Download Monday, January 25 th at 12:15PM

Common Core Algebra 2
Midterm Review Packet
This packet contains both multiple choice and free response
questions for all topics that will appear on the midterm.
The midterm will consist of multiple choice and free response
questions.
The midterm will count for 20% of the second quarter grade.
Midterm review will occur during class the week of
Jan. 19th
The midterm is on:
Monday, January 25th at 12:15PM
Please bring pens or pencils and your calculator to the test.
No calculators will be loaned for the test.
Unit 1: Intro to Functions, Factoring and Equations
1. What is the solution set of the equation x  6  4  10 ?
(1) {0, 12}
(2) {-8, 12}
(3) {-12, 0}
(4) {-12, -8}
2. The domain for f ( x)  x 2  1 is  3  x  3 . The smallest value of the range of f is
(1) -1
(2) -2
(3) -3
(4) 0
3. Which is the range of the relation y  2 x 2  3x if the domain is
the set {-2, -1, 0}?
(1) 2, 1, 0
(2) 2, -1, 0
4. If f ( x) 
(1) 
1
2
(3) -1, -5, 0
(4) 10, 1, 0
x2
, then f ( n  1) is equal to
x 1
(2)
n 1
n2
(3)
n 1
n2
(4)
n2
n 1
5. If f ( x)  kx2 and f (2)  12 , then k equals
(1) 1
(2) 2
(3) 3
(4) 4
6. The minimum point on the graph of the equation y  f (x) is (-1, -3). What is the
minimum point on the graph of the equation y  f ( x)  5 ?
(1) (-1, 2)
(2) (-1, -8)
(3) (4, -3)
1
(4) (-6, -3)
7. The graph below shows the function f(x).
Which graph represents the graph of f(x - 2)?
(1)
(3)
(2)
(4)
8. Determine which of the following has the largest maximum:
(1) x 2  y  4 x  8
(3) y  ( x  3)2  5
(2)
(4)
2
x
3
4
5
6
7
y
-1
2
3
2
-1
9. Solve: 5 c  2  30
10. Solve: x  2  2 x  3
11. Solve: |𝑥 + 2| − 1 < 4
2
12. Solve: 3x  3x  0
13. Solve: x 2  10 x  29  0
14. Solve: x 2  12 x  27  0
 3 x  7 x  1
15. (a) Graph p ( x)  
 x  5  1 x  1
(b) find: i. p (5)
ii. p (1)
iii. p(3)
Unit 2: Quadratic Functions and Powers of i
1. The equation 2 x 2  5  0 has
(1)
(2)
(3)
(4)
no real roots
1 repeated root
2 distinct roots
There is not enough information
2. The equation 2 x 2  5 x  6  0 has
(1)
(2)
(3)
(4)
2 distinct roots
1 repeated root
no real roots
not enough information
3. The equation 2 x 2  8 x  4  0 has
(1)
(2)
(3)
(4)
no real roots
1 repeated root
2 distinct roots
not enough information
3
4. The roots of the equation 2 x 2  7 x  3  0 are
(1) 
(2)
 7  73
4
7  73
(4)
4
1
and -3
2
(3)
1
and 3
2
5. The roots of the equation x 2  3x  7  0 are
3  19
2
3  i 19
(2)
2
19
2
i 19
(4) 3 
2
(3) 3 
(1)
6. The solutions of the equation y 2  3 y  9 are
3  3i 3
2
3  3i 5
2
(1)
(2)
33 5
2
33 5
(4)
2
(3)
7. What is the product of the roots of the equation 2 x 2  x  2  0 ?
(1) 1
(2) 2
(3) -1
(4) -2
8. What is the sum of the roots of the equation 2 x 2  6 x  7  0 ?
(1) 
7
2
(2) -3
(3) 3
x2
 5 x  17 ?
2
(3) x 2  10 x  34  0
9. Which equation has the same solution as
(1)
x  52  9
(2) x 2  10 x  34  0
(4)
(4)
x  52  9  0
4
7
2
10. Which quadratic equation has the roots 3  i and 3  i ?
(1) x 2  6 x  10  0
(2) x 2  6 x  8  0
(3) x 2  6 x  10  0
(4) x 2  6 x  8  0
11. Which equation has the complex number 4  3i as a root?
(1) x 2  6 x  25  0
(2) x 2  6 x  25  0
(3) x 2  8 x  25  0
(4) x 2  8 x  25  0
12. Which of the following products is equal to a real number?
(1)
(2)
8  2i 8  2i 
2  8i 8  2i 
(3) 8  2i 8  2i 
(4) 2  8i 8  2i 
13. Write 2  i  in standard a  bi form
3
(1) 2  11i
(2) 2  13i
(3) 8  17i
(4) 8  19i
14. Electrical Engineers find the voltage, E, produced by a battery by finding
the product of the impedence, I, and the current, Z. E  I  Z where E
represents the voltage, I represents the impedence, and Z represents the
current. What is the voltage if the impedence is 4  8i and the current is
2  3i ?
(1) 32  4i
(2) 32  28i
(3)  16  4i
(4) 8  20i
15. If x 2  2  6 x is solved by completing the square, an intermediate step
would be
(1)  x  3  7
2
(2)  x  3  7
2
(3) x  3  11
2
(4) x  6  34
2
5
16. What are the focus and directrix of the parabola represented by the equation:
8( y  6)  ( x  2)2 ?
(1) Focus: (0, 6)
Directrix: x  4
(2) Focus: (-2, 8)
Directrix: y  4
(3) Focus: (-2, 4)
Directrix: y  8
(4) Focus: (-2, 6)
Directrix: y  4
17. What is the equation of a parabola with a focus of (8, 2) and directrix x  6 ?
(1) 4( x  7)  ( y  2)2
(2) 4( x  7)  ( y  2)2
(3) 4( x  2)  ( y  7)2
(4) 4( x  2)  ( y  7)2
18. Multiply ( 2  3i ) by its conjugate and express in simplest a  bi form.
19. Factor each expression completely:
(a) 5 x3  20 x 2  60 x
(c) 16 x 4  81
(e) 27 x 3  125
(b) 4 x3  12 x 2  40 x
(d) 2 x 4  10 x 2  12
(f) x 2 ( x  2)  4( x  2)
20. Determine all values of k for which the roots of the equation x 2  6 x  k  0
are complex. Justify your answer.
21. Solve x 2  10  6 x by the quadratic formula and completing the square. Explain how
the two methods are related.
22. Solve the following equation: 4 x 2  8 x  7  0 . Express your answer in simplest
a  bi form.
23. Express i 4  2i 3  3i 2  4i in simplest a  bi form.
24. A boy standing on the top of a building in Albany throws a water balloon up
vertically. The height, h (in feet), of the water balloon is given by the equation
h(t )  16t 2  64t  192 , where t is the time (in seconds) after he threw the water
balloon. What is the value of t when the balloon hits the ground? Explain and show
how you arrived at your answer.
6
Unit 3: Functions
1. What is the inverse relation of the function whose equation is y  2 x  3 ?
(1) y  2 x  3
x3
(2) y 
2
(3) y  3 x  2
(4) y  x
2. The inverse function of {(2, 6), (-3, 4), (7, -5)} is
(1) {(-2, 6), (3, 4), (-7, -5)}
(2) {(2, -6), (-3, -4), (7, 5)}
(3) {(6, 2), (4, -3), (-5, 7)}
(4) {(-6, -2), (-4, 3), (5, -7)}
3. The accompanying diagram represents the graph of f(x).
Which graph represents f 1 ( x) ?
(1)
(2)
(3)
4. What is the domain of the function f ( x)  x  2 ?
(1) x x  0
(2)
x x  2
5. The domain of the relation y 
(1) x x  1
(2) x x  1
(3) x x  2
(4) x x  2
4
is
x 1
(3) x x  1
(4) x x  2
7
(4)
6. The function f ( x) 
(1) 
1
is defined for all real numbers except when x is
x 3
1
3
(2) -3
(3) 3
7. Use the functions below :
h( x)  x 2  16
2x  2
j ( x) 
4
f ( x)  2 x  5
g ( x) 
x  16
a. Write the rule for the following
(i) f ( g ( x))
(iii) g (h( x))
(ii) ( g  f )( x)
(iv) f ( j ( x ))
b. Evaluate the following
(i) g (h(5))
(iii) j ( f (3))
(ii) f ( j (3))
(iv) f ( f (1))
8. Determine the domain of each function
(a) f ( x) 
3x  5
x  2 x  24
(b) g ( x) 
x 1
x6
2
(c) h( x)  x 2  2 x  8
8
(4) 0
9. For each, determine if
a. the relation is a function (yes/no)
b. it is a function, its domain and range
c. it is a one-to-one function
10
(i)
(ii)
8
10
8
6
6
4
4
2
2
-10 -8 -6 -4 -2
2
4
6
8 10
-10 -8 -6 -4 -2
-2
-4
-4
-6
-6
-8
-8
-10
-10
(iii)
2
4
6
8 10
2
4
6
8 10
-2
(iv)
10
10
8
8
6
6
4
4
2
2
-10 -8 -6 -4 -2
2
4
6
8 10
-10 -8 -6 -4 -2
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
9
Unit 4: Polynomial Functions
1. Determine the points of intersection for x 2  y 2  1 and y  x  1 .
(1) (0, -1) and (1, 0)
(2) (-1, 0) and (0, 1)
(3) (1, 0) and (0, 1)
(4) (-1, 0) and (1, 1)
2. Find the points of intersection of the line y  x  1 and the circle x 2  y 2  25
algebraically.
3. What is the solution to the system of linear equations?
x  2 y  3z  7
2x  y  z  4
 3 x  2 y  2 z  10
4. Solve each system algebraically:
(a)
y  x2  2x  3
x  y 1
(b)
x 2  y 2  16
y  2x
5. Solve the system graphically:
x 2  3x  3  y  0
y  2 x  1
6. Solve each equation algebraically and graphically:
(a) 2 x3  2 x 2  4 x  0
(b) x 4  5 x 2  4  0
(c) 2 x3  3x 2  2 x  3  0
(d) 2 x 2  3x  2  0
10
7.
f ( x)  x 3  6 x 2  11x  6
(a) Show ( x  1) is a factor of f (x ) using synthetic division
(b) Find the remaining zeros
(c) Write f (x ) as the product of linear factors
(d) Determine the y-intercept
(e) Sketch f (x )
Unit 5: Rationals
A. Undefined Fractions – set denominator = 0 and solve.
1. For what values of x is
x4
undefined?
x2  6x
(1) x = 0, x = 4, x = 6
(2) x = 4
2. For what values of x is
(3) x = 6
(4) x = 0, x = 6
x 2  25
undefined?
x2  6 x  8
B. Simplifying
1. The expression
10  2 x
is equivalent to
x 2  25
2
x5
2
(2)
x5
2
5 x
2
(4)
5 x
(1)
2. Simplify:
(3)
x 4  16
x2
7 x  x2
3. Simplify: 2
x  10 x  21
11
C. Multiplying/Dividing – express in simplest form
4 x  16
x2  9
1. The product of
and 2
is
x  7 x  12
3 x
(1) -4
(2) 4
(3) 1
(4) -1
2. Simplify:
x 2  x  12 x 2  11x  30

x 2  x  20 x3  9 x 2  18 x
3. Simplify:
x2  9 x2  5x  6

9  3x
6 x  12
D. Complex Fractions
1
x ; x  0 , is equivalent to
1. When simplified, the complex fraction
1
1
x
1
(1) 1
(2)
(3)
x 1
1 x
2. Simplify:
1
1 x
(4)  1
4x  9
6x
4x 2  9
9x
1
x
3. Simplify:
x 1
x
E. Equations
1. What is the solution of the equation
(1) {-3}
(2) {-3, 0}
(3) {0}
(4) { }
12
x
2
6

 2
?
x  3 x  1 x  4x  3
2. Solve:
4 9  4x x


x
3x
3
5
3
21

 2
3. Solve: x  3 x  4 x  7 x  12
Unit 6: Radicals
1. If
x  4  7 , what is the value of x?
(1) 11
2. In simplest form,
(1) 3i 10
(2) 18
(3) 45
 300 is equivalent to
(2) 5i 12
3. Which expression is equivalent to
14  5 3
11
17  5 3
(2)
11
(1) 
(4) 53
(3) 10i 3
(4) 12i 5
3 5
?
3 5
14  5 3
14
17  5 3
(4)
14
(3) 
4. The expression 4ab 2b  3a 18b3  7ab 6b is equivalent to
(1) 2ab 6b
(2)  5ab  7ab 6b
(3) 16ab 2b
(4)  5ab 2b  7ab 6b
5. Which of the following is equivalent to  4  48 ?
(1) 8 12
(3)  8i 12
(2) 16 3
13
(4)  16i 3
6. Solve for x:
x 1  x  7
(1) {10, 5}
(2) {5,
2}
(3) {-7, 7}
4  2 y  y2  2  y
7. Solve for y in the following equation:
(1) No solution
(4) {10}
(2) {-3}
(3) {0}
(4) {-3, 0}
3
8. Simplify: √16𝑎4 𝑏 5 − 8𝑎6 𝑏 3
9. What is the solution set of the equation
10. The solution set of the equation
x 2  3x  3  1 ?
x  3  3  x is
11. Express 5 3x3  2 27 x3 in simplest radical form.
12. Find the sum of 12a3 and
13. Simplify:
14. Express
15. Express
27a5b2 .
4 1
1

20 
45
5 2
5
5
with a rational denominator, in simplest radical form.
3 2
108 x 5 y 8
6 xy5
in simplest radical form.
16. Multiply (3  5 2 ) by its conjugate and express in simplest radical form.
14