrational roots)
19
3. Solve by completing the square: x 2  6 x  1
4. Solve by the quadratic formula: 9x 2 - 3x = -1
5. Solve by using square roots:
4(x - 3)2 + 5 = 1
6. Solve using Perfect Trinomial Squares if possible: 25x 2 - 40x +16 = 8
7. Solve 25x 2 + 4 = 0
8. Factor by grouping. Then solve!
4 x 3  8 x 2  5 x  10  0
20
9. Solve each quadratic equation. Leave answers in simplest radical form.
2
a) 5x = 6 - 13x ______________________
2
b). x2 - 100 = 0 ______________________
c) 2x + 3x = 1 _______________________
d) (2x - 5)(x + 1) = 2 _________________
2
e) (2x + 7) = 25 ______________________
f) 45x - 30x + 5x = 0 ________________
g)
2
3x + 27 = 0
________________________
2
h)
3
2(x - 3)2 + 3 = 5
21
i) x 3  4x 2  x  4  0 ______________________
j) x 3  2x 2  4x  8  0 _________________
k) 4x2 + 20x + 25 = 6
l)
x 2 + 6x = 4
10.
a. 2 4
,
b. 1 2
,
5
c. 56 13
,
5
5
d. 2 1
,
5
11. Solve:
a.
b.
4 4
 i, i
3 3

16
9
i,
16
9
i
c.
3 3
 i, i
4 4
d.
4 4
 ,
3 3
12.
a. 3, –3
c. 3, –7
b. 7, –3
d. 7, –7
13.
a.
Solve for x:
or
b.
or
c.
or
d.
or
22
2
14. Solve: 4x = 16
A.) x = ±4
B.) x = ±2
C.) x = 2
D.)
x=± 2
Solve by factoring
5 x3  25 x 2  120 x
15.
a. 8, 3
b. 3, 8
c. 8, 3,0
d . 3, 8,0
Solve the equation by factoring.
16.
9b2  21b  1  9
5 2 
3 3 
A)  , 
 3 2
B)  , 
 5 3
3 3 
C)  , 
5 2 
 5 2
D)  , 
 3 3
Use the Quadratic Formula to solve the equation.
17.
a. 3 , 1
b. 1 , 3
c. 2 , 6
d. 2 , 6
18.
a. 6
c. 7
7
6
Use the quadratic formula to solve the equation.
19.
7 x 2  6  5x
B)
5  143 5  143
,
14
14
B)
5  i 143 5  i 143
,
14
14
C)
5  i 143 5  i 143
,
14
14
D)
5  143 5  143
,
14
14
23
_____ 20.
If b - 4ac < 0, then the type and number of zeros for the quadratic equation
are which of the following?
A. One real and one imaginary
B. One real
C. Two imaginary
D. Two real
_____
If ax + bx + c is a perfect square trinomial, then how many
solutions/roots/zeroes does it have?
A. Three
B. Two
C. One
D. None
Which of the following is a perfect square trinomial?
21.
_____ 22.
2
2
1
1
x+
4
8
2
1
2
C. x + x +
3
9
A. x +
2
B.
25x 2 + 5x +1
D.
9x 2 + 16
______ 23. A quadratic equation can be used to model which of the following situations?
A. The path of Peyton Manning throwing a football
B. A ground to air missile being shot at a target
C. The revenue generated by renting apartments in a complex at different rates.
D. A and C only
E. A, B, and C
_________24.
A graph of a quadratic that never intersects the x axis would have:
A. Two imaginary or complex solutions.
B. Two real solutions
C. One real solution
D. cake for breakfast
_________25. The solutions of a quadratic equation are 2 and -2. Which equation would
match this solution?
A. x 2 +16 = 0
B. 4x 2 -1 = 0
C. 200x 2 - 72 = 0
D. 25x 2 -100 = 0
24
TOPIC 10 FUNCTIONS
Given f ( x)  x 2  4 and, determine each of the following.
1.
4.
f (3)
2. f(-3)
If f(x) =
A.) –10
3.
f(-6)
x2  4
, find f(–4).
x2
B.) –6
C.) 6
D.) 10
TOPIC 11 Graphing QUADRATICS
1. Sketch a graph of f ( x)  x 2  6 x  6 . Then complete the characteristics below.
DOMAIN
RANGE
AXIS OF
SYMMETRY
X-INTERCEPTS
Y-INTERCEPTS
vertex
y = 2(x + 1)2 - 4
2. Graph
DOMAIN
RANGE
AXIS OF
SYMMETRY
X-INTERCEPTS
Y-INTERCEPTS
vertex
25
3. Sketch the graph of
y = (x – 3)(x + 1)
roots = ________________ axis of symmetry: x = ________________
Vertex = _______________ y intercept: _________________
Point symmetrical to y intercepts: ________________________
4. Write in vertex form by completing the square
y = x 2 + 6x - 3
STEP 1: move the constant term
Step 2: complete the square!
Step 3: Whatever you added to the right side to complete the square you must add to the left side as well to
balance the equation!
Step 4: Now, write the right side as a quantity squared
Step 5: Isolate the y and you have it!
26
5. Use a graphing calculator to solve the equation
nearest hundredth.
a. –0.81, 2.21
b. –2.21, 0.81
. If necessary, round to the
c. –1.63, 4.43
d. –1.51, 1.51
6. Solve using a graphing calculator
2x 2 - 3x - 4 = x 2
7. Which graph represents f  x     x  2   1 ?
2
A.
B.
27
TOPIC 12: Writing Equations of Quadratics
A. When given the vertex and a point

Plug the vertex in for (h, k) in y  a( x  h) 2  k

Plug in the given point for (x, y)

Solve for a. Plug in a, h, k into y  a( x  h) 2  k
1.) Write a quadratic equation in vertex form
for the parabola shown.
When



given the x-intercepts and a third point
Plug in the x-intercepts as p and q into y = a(x  p)(x  q)
Plug in the given point for (x, y)
Solve for a. Plug in a, h, k into y = a(x  p)(x  q)
2.) Write a quadratic function in intercept form
for the parabola shown.
28
When given three points on the parabola
 Label all three points as (x, y)

Separately, plug in each point into y  ax 2  bx  c

You now have 3 equations with three variables: a, b, c

Solve for a, b, and c using elimination (see notes #13). Plug back into y  ax 2  bx  c
3.) Write a quadratic function in standard form for the parabola that passes
through the points (2, 6), (0, 6) and (2, 2).
TOPIC 14 : Simple APPS of Quadratics
1. A football’s flight is described by the equation:
h = -5(t – 5)^2 + 125
A. SKETCH and label the axes of the graph (you may use your calc).
B. If h is measured in feet, how high does the football reach and how long does it take to
get that high? (THINK: I need to find the ________________)
29
C. How long will it take for the football to hit the ground if no one catches it? (Think,
GROUND means y = ________)
D. If the receiver can catch the ball at a height of 7 feet, after how many seconds might the receiver
be able to catch the ball on its way down? (Think: y1 = EQ y2 = 7 CALC INTERSECT)
2
2. The function y = -16t + 340 models the height y in feet of a stone t seconds after it is dropped
from the edge of a vertical cliff. How long will it take the stone to hit the ground? Round to the nearest
hundredth of a second.
3. Ryan and Gino have engine problems while fishing from their boat out in Long Island Sound. They set
off an emergency flare with an initial velocity of 30 meters per second. The height of the flare can be
modeled by h(t) = - 5t2 + 30t, where t represents the number of seconds after launch. What is the
maximum height of the flare?
30
TOPIC 1: EXPONENTS
Laws of Exponents:
(a )  a
m n
n
mn
an
a

 
bn
b
an
 a nm
m
a
1
an  n
a
(ab)n  a nb n
m
a n  n a m or
 a
n
m
Simplify each expression.
1.
(5x 2 )(2 x 5 )
2.
3.
(t 3 )(t n  3 )
4.
(2c 3 ) 2
10  2 6
8  2 2
5. Simplify:
p 2 ip 3 ip
(a)
(b) (m3 ) 4
(c) (a 2 ) 3
a4
(d) 9
a
3xy 5
(e)
12 x 2 y 0
2
(f)  
5
(g) (3x y)
2
2
3
2
4
5
(h) (2 x y )(3x y )
3
3x3 y 2
(i)
6 x 2 y 1
31
k). 27
æ 1ö
l). ç ÷
è 8ø
2
3
-
2
3
Simplify the expression.
4 p  4 p 
3
m).
C)
p7
16
4
B) 16 p12
C) 16 p12
D) 16 p7
5. Use completing the square to write the equation in vertex form. State the vertex.
A. f(x)= x2 - 10x + 11
B. f(x)= 3x2 - 12x + 4
32