NAME ______________________________________________________ DATE____________ PD. ___
INTEGRATED MATH 3 END OF COURSE REVIEW
1. The table displays the ages of the people attending a family reunion.
Age (years)
Number of
Employees
21–30
11
30–40
15
40–50
13
50–60
7
60–70
4
Relative Frequency
a.
Determine the relative frequency of each interval to complete the table.
b.
Construct a relative frequency histogram for the data.
c.
Does the histogram approximate a normal distribution? Explain.
2. The mean battery life of one model of laptop computer is 6 hours with a standard deviation of 0.6 hour. Assume
that the battery life is normally distributed.
a.
Determine the percent of laptop computers with a battery life between 1 standard deviation below the
mean and 1 standard deviation above the mean.
b.
Determine the percent of laptop computers with a battery life between 4.8 hours and 7.2 hours.
3. The heights of 12-year-old boys in a large school district are normally distributed with a mean of 61 inches and
a standard deviation of 2.5 inches.
a.
Calculate the z-score for Logan, a 12-year-old boy in the school district who is 57 inches tall.
b.
Explain the meaning of the z-score that you found in part (a).
c.
What percent of 12-year-old boys in the school district are shorter than Logan? Explain your method.
4. A survey of 575 high school students reports that 76% of the students would like to have more time for lunch.
a.
Calculate the standard deviation of the sampling distribution to the nearest thousandth.
b.
Determine the 95% confidence interval for the population proportion.
5. Calculate each power of i.
a.
b.
c.
6. Simplify each expression. Show all of your work.
a.
b.
c.
7. Solve the equation for all real or imaginary roots.
8. Umi is making open nets of rectangular prisms out of construction paper as part of an art project. She cuts
squares from each of the four corners of the construction paper, and then folds up the remaining sides to form a
box without a lid. Each sheet of construction paper is 12 inches long by 8 inches wide.
a.
Let x represent the side length of a corner square in inches. Write a function V(x) to represent the
volume of a box in terms of x.
b.
What is the greatest possible integer value of x? Explain your reasoning.
c.
For what integer value of x is the volume of the box a maximum? State the volume and dimensions of
this box.
9. Determine the product, h(x), of the linear and quadratic factors.
and
10. The graph of the basic cubic function
is shown. Suppose that
. Use reference
points and symmetry to complete the table of values for h(x). Then, graph k(x) on the same coordinate plane as
k(x) and label it.
Reference
Points on h(x)
Corresponding
Points on k(x)
(0, 0)
(1, 1)
(2, 8)
11. Use sigma notation to rewrite the finite series, and then compute.
4, 1, 2, 5, 8, 11
________________________
12. A farmer plants fruit trees in the corner of a field. The first row has 2 trees, the second row has 5, the third row
has 8, and so on, as shown in the diagram.
a.
Calculate the number of trees in the 11th row.
b.
Use Gauss’s formula to calculate how many trees the farmer would need for 11 rows.
13. Compute the series
.
14. A local charity is selling raffle tickets. The first day they sell 23 tickets. They expect their sales to increase by
8% each day as more people hear about the raffle.
a.
At this rate of increase, how many tickets will they sell on the 10th day? Round to the nearest whole
number.
b.
At this rate of increase, what is the total number of tickets they will sell by the 10th day?
15. Consider the function
.
a.
What type of function is f(x)? Explain your answer.
b.
Determine the vertical and horizontal asymptotes of f(x).
c.
Determine the domain and range of f(x).
d.
Describe the end behavior of f(x).
e.
In which quadrant(s) does the graph of f(x) exist?
16. Consider the function
.
a.
Describe how you would obtain the graph of g(x) from the graph of
b.
Determine the vertical and horizontal asymptotes of g(x).
c.
Determine the domain and range of g(x).
d.
Determine the y-intercept of g(x).
e.
Sketch the graph of g(x). (USE GRAPH ON NEXT PAGE)
.
17. Determine whether the graph of the rational function
has a vertical asymptote, a
removable discontinuity, or both. List any discontinuities and explain your reasoning.
18. Consider the functions
and
.
a.
Graph both functions on the same coordinate plane.
b.
Use your graph to solve the equation
. Explain your reasoning.
19. Combine like terms, if possible, and write your final answer in radical form.
20. Solve the equation and check for extraneous solutions.
21. Consider the logarithmic equation
a.
Identify the base, the argument, and the exponent in this equation.
b.
Rewrite the equation in exponential form.
c.
Solve the equation.
22. Suppose
and
23. Consider the exponential equation
. Write an algebraic expression for
.
1000.
a.
Solve the equation by using equivalent bases.
b.
Solve the equation by converting it into a logarithmic equation and then using the Change of Base
Formula.
24. Consider the logarithmic equation
a.
Solve the equation.
b.
Check your work.
Classify thesituation as a sample survey, an observational study, or an experiment. Then identify the population,
the sample, and the characteristic of interest.
25. In a poll of 1500 registered voters, 705 people or 47% said they would vote in favor of a referendum to impose
term limits for members of the city council. The margin of error for this poll is ±4%.
a.
Does the poll represent a sample survey, an experimental study, or an experiment?
b.
What does this poll information suggest about the outcome of the referendum?
c.
Based on the poll, can you say that the referendum will fail?
26. Consider the function
a.
.
Graph f(x) and its inverse,
, in the same coordinate grid. Label each graph.
b.
What are the domain and range of f(x)?
c.
What are the domain and range of
d.
What are the asymptotes of each graph?
e.
Write an equation for
f.
What are the intercepts of each function?
g.
What are the intervals of increase and decrease for each function?
h.
What is the end behavior of both functions?
?
.
27. A geometry teacher distributes sheets of construction paper to her class so that they can create open nets of
rectangular prisms. She instructs her students to cut squares from each of the four corners of the construction
paper, and then fold up the remaining sides to form a box without a lid. Each sheet of construction paper is 24
inches long by 18 inches wide.
a.
Let x represent the side length of a corner square in inches. Write a function V(x) to represent the
volume of a box in terms of x.
b.
What is the greatest possible integer value of x? Explain your reasoning.
c.
For what integer value of x is the volume of the box a maximum? State the volume and dimensions of
this box.
28. Determine the product, h(x), of the linear and quadratic factors.
and
29. The graph of the basic cubic function
is shown. Suppose that
. Use reference points
and symmetry to complete the table of values for k(x). Then, graph k(x) on the same coordinate plane as h(x) and
label it.
Reference
Points on h(x)
(0, 0)
(1, 1)
(2, 8)
Corresponding
Points on k(x)
30. Identify the sequence as arithmetic, geometric, or neither. If possible, determine the next term of the sequence.
5, 20, 80, 320, 1280 …
Type of sequence: _______________
Next term: _______________
31. Use sigma notation to rewrite the finite series, and then compute.
6, 7, 8, 9, 10, 11
_________________________
32. The chairs in an auditorium are arranged as shown in the diagram. The first row has 3 chairs, the second row has
5, the third row has 7, and so on.
a.
Calculate the number of chairs in the 9th row.
b.
Use Gauss’s formula to calculate how many chairs would be needed for 9 rows.
33. The first three terms of an infinite sequence are represented by the figures shown. Write a formula to compute
the first n terms of the series.
34. Compute the series
35. Determine whether the series is convergent or divergent, and explain how you know. If the series is convergent,
use the formula to compute the sum. If it is divergent, write infinity.
Convergent or divergent? _______________
Explanation:
S = _______________
36. Consider the function
.
a.
Describe how you would obtain the graph of g(x) from the graph of
b.
Determine the vertical and horizontal asymptotes of g(x).
c.
Determine the domain and range of g(x).
d.
Determine the y-intercept of g(x).
e.
Sketch the graph of g(x).
.
37. Determine whether the graph of the rational function
has a vertical asymptote, a
removable discontinuity, or both. List any discontinuities and explain your reasoning.
38. Consider the functions
and
.
a.
Graph both functions on the same coordinate plane.
b.
Use your graph to solve the equation
. Explain your reasoning.
39. Determine the least common denominator (LCD) of the rational expressions.
40. Explain why the function
is not invertible.
41. The graph of the square root function,
, is shown.
a.
Use inequality statements to write the domain and range of f(x).
b.
Sketch the graph obtained by shifting the graph of f(x) to the left 3 units. Label the new function g(x).
c.
Write an an equation for the function g(x).
d.
Use inequality statements to write the domain and range of g(x).
42. Rewrite the radical by extracting all possible roots, and write your final answer in radical form.
43. Combine like terms, if possible, and write your final answer in radical form.
44. Solve the equation and check for extraneous solutions.
45. The graph of the function f(x) is shown.
a.
Draw a graph of the inverse function,
, on the same coordinate grid. Show the three
corresponding reference points on your graph.
b.
Write an equation for the function
c.
What are the asymptotes for f(x) and
46. Describe the graphical transformations on
a.
b.
.
?
that produce each function.
47. Consider the logarithmic equation
a.
Identify the base, the argument, and the exponent in this equation.
b.
Rewrite the equation in exponential form.
c.
Solve the equation.
48. Suppose
, logb
, and
Write an algebraic expression for
49. Consider the exponential equation
a.
Solve the equation by using equivalent bases.
b.
Solve the equation by converting it into a logarithmic equation and then using the Change of Base
Formula.
50. Consider the logarithmic equation
a.
Solve the equation.
b.
Check your work.
51. A function and one of its factors is given. Use long division to determine another factor.
52. Given
, use synthetic division to determine f(5).
53. Use the Factor Theorem to determine the unknown coefficient, a, if
.
is a factor of
54. Factor the polynomial completely.
55. The graph of
is shown. Determine all zeros of the function.
56. Use the Binomial Theorem and substitution to expand the binomial
.
57. A function and one of its factors is given. Use long division to determine another factor.
58. Factor the polynomial completely.
59. Use the Binomial Theorem and substitution to expand the binomial
.
For problems 60 – 63, add, subtract, multiply, or divide as indicated. List any restrictions for the variable(s) and
simplify the answers when possible.
60.
61.
62.
63.
Solve each equation.
64.
65.
66. The speed of the current in a river is 2 kilometers per mile. Bob kayaks 24 kilometers upstream then 24
kilometers downstream in a total of 5 hours.
a.
Let x represent Bob’s speed (rate) if there were no current. What is the upstream rate, in terms of x?
What is the downstream rate in terms of x?
b.
A formula for distance d, rate r, and time t is
c.
An expression for the upstream time is
d.
Since the total time is 5 hours, the sum of the two expressions is 5. Write and solve an equation to
calculate Bob’s speed in the kayak if there were no current.
. Solve this formula for t.
. Write an expression for the downstream time.
INTEGRATED MATH 3 END OF COURSE REVIEW
Answer Section
1.
a.
Age (years)
Number of
Employees
Relative Frequency
21–30
11
0.22
30–40
15
0.30
40–50
13
0.26
50–60
7
0.14
60–70
4
0.08
See table.
b.
c.
No. The histogram does not approximate a normal distribution because it is neither bell-shaped nor
symmetrical about the mean.
a.
Approximately 68% of the laptop computers have a battery life between 1 standard deviation below the
mean and 1 standard deviation above the mean.
b.
Approximately 95% of the laptop computers have a battery life between 4.8 hours and 7.2 hours.
2.
A battery life of 4.8 hours is 2 standard deviations below the mean and a battery life of 7.2 hours is 2
standard deviations above the mean. I know that approximately 95% of data in a normal distribution is
between 2 standard deviations below the mean and 2 standard deviations above the mean.
3. a.
b.
Logan’s z-score means that his height is 1.6 standard deviations below the mean.
c.
Approximately 5.48% of the 12-year-old boys in the school district are shorter than Logan.
I used a graphing calculator and entered normalcdf(0, 57, 61, 2.5).
5. a.
b.
c.
6. a.
b.
c.
7.
9.
10.
Reference
Points on h(x)
Corresponding
Points on k(x)
(0, 0)
( 1, 1)
(1, 1)
(0, 0)
(2, 8)
(1, 7)
11.
12.
a.
There are 32 trees in the 11th row.
b.
The farmer would need 187 trees for 11 rows.
13.
14. a.
b.
At this rate of increase, they will sell 46 tickets on the 10th day.
At this rate of increase, they will sell 334 tickets by the 10th day.
15.
a.
The function f(x) is a rational function because both the numerator, 3, and the denominator,
polynomials.
b.
The vertical asymptote is
The horizontal asymptote is
.
.
c.
The domain is all real numbers except 0.
The range is all real numbers except 0.
d.
As x approaches negative infinity, f(x) approaches 0.
As x approaches infinity, f(x) approaches 0.
e.
The graph exists in the first and second quadrants.
, are
16.
a.
Translate (shift) the graph of f(x) 1 unit to the left and 2 units down.
b.
The vertical asymptote is
The horizontal asymptote is
c.
The y-intercept is (0, 1).
19.
.
The domain is all real numbers except .
The range is all real numbers except .
d.
e.
.
20.
Check:
Extraneous Solution

So, x=7 is the only solution
21.
a.
b.
c.
22.
The base is 81, the argument is 27, and the exponent is n.
23. a.
b.
24.
a.
b.
:

The logarithms of negative numbers are undefined, so
So, the only solution for this problem is x=5
is extraneous.
26.
a.
b.
The domain is
The range is
c.
d.
.
.
The domain is
.
The range is
.
The graph of f(x) has a horizontal asymptote at
The graph of
has a vertical asymptote at
.
.
e.
f.
For f(x), the y-intercept is (0, 1). There is no x-intercept.
For
, the x-intercept is (1, 0). There is no y-intercept.
g.
Both f(x) and
increase over all real numbers.
h.
As
As
.
As
As
.
from the right,
.
.
28.
29. ANS:
Reference
Points on h(x)
Corresponding
Points on k(x)
(0, 0)
(0, 1)
(1, 1)
(1, 0)
(2, 8)
(2, 7)
30.
Type of sequence: geometric sequence
31.
32. a.
b.
33.
34.
There are 19 chairs in the 9th row.
The auditorium would need 99 chairs for 9 rows.
35.
Convergent or divergent? convergent
Explanation: The series is convergent because the common ratio is between 0 and 1.
36.
a.
Translate (shift) the graph of f(x) 1 unit to the right and 3 units up.
b.
The vertical asymptote is
The horizontal asymptote is
c.
.
The domain is all real numbers except 1.
The range is all real numbers except 3.
d.
The y-intercept is (0, 1).
e.
.
39.
40.
The Horizontal Line test shows that two different values of x may correspond to the same value of y, so the
inverse of
is not a function, which means that
is not invertible.
41.
a.
The domain is
. The range is
.
b.
c.
d.
42.
43.
The domain is
. The range is
.
44.
Check:
Extraneous Solution

So, the only solution is x=3
45.
a.
b.
c.
The function f(x) has a horizontal asymptote at
The function
has a vertical asymptote at
.
.
46.
a.
The transformation is a reflection across the x-axis.
b.
The transformations are a horizontal translation left 2 units, a vertical dilation by a factor of 6, and a
vertical translation up 4 units.
a.
The base is 16, the argument is 32, and the exponent is n.
47.
b.
c.
48.
49.
a.
b.
50.
a.
b.

The logarithms of negative numbers are undefined, so
So, the only solution is x=5
51.
Another factor of f(x) is
.
52.
So,
is extraneous.
53.
If
is a factor, then by the Factor Theorem,
By the Transitive Property,
.
.
54.
55.
From the graph, I can determine that one of the zeros is
.
56.
Let
and let
.
57.
Another factor of f(x) is
58.
59.
Let
60.
and let
.
.
61.
62.
63.
64.
65.
66. .
a.
upstream:
; downstream:
b.
c.
d
Bob’s speed would be 10 km per hour if there were no current.