Study Guide for 3rd 9 Weeks Assessment
Math 3
Matrices
1. What is the identity for a 3 x 3 matrix?
1 0 0 
0 1 0 


0 0 1 
2. Write a matrix and find its inverse?
3. Solve X  2 A  B if A =  7
X = B – 2A
X = [3 4] – 2[-7
X = [3 4] – [-14
X = [17 0]
2 and B = 3 4 .
2]
4]
2 4
 1 0 5  
4. Multiply 
  3 1  .

 1 0 2 
 4 2
18 6 

 6 0
= 
5. You are planning a birthday party for your younger brother at a skating rink. The cost of admission is $3.50 per adult
and $2.25 per child, and there is a limit of 20 people. You have $50 to spend. Write a matrix equation that could be used
to determine how many adults and children you can invite.
x + y = 20
3.50 x + 2.25y = 50
1   x   20
 1
3.50 2.25   y   50 

    
 5 3   x  6
    .
 3 2   y  6 
6. Solve 
5
3
3 2
 10  9  10  9  1
 d b 
 c a 


1  2 3  2 3 
A1  

1  3 5   3 5
 x   2 3   6   12  18   30 
 y    3 5   6    18  30    48
  
   
 

A1 
1
A
5 x  2 y  1
.
 x  4 y  35
7. Write a matrix equation to represent the system of equations 
5 2  x   1
1 4   y   35

   
8. Solve:
x 10
5
4
 6
4x -50 = -6
4x = 44
x = 11
9. Write a system of equations to represent the matrix equation.
4 5 1   x   4 
 5 3 1  y    7  .

   
 4 3 2   z   2 
4x + 5y + 1z = 4
5x – 3y – 1z = -7
4x - 3y + 2z = 2
0 3
.
2 1 
10. Find A1 if A  
 d b 
 c a 


A  0  6  0  6  6
A1 
1
A
1
6
1
3


1
A1  


6  2 0   1
 3
1
2

0

 x  5   3 10
2
   
 y  6  8   7 
11. Solve for x and y.
x = _______
 2 x  10   3 10 
 2 y  12    8    7 

    
 2 x  13  10 
 2 y  20    7 

  
2 x  13  10
2 y  20  7
2 x  23
x
23
2
2 y  13
y
13
2
12. What are the dimensions of matrix M if M  A32  B42 ?
___4x3_______________
y = ___________
 2 1 3 
Let A  

 0 1 2 
 4 2 
B

 1 1
 2 3
C

 1 4
 1 3
D   1 2 
 2 3 
Evaluate. Show work. If not possible, state why.
13. 2B  C
 4 2   2 3 
2


 1 1  1 4 
 8 4   2 3 
 2 2    1 4  

 

 10 1 
 3 6 


14. AD  C
 1

1  1 2  1  1 2   2
1
15. B =


4  2  1 4  2  1 4   1

 2
3
1
 2 1 3  
 2 3
  1 2   

0 1

2 
1 4 


 2 3 
 3 1   2 3 
 3 4    1 4  

 

 1 4 
 2 0



1

2 

16. Use the inverse matrix to solve the system of equations. You must show A1 and how to set up the problem along
with your solutions for x and y in order to receive full credit.
 3 2   x   4 
 2 4   y    16 

  


2
1  4 2  1  4 2  
1
A 


12  4  2 3  8  2 3   1

 4

 2

 1
 4
1
 
4  4   8  4   4 



3   16  1  4   3
8 
1
 
4

3 
8 
3 x  2y  4

2x  4y  16
17. What is the area of the triangle with vertices (-3, 4), (6, 3), (2, -1) in square units?
x1
1
A   x2
2
x3
y1 1
y2 1
y3 1
3 4 1
1
A 6 3 1
2
2 1 1
3
4
6
1   3 3 1   4 1 2   1 6 1    2 3 1   1 1 3   1 6 4  
1 1
2
1
3
 (9  8  6)  (6  3  24)  (7  33)  (40)
A
1
40  20
2
 5 2 1


18. What is the determinant of 3 1 2 ?


 2 0 4 
5 2 1 5 2
3 1 2 3 1   5 1 4    2 2 2   1 3 0     2 1 1   0 2 5    4 3 2  
2 0 4 2 0
 (20  8  0)  (2  0  24)  (28  22)  (28  22)  6
19. A local theater charges $5 for child tickets and $9 for adult tickets. One Friday night they sold
165 tickets and collected $1185 in sales. How many of each kind of ticket were sold?
x  y  165
5 x  9 y  1185
1 1   x   165 
5 9   y   1185

  

A  95  4
 9 1 
9

1
 4
1
4
A1  



4  5 1   5 1 
 4
4 
 9 1 
 x  4
4   165   371.25  296.25   75
 y    5 1  1185   206.25  296.25  90 
  
 
  

 4

4
Linear Programming and Vertex Edge Graphs
1. Find the maximum value of the objective function z  5 x  6 y subject to the following constraints.
x0
y 0
x  2y  8
3 x  3 y  15
(0, 4)
(2, 3)
(5, 0)
z = 5(0)+6(4) z = 24 Min
z = 5(2) + 6(3) z = 28 Max
z = 5(5) + 6(0) z = 25
2. Find the maximum value of the objective function z  10 x  8 y subject to the following constraints.
x0
y 0
xy 5
3 x  y  12
2 x  y  2
(0, 2)
(1, 4)
(3.5, 1.5)
(4, 0)
z = 10(0) +8(2)
z = 10(1) +8(4)
z = 10(3.5) +8(1.5)
z = 10(4) +8(0)
z = 16
z = 42
z = 47 Max
z = 40
3. Find the minimum value of the objective function C  x  y subject to the following constraints.
x  10
(0, -2)
x0
y  x 2
(-10, -12)
C = -10+-12 = -22 Min
C= 0 + -2 = -2
Use the following problem to answer questions 4 and 5.
Your factory makes fruit filled breakfast bars and granola bars. For each case of breakfast bars, you make $40 profit. For
each case of granola bars, you make $55 profit. The table below shows the number of machine hours and labor hours
needed to produce one case of each type of snack bar. It also shows the maximum number of hours available. Assume x
= the number of cases of breakfast bars and y = the number of cases of granola bars.
Production Hours
Breakfast Bars
Granola Bars
Maximum Hours
Machine Hours
2
6
150
Labor Hours
5
4
155
4. Which system of inequalities represents the constraints?
x0
A.
y 0
B.
2 x  6 y  150
5 x  4 y  155
x0
C.
x0
y 0
2 x  6 y  150
5 x  4 y  155
x0
y 0
D.
2 x  5 y  150
6 x  4 y  155
y 0
2 x  6 y  150
5 x  4 y  155
5. Write an equation that represents the profit (objective function)?
40x + 55y
6. Given the following matrix, draw a representative planar graph. (Vertex Edge Graph)
A
1
0
1
0
1
A
B
C
D
E
B
0
1
0
1
0
C
1
0
1
1
1
D
0
1
1
1
1
E
1
0
1
1
0
A
B
E
C
D
7.The following planar graph represents Tara’s class schedule. Write a matrix representing the graph.
A
B
A B C
A 0
B  0
C 0

D 2
D
0 0 2
0 1 1 
1 0 1

1 1 0
D
C
Polynomial Functions:
f(x) = x – 3x2 + 2
1. Is this a polynomial function?
What is the degree?
___2_____
__YES____
Name the lead coefficient.
___-3_____
Perform the indicated operations and write the answer. Use Synthetic Division
2. (2x4 + 3x3 + 5x – 1)  (x + 2) = 2x3 –x2 + 2x + 1 +
2
3 0
-4 2
-1 2
-2
2
5
-4
1
-1
-2
-3
3. (10x3 + 27x2 + 14x + 5)  (x + 2) = 10x2 +7x +
10
-2
4.
27 14
-20 -14
10 7
0
3
x2
5
x2
5
0
5
List all possible rational zeros for this polynomial.
p = ±1, 2, 4, 8
q = ±1, 3
f(x) = 3x3 – 2x2 + 4x – 8
p
1 2 4 8
 1, 2, 4,8, , , ,
q
3 3 3 3
Factor the polynomial completely.
4. f(x) = 6x2 + x – 2
-12
4
-3
4 
2

 x     x     3x  2 
6 
3

3 
1

 x     x     2 x  1
6 
2

6.
1
f(x) = x3- 4x2 + 3x = x(x2 – 4x + 3) = x(x – 3)(x – 1)
7.
f(x) = 2x3 + 15x2 + 22x – 15 =
8.
f(x) = 64x3 + 125 =
Find all the zeros of the polynomial equations.
9.
f(x) = x3 + 3x2 + 25x + 75
10.
Use complete sentences to describe what the zeros of a function represent graphically.
Zeros are the x intercepts on the graph
11.
How many zeros and the number of turns does this polynomial
have?
f(x) = 10x3 + 27x2 + 14x + 5
This polynomial has 3 zeros and 2 turns.
12.
Use the graph to give information about the degree, lead coefficient, x-intercepts, domain, range, interval of
increase, interval of decrease and end behavior of the function.
Degree : 3
Leading Coefficient is negative.
x intercept: (-1.25,0)
Domain: all real numbers
Range: all real numbers
Interval of increase: there isn’t one
Interval of decrease: ( -∞, +∞)
End Behavior: left side is going up, right side is going down
Given a function f(x) = x3, write the equation of the new function and graph it given the
following translation:
a) shifted 4 units up : x3 + 4
b) shifted 2 units left: (x + 3)3
c) reflected over the x –axis: -(x3)
d) stretched vertically by a factor of 5 5x3
e) shrinks vertically by a factor of ¼
¼ x3
f) reflected over the x-axis, shifted right 1 unit and shifted down 2 units: –(x - 1)3 - 2
Write a polynomial function of least degree that has 2, -2 and 1 as its zeros.
x=2
x = -2
x=1
x–2=0
x+2=0
x–1=0
(x – 2)(x + 2)(x – 1) = 0
(x2 – 4)(x – 1) = 0
x3 – x2 – 4x + 4 = 0
13.
14.
15.
Explain what do you mean by the multiplicity of zeros of a polynomial function means and how can it help us in
deciding about the graph?
When do we call a function as even or odd and draw an example to show that the function is even or odd.
Even functions have all even exponents and are symmetrical about the y axis.
Odd functions have all odd exponents and are symmetrical about the origin.
16.
Three-Dimensional Space
1. The following graph is of which point?
a.   8,  6,4 b.   8,  6, 4
c.
 8, 6,4
d.
  8, 6,4
2. Write the equation of a sphere with the center at
(x + 5) + (y) + (x – 3) = 16
2
2
2
  5,0, 3 and radius 4?
Graph the following.
3.
(4, 7, 3)
4. 2 x - 3 y + 4 z = 12
5. Find the distance between the points
(- 2, 9, 5) and (4, - 7, - 3).
D
 4  2    7  9    3  5
D
 6
2
2
2
2
 (16) 2  (8) 2
D  36  256  64
D  356  18.87
6. Suppose a spherical parade balloon is sketched in a three-dimensional coordinate system with the origin at the
center of the balloon. Each axis is given in feet. The diameter of the balloon is 40 feet. Write an equation that
represents the spherical balloon.
x2 + y2 + z2 = 400
Rational Exponents and nth Roots
1. Rewrite the expression using rational exponent notation.
a.
 11
8
7
8
7
__ 11 ______
b.

d.
17
2
9

4
9
4
___ 2 _____
Rewrite the expression using radical notation.
c.
 28 5
7
_ 5 287 

5
28

7
1
__ 3 17 ______
3
2. Simplify the following using the properties of exponents
2
a.
(8p q )
c.
3
6 3
16x 4
3
2
3
3
4 p q ________
b.
(4
__ 2 x 3 2 x ______
d.
2 27 - 3 48 =
4 2
3
g5
4
)
2
3
4
4 × 5 ________
2 27  3 48  6 3 12 3  6 3
3. Simplify the following using the properties of exponents.
_= 𝑒 10_____
a. e3 e 7
d. 5
h.
1
4
x
x
7
4
9
5
5
1
__ 2
5
4
3
=
b.
1
____
25
5
e.
42e4 e2
21e3e
7
7
1
2
𝑒
__ 2 ____
4
5
__=7 =
5
√74
 3e 
2 3
c.
5
4
= ( √7) ____f.
5
6 
2
3
3
4
__−27𝑒 6____
6
1
_612 = 62 = √6_____
3
5
_=_𝑥 5 = √𝑥 3 = ( √𝑥 ) ____
5
Graph functions as transformations of f(x) = ax, log a x, ex, or ln x. Investigate & explain charac. of exponential & log.
Functions
Growth/Decay
1. f ( x)  2 x  1
Asymptote
Intercept
Domain
Range
decay
y = -1
(0, 0)
(−∞, +∞)
[−1, ∞)
decay
y=3
(0,10.59)
(−∞, +∞)
[3, ∞)
growth
Y = -1
(0, -2)
(−∞, +∞)
(−∞, −1]
decay
Y=5
(0, 19.78)
(−∞, +∞)
[5, ∞)
5. f  x   log3  x  2  4
Neither
X=2
(2.01,0)
[2, ∞)
(−∞, +∞)
6. f  x    ln( x)  3
neither
X=0
(1,0)
[0, ∞)
(−∞, +∞)
 3
2. f ( x)  2
x 5
3
3. f ( x)  e x  1
4. f ( x)  2e
4.
 x  2 
5
Describe the shifts of the following graphs:
A. y  e x 2  2
shift up 2, shift right 2, reflect across x axis
B. y   log7  x  4
shift left 4, reflect across x axis
 3
C. y   1
2 x
7
D. y  1  ln  x  3
E. y  2  5 
x 3
3
shift up 7, reflect across x axis, reflect across y axis
Shift up 1, Shift right 3, Reflect across x axis
Shift down 3, shift right 3, stretch by factor 2
5. If the population of Emerson, GA was modeled by P (t )  104e0.17t in 1930 (this means where t = 0 is
1930), then what was the population in 1958? t = 1958 – 1930 = 28
𝑃(28) = 104𝑒 0.17∗28 ≈ 12141.58 Since this is a population question and you can’t have a fraction of a person we
round this to 12,141.
r

6. Suppose that $ 2000 is invested at 5% compounded quarterly. Use A  P  1  
n

a. What will the investment be worth
after 3 years?
𝐴 = 2000 (1 +
.05 4∗3
)
4
nt
or A  Pe r
t
b. How much will the amount be after 3 years
if it is instead compounded continuously?
𝐴 = 2000𝑒 .05∗3 = 2323.67
= 2321.51
Define logarithmic functions as inverses of exponential functions
7. Find the inverse of the following functions
y  log4 12 x
y  ln  x  2
y  log x  4
Swap x and y 𝑥 = log 4 12𝑦
𝑥 = ln(𝑦 − 2)
𝑥 = log 𝑦 − 4
𝑒𝑥 = 𝑦 − 2
𝑥 + 4 = log 𝑦
𝑦 = 𝑒𝑥 + 2
𝑦 = 10𝑥+4
Solve for y
4𝑥 = 12𝑦
𝑦=
4𝑥
12
Solving Exponential and Logarithmic Equations
1. Expand the following
a. log 6x3 yz
log 6 + 3 log 𝑥 + log 𝑦 + log 𝑧
64 x 2
b. log8
y
log 8 64 + 2log 8 𝑥 − log 8 𝑦 = 2 + 2log 8 𝑥 − log 8 𝑦
c. ln 4 7x3
ln(7𝑥 3 )4 = 4 (ln 7 + 3 ln 𝑥) = 4 ln 7 + 4 ln 𝑥
d. log
x5 y 2
2y
1
1
1
3
(5log 𝑥 + (−2) log 𝑦) − (log 2 + log 𝑦)
2. Condense the following
a. 3log 6 4  log 6 2
log 6 (43 × 2) = log 6 128
b. 2 log 5  2 log x  log y
log 52 − log 𝑥 2 + log 𝑦 = log 𝑥 2 + log 𝑦 = log
c. 2ln  x  3  ln 6  3ln  x  2 
ln(𝑥 − 3)2 − ln 6 − ln(𝑥 + 2)3 = ln
25
(𝑥−3)2
6
25𝑦
𝑥2
(𝑥−3)2
− ln(𝑥 + 2)3 = ln 6(𝑥+2)3
3. Evaluate the following
a.
log 16
log 5
c. ln  -e 
b. log 1 625
a. log5 16
25
≈ 1.72
d. log 3232
b.
4. Solve the following
a. 5e  x  9  6
b.
1
2x
 4  1  5
4
c.
1
10 x 6   5  14

2
d. 73 x 4  3  18
5. Solve the following, also
a. 2log7 1  2 x   12
b. 15  2log2  x  4  31
c. log7  x  4  2
6. Solve the following inequalities:
x
2
a.    4
3
b. 5x 2  6280
d. log5  x  2   1
c. 1 log3 x  5  4.5
2
Solve Radical Equations:
7. Solve the following
3
a.
c.
6
2a  3  2
2x  4  2
b.
5
2x  1  3
d. 2 x  8
3
e.
8  3x  5  6
f. 3 x  5  3  6
8. Solve the following exponential equations:
a.
4 x- 4 = 512
b. 49 4 x- 1 = 3433 x
c. 1252 x+ 1 = 625- 3 x+ 4