Download Departmental Review Sheet, College Algebra

College Algebra--Departmental Review Sheet
Fall 2005
(College Algebra, 9th Edition. Lial, Hornsby, and Schneider. Pearson, Addison Wesley, 2005.)
Unless otherwise stated, all work should be done algebraically, all work should be shown, and exact answers should be
given. No formula sheets or note cards are allowed for use during the final exam. Each problem on this review sheet is
matched with its corresponding objective in the course outline and the section of the textbook in which it appears.
[Obj 1, Sec 2.4]
1.
Write an equation for the line described. Give your answer in slope-intercept form.
a.
through (1, 3) with slope – 2
b.
through ( - 8, 4) with undefined slope
c.
through 2,3 and 5,1
d.
horizontal through (2, 5)
e.
through (2,6) perpendicular to 3x+2y=4
f.
through (2,6) parallel to 3x+2y=4
[Obj 2, Sec 2.6]
2.
Write an equation of a function with the same basic shape as y 
vertically stretched by a factor of 4.
3.
Explain how the graph of y  ( x  3)  2 can be obtained from the graph of y  x . List all appropriate
translations (be specific about direction and number of units), reflections, shrinks and/or stretches.
x but moved left 3 units, down 1 unit, and
2
2
[Obj 3, Sec 3.3]
4.
5.
Given
a.
b.
c.
Given
a.
b.
c.
f ( x)  x 3  5 x 2  2 x  8
List the possible rational zeros.
Find the zeros of f(x). Show all work and give exact values.
Factor f (x ) into linear factors.
f ( x)  15x 3  61x 2  2 x  8
List the possible rational zeros.
Find the zeros of f(x). Show all work and give exact values.
Factor f (x ) into linear factors.
[Obj 4, Sec 1.4]
6.
Solve algebraically, show all your work, and give an exact answer:
4x2 = 12x - 11
[Obj 4, Sec 1.6]
7.
Solve algebraically, show all your work, and give an exact answer:
[Obj 4, Sec 4.2]
8.
Solve algebraically, show all your work, and give an exact answer:
4x  3
2
1

 2
x 1
x
x x
27 2 x  9 2 x 3
[Obj 4, Sec 4.3]
9.
Solve algebraically, show all your work, and give an exact answer:
log
x
1
 2
16
[Obj 4, Sec 4.5]
10.
Solve algebraically, show all your work, and give an exact answer:
log 2 ( x  1)  log 2 x  log 2 5
11.
Solve algebraically, show all your work, and give an exact answer and an approximate answer to four decimal
places if the solution is irrational:
10e 3 x 7  5
[Obj 5, Sec 1.5]
12.
A rectangular piece of metal is 10 inches longer than it is wide. Squares with sides 2 inches long are cut from the
four corners, and the flaps are folded upward to form an open box. If the volume of the box is 832 in 3, what were
the original dimensions of the piece of metal?
1
[Obj 5, Sec 4.2]
13.
Find the required annual interest rate to the nearest tenth of a percent for $48,000 to grow to $78,186.94 when
compounded semiannually for 5 yr. Round your decimal answer to four decimal places and convert to a percent.
[Obj 5, Sec 4.6]
14.
Assume the cost of a loaf of bread is $2. With continuous compounding, find the time it would take for the cost to
triple at an annual inflation rate of 6%. Round to the nearest tenth of a year.
[Obj 6, Sec 5.2]
15.
Solve the following systems of equations using either Gauss-Jordan elimination with matrices or Gaussian
elimination with back-substitution and matrices. Show all work and list the row operations used at each step.
(For example:  2R1  R 2 ) You may use ROWOPS or MTRXOPS.
x  2y  z  0
b. 2 x  y  z  4
3x  y  z  1
3x  4 y  5
a.
9x  8 y  0
[Obj 7, Sec 1.7]
16.
Solve algebraically. Write your answer in interval notation and express your solution on the number line. Show all
of your work.
3x 2  x  4  0
[Obj 8, Sec 2.2]
17.
Algebraically find the domain of h( x ) 
[Obj 8, Sec 2.7]
18.
Given
f ( x)  x 2  2 x  3 find
x2
f ( x  h)  f ( x )
h
[Obj 9, Sec 2.7]
19.
Given
f ( x)  3x 2  2 x  1 , g ( x)  3x  2 , and h( x)  x  2 find the following:
a.
(f  g )( x )
b.
(g  h)( x  3)
d.
h
 (x )
f 
e.
 f  g x
c.
(gf )( x )
[Obj 10 & 8, Sec 3.1]
20.
Graph the quadratic function
Give exact answers.
f ( x)  x 2  6 x  5 by hand. Give the vertex, axis, domain, range, and intercepts.
[Obj 10, Sec 3.4]
21.
Describe the end behavior of the polynomial graph of f ( x )  ax 4  3x  2 , where a is a negative real number.
22.
Graph the polynomial function f ( x )  x 2 ( x  2)(x  3) by hand. Include the zeros and the end behavior in your
graph.
[Obj 10, Sec 2.5]
23.
Graph by hand:
2 x  1 if x  0
f (x)  
 4  x if x  0
2
[Obj 10 & 8, Sec 3.5]
24.
Given f ( x ) 
a.
b.
c.
d.
e.
x5
, algebraically find the following and show all of your work.
x3
Vertical asymptote(s)
Horizontal asymptotes(s)
x-intercept(s)
y-intercept(s)
Graph.
[Obj 10, Sec 4.2]
25.
Graph f ( x )  2 x  1 by hand. Include the horizontal asymptote in the graph.
[Obj 10, Sec 4.3]
26.
Graph f ( x )  log 2 ( x  1) by hand. Include the vertical asymptote in the graph.
[Obj 11, Sec 2.4]
27.
The data in the table shows the advertising expenditures x and sales volume y for a company for six
randomly selected months. Both are measured in thousands of dollars.
Expenditures, x
2.4
1.6
2.0
2.6
1.4
2.2
a.
b.
Sales, y
212
178
194
230
165
218
Use the regression capabilities of a graphing calculator to find a linear model for y as a function of x.
Write that equation here. Round all decimals to four places.
Use the model to estimate sales for advertising expenditures of $1700 . Show all of your work.
[Obj 11, Sec 3.1]
28.
A frog leaps from a stump 3 feet high and lands 4 feet from the base of the stump. We can consider the initial
position of the frog to be at (0, 3) and its landing position to be at (4, 0). It is determined that the height of the frog
as a function of its horizontal distance x from the base of the stump is given by h( x )  .5x 2  1.25x  3 , where x
and h(x) are both in feet.
a.
Use your graphing calculator to find at what horizontal distance from the base of the stump the frog
reached its highest point. Round your answer to the hundredth.
b.
Use your graphing calculator to find the maximum height reached by the frog. Round your answer to the
hundredth.
[Obj 11, Sec 3.4]
29.
For the polynomial f ( x )  x 4  2x 3  3x 2  6 , use a graphing calculator to approximate each zero as a decimal
to the nearest tenth.
30.
Use a graphing calculator to find the coordinates of the relative extrema of the graph of the polynomial
f ( x )  x 3  4x 2  8x  8 over its entire domain. Give answers to the nearest hundredth.
[Obj 12, Sec 4.1]
31.
Determine whether the following functions have an inverse function. If one exists, algebraically find the inverse
function. Show all steps for finding the equation. If the inverse function does not exist, give the reason.
a.
b.
f ( x )  2x 3  3
f (x)  x 2  4
3
[Obj 13, Sec 5.6]
32.
Solve graphically and show all your work: Find the minimum and maximum values of the objective
function z  2x  4y subject to the following constraints.
33.
3 x  2y  12
5x  y  5
x  0, y  0
Solve graphically and show all your work: A farmer has 100 acres of land available for planting corn and soybeans.
Labor costs $80 per acre for corn and $50 per acre for soybeans. Cost of agricultural products, such as seed and
fertilizer, is $40 per acre for corn and $70 per acre for soybeans. The farmer, who has budgeted at most $7550 for
labor and at most $6250 for agricultural products, anticipates a profit of $230 per acre of corn and $210 per acre of
soybeans. How many acres should be planted in each crop to maximize profit?
[Obj 14, Sec 7.2]
34.
Find a1 , d, and a n for the arithmetic sequence with
The following formulas may be helpful.
Sn 
an  a1  (n  1)d
n
a1  an 
2
S 20  1090 and a20  102 .








an  a1r n1
Sn 
a1 1  r n
1 r
an  a1r n1
Sn 
a1 1  r n
1 r
an  a1r n1
Sn 
a1 1  r n
1 r
an  a1r n1
Sn 
a1 1  r n
1 r
S 
a1
1 r
S 
a1
1 r
S 
a1
1 r
S 
a1
1 r
15
35.
Find the sum:
 4i  8
k 1
The following formulas may be helpful.
Sn 
an  a1  (n  1)d
n
a1  an 
2
[Obj 14, Sec 7.3]
36.
Find a5 and an for the geometric sequence: -2, 6, -18, 54, . . .
The following formulas may be helpful.
Sn 
an  a1  (n  1)d

37.
Find the sum:
n
a1  an 
2
k
 .3
k 1
The following formulas may be helpful.
an  a1  (n  1)d
Sn 
n
a1  an 
2
4