# review for 2nd semester final by xuyuzhu

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```									          Review for 2nd Semester Final – Answer Key
3. If the x in the rational parent function is
9. Graph                                          . If you translate the
replace with x + 8, how is the graph changed?
graph 3 units to the left and 3 units up, write
the domain and range for the new function.
The graph is shifted 8 units to the left, and
the vertical asymptote shifts from x = 0 to                                                                             2                  16

y=                           +3
x        (5       3)
16

x = –8. See graph below.
14

14
12

2
→y=                         +3
12
10

10
x        2           8

8
VA: x = 2  6

6
1                                                                                4
HA: y = 3
1                                    f(x) =
h(x) =
4
x                                                                                2

x+8                              2

25             20             15             10            5                    5            10       15   20

25         20         15          10        5                     5            10         15             20             25
HA: y = 0            2                     2
2                                                                                                           y=
4                 x        5
4

6

6                                                                                                           VA: x = 5
8

8

10
10

12

6. An automobile’s velocity starting from a
12
Domain: All real numbers, x 2
Range: All real numbers, y  2
complete stop is                                       , where v is

measured in feet per second. What happens                                                 11. Solve and check your answer.
to the auto’s velocity as time increases?
The             automobile’s                velocity           would
accelerate rapidly for about 2.5 seconds
then start to slow until it reached just
65

under 28 feet per second.
60

55

50

,  both sides by (x + 2)
45

40                                                                                                           ,  2 from both sides
35
,  by 4
30

 Check
25

20

15
140∙x
10       r(x) =
5∙x + 3
5
This solution does not work, because it
40         30         20          10                   10         20           30         40             50             makes the denominator of the original
60    70     80   90    100

5                                                                          problem = 0  NO SOLUTION!
10

15
18

16

13. Graph                      .                                                                                                14

18
21.                    has moved the parent function
16
in what way?
12

Locate where the graph of14
Solve for y  y 10 b – 5
= x
x+3                   12

y=        is > y = 0
x 7                   10
x+3
8

You will find 2 distinct parts
8                       y=
x 7                                                                  6
where x ≤ -3 and where x > 0
6

4
4
HA: y = 1
2

y = 2x
2
20      15     10     5                 5         10        15    20         25        30         35          40
A
2
y=0
4
20             15                    10             5                  5        10   15
6

VA: x = 7                                                                        2
HA: y = 0
8

10                                                             y = 2x                5             4

12

6

From the denominator
14

16
and        the                                                           HA: y = -5
numerator x = 3 and x = 7
8

23. Suppose that the wealth of a business owner
10

Factors              -3               7                                                   is increasing exponentially. In 1993, he had
\$20 million. In 2001, he had \$35 million. For
Test #         -4            0              8
the domain of 1993 to 2010, what is a
x+3                                                                                   reasonable range for this situation?
x–7                                      
                                                                   1) Create a table with the information given to
determine what the x value will be in 2010, if
value        >0            <0             >0                                           x = 0 is the year 1993.
15. Carl can do a particular job in 4 hours. It takes                                                              Domain             Range
Mike 6.5 hours to do the same job. Write an                                                        year                x              y
equation that shows how long it will take the boys                                                 1993                0          40,000,000
to complete the job if they work together.                                                         1994                1
1995                2
1996                3
1997                4
1998                5
1999                6
Possible variations:                                                                             2000                7
2001                8          55,000,000
2002                 9
2003                10
18. The cost of per person to rent a chartered bus                                                     2004                11
varies inversely to the number of people who                                                       2005                12
ride the bus. If 40 people pay \$9.50 each to                                                       2006                13
ride the chartered bus, what is the cost per                                                       2007                14
person if only 25 people go?                                                                       2008                15
2009                16
                                                                             2010                     17        \$78,696,103

So,                                                                                   2)  Put the data into the “STAT” list in the
calculator. Then STAT > CALC menu >
 k = 40(9.50) = 380                                                              0:ExpReg() > Enter > Enter
3) To put the equation into Y=: Y= > CLEAR >
VARS > 5:Statistics >EQ menu > 1:RegEQ
Then                                                                                  4) Go to the TABLE > Scroll to x = 17

y =\$15.20 per person                                                    Range: 40,000,000 < x < 78,696,103
25. Solve.                                             32. Which equation is the inverse of the function
Change 729 to base 9 raised to a power                 shown in the graph?
This is the parent exponential function which
Since the base is the same, then the                      is given in choice A. The inverse of an
exponents must have the same inequality.                  exponential equation is it’s logarithm equation
a–5<3         + 5 to both sides                   after exchanging x and y.
a<8                                                       x = 2y  log2x = y
27. The weight of a particular bacteria in a culture   NOTE: Equation colors match their graph colors
is tripling every 20 minutes. The weight of the            A.
bacteria was originally 17 grams.
B.
A.    Write an equation which expresses the               This is the inverse equation of A after
weight, w, in grams after t minutes.                 exchanging the x and y then converting to
a log
This is an exponential equations in the                   C.             This is the log form of A: x
form of y = Abx                                           and y have not been exchanged – same
A, Initial amount = 17 grams                      graph
b, the overall rate = 3                           D.
t, time in minutes, and                           The exponential form of this equation is
t/20 = the intervals                              xy = 2! See graph below!?!?!
Equation:
8
D
B. Calculate the weight of the bacteria after 2                                6

hours.
4

2 hours = 120 minutes
So,                       = 12,393 grams                                      2

A&C
10             5                       5           10

29. Merlin Industries bought a laptop for \$2100. It                                 2

is expected to depreciate at a rate of 14% per
year. What will the value of the laptop be in 5                                 4

years? Round to the nearest dollar.
B6

This problem uses the growth formula
y = A(1 – r)t                                                    8

A = \$2100
r = 14%  0.14
t = 5 years                                      33. What is the domain and asymptote of
Substitute into the formula,                                     ?

V(t) = 2100(1-.14)5 = \$947.90                             See graph B in # 32.

Domain: x > 0
40. Solve:                          ?                      52. The x-intercepts of a parabola are at (5, 0)
and (9, 0). What is the equation of the axis of
For logbX - logbY, when subtracting logs,                  symmetry?     What are the roots of the
you can write it as a single log by dividing               equation?
X/Y
Since a parabola is symmetric about a
midline, the the mid-point of the 2 x-
intercepts would give the point through
Convert to exponential form                                 which the axis of symmetry would pass

 20 = 1                                  So,

 both sides by (3x – 2)                   Axis of Symmetry is x = 7
8

3x – 2 = x    Solve for x                                 The roots are the x values of the x-
intercepts, x = 5, 9
x=1                                                        6

Check using change of base formula
4

x=7
2

47. What conic section is formed when the plane                                          (7,0)
intersects the cone an angle to the base but
does not intersect the base?
5

(5,0)
5
(9,0)
10         15

2

A circle
4

51. Graph:
6
2
From the denominator of the x ratio, a = 5                  54. Identify the conic section modeled by the
From the denominator of the y2 ratio, b = 3
Center is at (0, 2)                                              equation:
8
.
Since y leads the equations, the graph will open                 Hyperbola.
up and down.      10

58. Complete the square to find the coordinates
8                                       of the center of the following conic section.

6

BTW this is a circle
4

Add ½ of 4, squared and ½ of 6, squared
2
to both sides
5         3                               (x2 + 4x + 22) + (y2 + 6y +32) = 0 + 22 + 32
10            5                      5             10
Factor each ()s
2                                         (x + 2)2 + (y + 3)2 = 13
Center: (–2, –3)
4

6

8
60. Sonja made a pot of hot tea and recorded its      68 The area of a rectangle is 348 ft2. The length
degrees above room temperature for one               is 5 ft longer than twice the width. Which
hour. Find the most appropriate regression           system of equations can be solved to find the
given the following data.                            length, L, and the width, W, of the rectangle?

Enter data into the STAT List                          The length is 5 ft longer than twice the
STAT > Edit menu >1:Edit                             width  L = 2W + 5
Turn on Plot 1                                         The area of a rectangle is 348 ft2
2nd > Y= > 1: Plot1 > Enter                                 (L)(W) = 348
Graph the data: ZOOM > 9:ZoomStat                      Which is choice A
What kind of function does this look
like – Exponential Decay!                        69. Simone went to an outlet mall and found a
blowout sale. Sweaters were \$5, shirts were
Check out your choices:                              \$3 and a pair of socks was \$1 each. She
A This is a quadratic function                      spent \$70 and bought 26 items. She bought
one more shirt that she did sweaters.
B This is a linear function
Ignoring tax, set up the system of equations
C This is a cubic function ( notice x3)             that can be used to find the number of
D This is the only function that is an              sweaters, shirts and pairs of socks Simone
exponential function-where x is the                  bought. Solve the system.
exponent                                             Let x = the number of sweaters
And y = the number of shirts
You can calculate the exponential                    And z = the number of pairs of socks
function from the data.                                x + y + z = 26
STAT > CALC menu >                                   5x + 3y + z = 70
0:ExpReg>Enter > Enter                                 y = x + 1  standard form x – y = –1
ExpReg
Use the MATRIX in the calculator to solve.
Y = ab^x                                           Using 3x4 matrix
a=145.4910691                                 2nd > x1 > EDIT menu >[A] > 3 > ENTER > 4 >
b=.9380039207                                       ENTER
Enter data
64. The function y = 64(x – 2.50)2 + 400 models                  1     1     1  26
a store’s profits, in dollars, on potato chips                5     3     1  70
where x is the price of a bag of potato chips.                1    1     0  1
What should the store charge for a bag of         nd              nd   1
2 > MODE > 2 > x > MATH menu > B:rref(
potato chips to maximized its profits? What is         >2nd > NAMES menu > 1:[A] > ENTER
the maximum profit earned?                                    1     0     0   7
0     1     0   8
The vertex of this parabola is all we need.                   0     0     1  11
The maximum charge, x, is the x of the           Last column is your answers: x = 7, y = 8, z = 11
vertex, and maximum profit is the y of the
vertex. So from the equation the vertex is       72.  Solve the following system of equations by an
(2.50, 400)                                           appropriate method. (2, 4) or (4, 2)
So, maximum charge is \$2.50 and the                   xy = 8
st
maximum profit is \$400.                               y = x – 2 Substitute this into the 1 equation
x(x – 2) = 8
2
x – 2x = 8        8 from both sides
2
x – 2x – 8 = 0     Factor the quadratic
(x + 2)(x – 4) = 0 Set each factor = 0 and solve
x + 2 = 0  x = –2         x–4=0 x=4
Substitute –2 into xy = 8  (–2)y = 8  y = –4
Check: (–2)( –4) = 8 TRUE        –4 = –2 – 2 TRUE
Substitute 4 into xy = 8  (4)y = 8  y = 2
Check: (4)(2) = 8 TRUE        2 = 4  2 TRUE
74. The length of a rectangle is 3 inches longer           2a. Solve:      0 = 3x2  95x  984
than the width. The area is 54 in2. In solving
this problem, Aaron factors the resulting                  Using the quadratic formula
equation and gives 9 and 6 as answers for
the width. Is he correct? Why or why not?                    x=

This is a “no brainer” – distance can                         =                   8.22   or 39.89
not be negative!
3a. Surface area of a sphere is: SA = 4r2. Find
4 additional problems                                        the radius for a sphere having a surface area
of 462 in2.
1a. Bailey is building a rectangular pen for
animals using the side of a barn as one side.              Substitute the given into the equation:
He has 200 feet of fencing to use for the other
3 sides. What is the maximum area that he                   462 = 4r2  4 both sides then
can enclose?                                                 r =         6.06      in
BARN                                      4a. Write a system of equations that describe the
W               W                                   graph.

L                                                            10

2W + L = 200  L = 200 – 2W
Area = LW  A = (200 – 2W)(W)                                                8

Set = 0 and solve for W: (200 – 2W)(W) = 0
 200 – 2W = 0  2W = 200
6

 W = 100 and W = 0 these are the zeroes                                          4

 x-intercepts are (0,0) and (100,0)
The midpoint of these will give the x value of                                    2

the vertex of the parabola and the y value
of that vertex is the maximum                    10               5                        5          10    15

Midpoint =                                                                   2

Substitute x = 50
2                                              4

 (200 – 2*50)(50) = 5000 ft
Identify the zeroes of the parabola (2,0) and (5,0)
the maximum area that can be enclosed
 x=2 x–2=0
6

OR The most the 2W could = is 100                             (x – 2) is one of the factors of the quadratic
8

So maximum W = 100/2 = 50 ft                                        And  x = 5  x – 5 = 0
this leaves L = 100 ft                                            (x – 5) is the other factor
Area = 50(100) = 5000 ft
2                                   So, (x – 2)(x – 5) = 0 is the factor form
of the equation
Is the maximum area
FOILing  x – 7x + 10 = y one equation
2

Identify 2 points on the line. (–2, 5) and (1, 3)
Substitute into the slope formula:

Then substitute into the point-slope form
y–3=       (x – 1)
Simplify and solve for y
y–3=        x+
y=        x+         the other equation

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