MATHEMATICS-55A
Final Examination
Name
STUDY GUIDE
Quiz/Examination Procedures and Rules
1. CLOSED Book Exam; NO reference materials permitted
2. Examination Duration = 110 minutes (The Final Exam Period)
3. Permitted Tools
 Pen or Pencil
 Straight-Edge Ruler
 Scientific Calculator (NO Laptop Computers allowed)
4. Do NOT use a Calculator if so indicated by the problem statement
5. Partial Credit Awarded so SHOW ALL WORK  NoWork = NoCredit
6. Express all answers in SIMPLEST terms
7. Identify FINAL Answer by BOXING IT or by Placing it in the Answer SLOT
8. Academic dishonesty will not be tolerated
u  x   2x  3  5
2
Problem-00
Graph Below the function u(x) shown at right using
TRANSLATION TECHNIQUES. Be sure to
properly construct and label the graph
y
y
x
5
O
3
© Bruce Mayer, PE • Chabot College • 291237135 • Page 1 of 47
x
© Bruce Mayer, PE • Chabot College • 291237135 • Page 2 of 47
Problem-00  HAND GRAPH
 Note the DASHED Axes that indicate the translations
© Bruce Mayer, PE • Chabot College • 291237135 • Page 3 of 47
Problem-1
Solve the following Formula for the Variable P
Solve for P →
A  P  Prt
© Bruce Mayer, PE • Chabot College • 291237135 • Page 4 of 47
r x   x 2  6 x  5
Problem-2
Graph Below the function r(x) shown at right. Be sure to
properly construct and label the graph
10
y
9
8
7
6
5
4
3
2
1
0
O
-1
x
-2
-3
-4
-5
-6
-2
-1
0
1
2
3
4
5
Problem-2  Significant- Points Summary





“a” is positive so parabola opens UP
X-intercepts → (1,0) & (5,0)
Y-intercept → (0,5)
X-vertex = h = -b/2a = 3
Y-vertex = r(h) = -4
© Bruce Mayer, PE • Chabot College • 291237135 • Page 5 of 47
6
7
8
© Bruce Mayer, PE • Chabot College • 291237135 • Page 6 of 47
Problem-3
For the graph at right, identify
the domain and range.
Then use the VERTICAL LINE
TEST to determine whether
the
graph
represents
a
function (please show work
directly on the graph)
a. (3 Points) Domain.
Domain = {x|x 1}
b. (3 Points) Range
Range = {y|y is all Real Numbers}
c. (4 Points) State YES or NO as to whether the graph represents a function. Be sure to
the show the work that justifies the answer.
Function? =
NO – the Graph FAILS Vertical Line Test
Vertical Line Test
To determine whether a graphical relation is a function draw or imagine
vertical lines through each value in the domain. If each vertical line
intersects the graph at only one point, the relation is a function. If any
vertical line intersects the graph more than once, the relation is not a
function.
© Bruce Mayer, PE • Chabot College • 291237135 • Page 7 of 47
Problem-4
Solve the following equation for x by COMPLETING THE SQUARE
Complete-Sq & Solve →
8  4x  x2
© Bruce Mayer, PE • Chabot College • 291237135 • Page 8 of 47
Problem-5
Find (fg)(x) for functions f & g given as
f x   7  x 2
and
g x   x  3
© Bruce Mayer, PE • Chabot College • 291237135 • Page 9 of 47
Problem-6
UCBerkeley periodically mows the OutField grass at the Evans Diamond BaseBall Field. The
CAL Grounds crew has two riding mowers. The FAST mower, working alone, can cut the grass
in 39 minutes The SLOW mower takes one-third longer to complete the outfield cut. How
long will it take to mow the OutField Grass if BOTH mowers run at the same time?
OutField Mow-Time =
156/7 minutes ≈ 22.29 minutes (need UNITS)
© Bruce Mayer, PE • Chabot College • 291237135 • Page 10 of 47
© Bruce Mayer, PE • Chabot College • 291237135 • Page 11 of 47
© Bruce Mayer, PE • Chabot College • 291237135 • Page 12 of 47
Problem-7
FACTOR the following expression. If the expression is PRIME, then state this.
Factor →
a 2  b 2  10b  25
See Example 5.5-7 on pg 351 in TextBook
© Bruce Mayer, PE • Chabot College • 291237135 • Page 13 of 47
Problem-8
SOLVE the following Equation for t.
Solve for t →
3t 3  9t 2  30t
© Bruce Mayer, PE • Chabot College • 291237135 • Page 14 of 47
Problem-9
Solve the following inequality. Other than Ø, graph the Solution Set on the Number Line, and
State the Solution using INTERVAL NOTATION. Be sure to properly LABEL and Construct the
graph
Solve →
2x  3  5
© Bruce Mayer, PE • Chabot College • 291237135 • Page 15 of 47
Problem-10
Use the QUADRATIC FORMULA to Solve the following equation for x
Solve with Quadratic Formula →
2xx  2  5
© Bruce Mayer, PE • Chabot College • 291237135 • Page 16 of 47
Problem-11
Solve the Following Compound InEquality. State the answer in SET-BUILDER NOTATION.
Solve →
2x  7  3 and 5x  4  6
© Bruce Mayer, PE • Chabot College • 291237135 • Page 17 of 47
Problem-12
Solve the Following Rational Equation for t. If the equation has no solution, then state this and
justify the statement.
Solve for t →
2t  9 
18
t
© Bruce Mayer, PE • Chabot College • 291237135 • Page 18 of 47
Problem-13
Rationalize the denominator in the following Expression.
Rationalize
Denomination →
13
7  11
© Bruce Mayer, PE • Chabot College • 291237135 • Page 19 of 47
Problem-14
FACTOR the following expression. If the expression is PRIME, then state this and justify the
statement.
Factor →
18a 3  48a 2b  32ab 2
© Bruce Mayer, PE • Chabot College • 291237135 • Page 20 of 47
Problem-15
Carmelita set out on a 16 mile Bike Ride.
Unfortunately after 10 miles of Biking BOTH tires
Blew Out. Carmelita had to complete the trip on
Foot. Carmelita biked four miles per hour (4 mph)
faster than than she walked, and the Entire journey
took 2hrs and 40min. For this situation find
Carmelita’s Walking Speed (or Rate)
Walking Speed =
1  10  4.16 mph
© Bruce Mayer, PE • Chabot College • 291237135 • Page 21 of 47
© Bruce Mayer, PE • Chabot College • 291237135 • Page 22 of 47
© Bruce Mayer, PE • Chabot College • 291237135 • Page 23 of 47
g x   4  x  3
Problem-16
Graph Below the function g(x) shown at right. Be sure to
properly construct and label the graph
10
9
y
8
7
6
5
4
3
2
1
0
-1
x
0
-2
-3
-4
-5
-6
-7
-8
-9
-10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
X
-10
-9
-8
-7
-6
-5
-4
Y
-9
-8
-7
-6
-5
-4
-3
X
-3
-2
-1
0
1
2
3
0
1
Y
-2
-1
0
1
2
3
4
2
3
4
5
X
4
5
6
7
8
9
10
© Bruce Mayer, PE • Chabot College • 291237135 • Page 24 of 47
6
Y
3
2
1
0
-1
-2
-3
7
8
9
10
Problem-17
SIMPLIFY the following complex rational expression. Be sure to state the answering in FULLY
FACTORED form.
Simplify
→
5u 1  2v 1
25u  2  4v  2
© Bruce Mayer, PE • Chabot College • 291237135 • Page 25 of 47
Problem-18
Solve the following rational equation. If the equation has no solution, then state this and justify
this response.
Solve for x→
1
7
3
 
5x  5 5 x  1
© Bruce Mayer, PE • Chabot College • 291237135 • Page 26 of 47
Problem-19
Simplify the following rational expression. Include absolute value bars if needed.
Simplify →
5
 64z  13
5
Problem-20
Simplify the following rational expression. Write the answer in RADICAL Notation.
Simplify →
7
x2  x
© Bruce Mayer, PE • Chabot College • 291237135 • Page 27 of 47
Problem-21
Write the POINT-SLOPE equation for the Line that passes thru points (7, −11) and (−13, 5).
© Bruce Mayer, PE • Chabot College • 291237135 • Page 28 of 47
Problem-22
Find the result of the complex product below. Write complex answers in the form a + bi
Multiply →
6  i 7 6  i 7 
Problem-23
The equation of a certain line is given in GENERAL form below. ReWrite this line equation in
SLOPE-INTERCEPT form
Write in Slope-Intercept form →
11x  31 y  19
© Bruce Mayer, PE • Chabot College • 291237135 • Page 29 of 47
Problem-24
In lecture we discussed how a quadratic function contains within it FIVE pieces of information
that are useful in sketching the parabolic graph for the function. This problem asks that we list
these five info-points for the following Quadratic function.
Quadratic function →
f x   8  2x  3
2
a) (1 points) Circle the best answer of the two below. The graph of this function is:

Concave UP (bowl-shaped)

Concave DOWN (dome-shaped)
b) (4 points) Find the VERTEX of this function graph:
Vertex →
(h, k) = (3, 8)
c) (4 Points) Find the x-intercept(s) for this function graph:
x-intercept(s) →
1 & 5  (1. 0) & (5, 0)
d) (1 Points) Find the y-intercept(s) for this function graph:
y-intercept(s) →
−10  (0, −10)
© Bruce Mayer, PE • Chabot College • 291237135 • Page 30 of 47
Problem-25
A dependent quantity u varies directly as v, and inversely as the square of w. Also u = 7 when
v = 9 and w = 6.
a) (10 points) Write the EQUATION of Variation for this situation
Variation Equation →
u  28
v
w2
b) (5 points) Find u when v = 4 and w = 8
u=
7 4  1.75
© Bruce Mayer, PE • Chabot College • 291237135 • Page 31 of 47
Problem-26
Use Radical Notation to ReWrite the following Expression. Simplify if possible
Use Radical-Notation & Simplify →
5
2
4 8
2
3
© Bruce Mayer, PE • Chabot College • 291237135 • Page 32 of 47
Problem-27
Graph Below this Equation. Be sure to properly construct and label the graph
0  56  7 y  8 x  8 x  7 y  56  make T - Table of intercepts
© Bruce Mayer, PE • Chabot College • 291237135 • Page 33 of 47
Problem-28
Solve the Following formula for h
r
Solve for h →
3V
h
Problem-29
FACTOR the following expression. If the expression is PRIME, then state this and justify the
statement.
Factor →
u 4  1000u
© Bruce Mayer, PE • Chabot College • 291237135 • Page 34 of 47
Problem-30
For functions f & g as given below, find the DOMAIN of f+g. Write the answer
using INTERVAL Notation.
f x   
19 x
4 x
g x   
37
x 8
The Domain of f+g is the OVERLAPPING Region of the two individual
Domains, In this case
 f is NOT Defined (Div by Zero) for x = -8. Thus 8 is NOT part of
the domain of f
 g is NOT Defined (Div by Zero) for x = 4. Thus 4 is NOT part of
the domain of g
Then both -8 & 4 are excluded from the Domain of f+g
 Domain = {x|x is a real number with x ≠ -8 and x ≠ 4}
Domain  f x  g x   ,8   8,4  4, 
© Bruce Mayer, PE • Chabot College • 291237135 • Page 35 of 47
Problem-31
Solve the following Rational Equation for x. If there is No Solution, then state this and justify
the statement
Solve for x →
1 x  2 x  3  4
© Bruce Mayer, PE • Chabot College • 291237135 • Page 36 of 47
Problem-32
The total (sides plus bottom) Surface Area, S, of a
Right Circular Cone (see diagram at right) may be
calculated from the formula:
S  r r  h  r
2
2
2
h
A certain cone has a radius, r, of 130 mm, and a
surface area of 147,115.3 mm2.
 Find the HEIGHT, h, for this cone
r
h = 190 mm
© Bruce Mayer, PE • Chabot College • 291237135 • Page 37 of 47
Problem-33
Use Rational Exponents to Simplify the following expression. Assume that p & q represent
positive numbers
Simplify →
3
p 2 q  6 pq 2
© Bruce Mayer, PE • Chabot College • 291237135 • Page 38 of 47
Problem-34
Perform the following Rational Addition. Assume that p & q represent positive numbers
Add →
q  3 405 p  3 120 pq 3
© Bruce Mayer, PE • Chabot College • 291237135 • Page 39 of 47
Problem-35
Write a QUADRATIC EQUATION with the SOLUTION SET shown at right
© Bruce Mayer, PE • Chabot College • 291237135 • Page 40 of 47
 2 7
 , 
 3 5
Problem-36
The Heat Loss, Q, of a window varies
JOINTLY as the window’s Area, A, and the
difference between the OutSide and InSide
air-Temperatures, ΔT.
A window 300 mm wide by 600 mm long loses
1200 Joules per hour when the OutSide
Temperature is 20 °C colder than the
temperature inside.
Find Q for a 600mm by 900mm window
when the OutSide Temperature is 10 °C
colder than the Inside Temperature.
Q = 1800 J/hr
Q  UAT  U  Q /  AT 
U  1200 / 300  600  20   1 3000
1 J hr
U
3000 mm 2  C
 Qnew
 Qnew


1 J hr
600mm  900mm  10 C


2
 3000 mm  C 

1 J hr 5 400 000 mm 2  C 

 1800 J hr
2
3000 mm  C
© Bruce Mayer, PE • Chabot College • 291237135 • Page 41 of 47
Problem-37
Simplify the following expression
Simplify →
i 28  i 30
Problem-38
Perform the following Complex Division. Write the result in the form a + bi
Divide →
6  3i
4  2i
© Bruce Mayer, PE • Chabot College • 291237135 • Page 42 of 47
Problem-39
When you take Physics 4A you will
learn that “Ballistics” describes the
motion of “Launched” objects that then
fall under the influence of the Earth’s
gravity.
The Diagram at right describes the
major elements that determine the
Ballistic (or Parabolic) trajectory of an
object launched:
 At Height yo above the ground
 With original velocity, vo
 At a Take-Off angle θ relative to
the (horizontal) ground
Given yo, vo, and θ then calculations can be made for the:
 Flite-Height, y(t)
 Down-Range Distance x(t)
 The Maximum Height, h
 The Overall Range Distance at “Crash-Down”, R
 The final velocity at Crash-Down, vf
The Ballistics Equation below approximates the Flite-Height, y(t), of a projectile launched from
the top of an 155 foot tall tower, at 300 feet per second, with a TakeOff angle of 40°:
yt   16t 2  192t  155
Use the Ballistics Equation above to
determine the time(s) for which the FliteHeight, y, is 695 feet
ANS:
9
15
t s & t s
2
2
Mr. Scott Hildreth – Chabot
College Physics 4A instructor
© Bruce Mayer, PE • Chabot College • 291237135 • Page 43 of 47
© Bruce Mayer, PE • Chabot College • 291237135 • Page 44 of 47
© Bruce Mayer, PE • Chabot College • 291237135 • Page 45 of 47
Problem-40
Among ALL pairs of numbers whose difference is 24, FIND a pair whose PRODUCT is as
SMALL as possible. Also determine this smallest product
Number Pair → 12 & −12
Product = −144
© Bruce Mayer, PE • Chabot College • 291237135 • Page 46 of 47
Problem-41
You have 80 feeet of fencing to enclose a rectangular region. Find the Dimensions of the
Rectangle that MAXIMIZE the enclosed area. Also determine this maximum area
Dimensions = 20ft Wide by 20ft Long
Maximum Area = 400 sq-ft
Print Date/Time = 29-May-16/04:03
© Bruce Mayer, PE • Chabot College • 291237135 • Page 47 of 47