Permitted resources: 2016 – 2017 Algebra 2 Midterm Review
1. Simplify the following expressions.
a.  2(3 y  6)  5 y

FSA Approved calculator 
Algebra 2 FSA Reference Sheet 4. Use the following graph to answer parts a – g.
e. (9g 2  100 )(3g  10 )
b.
1
( 6c  15 d )  11( 4c  2d  7 )
3
f. ( x 4  2 x 3  3 x 2  2 x  1)( x 2  1)
c.
2
4
(8w  1)  ( 3w  7 )
5
5
g.
x 6  2x 4  6 x  9
x3  3
h.  2( 4f  3 p  1)  3( 3  p  7f )
d. ( 4 x 3  2 x 2  3 x  4 )  ( x  4 )
a. Over what interval is the graph increasing?
b. Over what interval is the graph decreasing?
2. Find the inverse of the relation if possible, otherwise write not invertible.
a. f ( x )  3 x  6
d. f ( x )  5 x  1
b. f ( x )  5 x 2  8
e. f ( x )  x 2  16 x  20
2
c. f ( x )  x  7
3
x 1
f. f ( x ) 
x
3. Designate the following as even, odd or neither.
a. f ( x )  3 x  6
e. f ( x )  2 x 3  5 x
b. f ( x )  5 x 2  8
f. f ( x ) 
c. f ( x )  x 4  3 x 2
g.
c. Over what interval(s) is the graph positive?
d. Over what interval(s) is the graph negative?
e. What is the relative minimum? Relative maximum?
f.
What is the domain?
g. What is the range?
5. The following graph represents a family’s distance from home as they return from vacation.
2
x7
3
a. Over what time interval(s) is the graph increasing?
b. Over what time interval(s) is the graph decreasing?
c. What is the domain?
d. What is the range?
ଵ
d.
h.
6. Let ݂ሺ‫ݔ‬ሻ ൌ ‫ ݔ‬ଶ . The graph of ݃ሺ‫ݔ‬ሻ is vertically compressed by a factor of , translated 5 units to the
ଶ
right and 4 units down. Write the function rule for ݃ሺ‫ݔ‬ሻ.
7. Describe the transformations applied to f ( x )  x 2 that yields the following functions. Then graph
each function.
b. g ( x )  ( x  9)2
c. y  ( x  3)2  2
a. g ( x )  x 2  4
8. Solve each equation for the designated variable.
a. S  6r 2t ; solve for t. Then solve for r.
b. A 
15. Marco is buying hamburgers and hot dogs for a party and only has $40 to spend. Hamburgers cost
$6.49 per pound. Hot dogs cost $4.99 per pound and Marco needs at least 5 pounds of hot dogs.
Write a system of inequalities to represent this situation.
1
(b1  b2 ) ; solve for b1. Then solve for b2.
2
9. Describe the meaning of the slope and y-intercept in each situation.
a. The cost for prom can be represented by the function c  25 p  110 where c is the total cost
for prom and p is the number of people attending.
b. Andrew has planted flowers in his yard and measures their height each week. The height of
the flowers can be represented by the equation h  12  4.2w where h is the height in cm
and w is the number of weeks since he planted the flowers.
10. Solve the following systems of equations:
3 x  y  9
a. 
4 x  y  5
16. The boys and girls golf teams are trying to raise $750 for an out-of-county tournament. The boys
team is selling candy bars for $2 each. The girls team is selling candles for $4 each. The girls need
to sell at least 100 candles to clear out already purchased inventory. Write a system of inequalities to
represent this situation.
17. Given the graph of the function shown below, identify the domain of each of the rules on the
piecewise function.
 2 x 2  8 x  5
f (x)  
3 x  1
 x  y  4
b. 
2 x  2y  8
if ____________ if ____________ 11. Identify the system of equations shown in the graph with the solution indicated.
18. Use the piecewise function shown in the graph below to answer parts a – d.
12. At a bookstore, used hardcover books sell for $8 each and used paperback books sell for $2 each.
Sheila purchased 36 used books and spent $144. Write a system of equations that can be used to
find how many hardcover and paperback books Sheila bought. Then state how many paperback
books Sheila actually purchased.
13. A group of 52 people attended a ball game. There were three times as many children as adults in
the group. Set up a system of equations that represents the numbers of adults and children who
attended the game and solve the system to find the number of children who were in the group.
a. Define the piecewise-defined function.
b. Over what interval(s) is this function decreasing?
14. Write a system of equations to model the following situations.
a. A candy store makes a 13-pound mixture of gummy worms, candy corn, and sourballs. The
cost of gummy worms is $2.00 per pound, candy corn cost $1.00 per pound, sourballs cost
$1.00 per pound. The mixture calls for three times as many gummy worms as candy corn.
The total cost of the mixture is $19.00.
b. A florist is planning her next order from a local nursery consisting of mums, orchids, and
roses. Mums cost $2.25 each, orchids cost $10.75 each, and roses cost $4.50 each. She
has $532 to spend on her order and would like to buy 116 flowers. She also plans to have 20
more roses than mums.
c. What is the relative minimum of this function?
d. What is the y-intercept of this function?
19. The local pet store charges for grooming according to your pet’s weight. If your pet is 10 pounds or
less, they charge $30. If your pet is between 10 and 30 pounds, they charge $35. If your pet is 30
pounds or more, they charge $40, plus an additional $2 for each pound over 30.
a. How much does it cost to groom a pet who weighs 47 pounds?
b. Over what interval is this piece-wise defined function increasing?
c. What does the y-intercept of this graph mean in context?
20. Graph the equations:
1
b. y  x  2  4
3
a. y  2 x  5
21. What is the equation of the absolute value function shown below?
24. A tee-ball is hit so that its height above ground is given by the equation h  
1 2
5
t  2t  , where h is
2
2
the height in feet and t is the time in seconds after the ball is hit.
a. Factor the equation to reveal its zeros.
b. How many seconds does it take for the ball to reach the round after it is hit?
25. Jesse has a square rug whose area can be represented with the function A(L )  L2  16L  64 ,
where A(L ) is the total area and L is the length of the rug in feet.
a. What expression can be used to find the length of one side?
b. What is the length of the rug?
26. Factor the following equations and identify their zeros.
a. y  10 x 2  11x  6
b. y  x 2  2 x  15
22. Fill in the missing properties for the solution of the equation given below.
Step
Property Used
2x 2  10 x  9  5 x  2
Given
2x 2  15 x  7  0
c.
3 
 
 18  12   27
b.  54


2x  1  0
x 7  0
2x  1
x 7
1
2
d. ( 5  2i )(3  4i )
b. (8  i )  (2  7i )
e. (3  4i )  ( 5  2i )
c. ( 10  i )(3  9i )
f. ( 10  i )  (3  9i )
29. Find the solutions to the following:
a. 25 x 2  49  0
23. Select all of the following justifications NOT used in the solution of the equation shown below.
Solution
 
d. 8   20   21   45
28. Simplify the following.
a. ( 5  2i )  (3  4i )
(2x  1)( x  7)  0
x
27. Simplify the expressions.
a.
 25
c. y  3 x 2  33 x  90
d. 3 x 2  3 x  2  0
b.
x 2  2x  2  0
e. 5 x 2  120
c.
x 2  6x  1
f. 3 x 2  10  4 x
3 x 2  75  0
30. Identify the vertex and axis of symmetry for each of the following graphs.
3 x 2  75
x 2  25
x 2  25
x  5
a.
b.
c.
d.
e.
Addition Property of Equality
Subtraction Property of Equality
Zero Product Property
Division Property of Equality
Multiplication Property of Equality
f.
g.
h.
i.
Factor
Distributive Property of Equality
Square root both sides of the equation
Square both sides of the equation
a.
b.
c.

31. For parts a and b, identify whether the function has a maximum or minimum value. Then state that
value.
37. Find the inverse of the relation if possible. (Refer to Question #2.) For those that were previously not
invertible, give a restriction on the domain of the function in order to make it invertible.
a. f ( x )  3 x  6
b. f ( x )  5 x 2  8
c. f ( x ) 
2
x7
3
d. f ( x )  5 x  1
a. y  ( x  2)2  11
b.
e. f ( x )  x 2  16 x  20
c. Which of the functions in parts a and b has a smaller maximum or minimum?
f.
32. Which of the following functions has the largest maximum?
f (x) 
x 1
x
38. Name all of the operations under which polynomials are closed.
39. Is the set of whole numbers closed under
a. addition?
b. subtraction?
a. y   x  x  3
2
x –2 0 2 4 6 y –12 –6 –4 –6 –12
b.
c.
c. multiplication?
d. division?
40. Show why the polynomial identity a 2  b 2  (a  b )( a  b ) is true.
33. Give the equation for each of the following graphs. (Use vertex or standard form.)
41. Show why the polynomial identity a 3  b 3  (a  b )( a 2  ab  b 2 ) is true.
42. Describe the end behavior of the following polynomials. Then sketch a graph of each.
1
a. y  2 x 4  4 x 3  6 x 2  8 x  8
b. y  ( x  3)3  2
4
a.
b.
34. What is the vertex form of the equation?
a. y  x 2  2 x  8
c.
b. y  2 x 2  8 x  2
35. What is the equation of a parabola given the following information?
a. focus (6, 13) and directrix y  7
b.
x 3  125
c.
x 4  40 x 2  144  0
d. 2x 3  3 x 2  18 x  27
44. Factor completely.
a. 27 x 3  125
b. focus (-3, 1) and directrix x  1
36. Solve the system.
y   x 2  4 x  5
a. 
y  2 x  3
43. What are the solutions (zeros) to the following equations?
a. 8 x 3  1  0
y  x 2  16 x  32
b. 
x  y  2
b. x 3  64
45. Write the equation of the polynomial graphed below.
a.
46. Graph the functions.
1
a. y  ( x  3 ) 2 ( x  1)( x  1)
2
b.
b. y  3( x  4 ) 3  1
47. Use the Remainder Theorem to determine if the given value is a zero of the polynomial.
a. Is 2 a zero of P ( x )  x 3  2 x 2  6 x  4 ?
b. Is –1 a zero of P ( x )  x 3  x 2  3 x  8 ?
48. Find the solutions to the following equations and name any extraneous solutions that arise.
x
1
2


a.
x  2 x  4 x 2  6x  8
b.
5x 2  8x  4
1
x


3 x 2  11x  6 x  3 3 x  2
49. Find the value(s) of x for which f ( x )  g ( x ) . Name any extraneous solutions that arise.
3
1
1
 and g ( x ) 
a. f ( x ) 
x2 x
5x
b. f ( x ) 
4
3
and g ( x ) 
x
x 2
50. Name any vertical and horizontal asymptotes of the graphs of the following equations.
x7
a. y  2
x  3x  2
b. y 
( x  2)( x  3)
( x  3)( x  1)
y
2x 2  x  3
3x 2  8x  5
d. y 
x  12
x 2  23 x  1
c.