Download Module 8 Study Guide - Valley Oaks Charter School Tehachapi

Algebra 2 Module 8 Study Guide
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Rewrite the function
in the form
. Then graph the function.
A
C
y
–10 –8
–6
–4
y
10
10
8
8
6
6
4
4
2
2
–2
–2
2
4
6
8
10
x
–10 –8
–6
–4
–2
–2
–4
–4
–6
–6
–8
–8
–10
–10
B
–10 –8
6
8
10
x
–6
–4
2
4
6
8
10
x
y
10
10
8
8
6
6
4
4
2
2
–2
–2
2
4
6
8
10
x
–10 –8
–6
–4
–2
–2
–4
–4
–6
–6
–8
–8
–10
–10
2. Which function is continuous?
A
C
B
____
4
D
y
____
2
3. Identify all asymptotes of
D
.
A
B
C
D
____
vertical asymptote:
; horizontal asymptote:
vertical asymptote:
; horizontal asymptotes:
no vertical asymptote; horizontal asymptote:
no vertical asymptote; horizontal asymptote:
and
4. Using the graph of
as a guide, describe the transformation and graph
A Translate
C Translate
left 3 units.
down 3 units.
y
–10 –8
–6
–4
B Translate
y
10
10
8
8
6
6
4
4
2
2
–2
–2
.
2
4
6
8
10
x
–10 –8
–6
–4
–2
–2
–4
–4
–6
–6
–8
–8
–10
–10
D Translate
up 3 units.
2
4
6
8
10
x
6
8
10
x
right 3 units.
y
y
10
9
8
6
6
4
3
2
–10 –8
–6
–4
–2
2
4
6
8
10
x
–10 –8
–6
–4
–2
–2
2
–3
–4
–6
–6
–8
–9
–10
____
5. Identify the asymptotes, domain, and range of the function
A Vertical asymptote:
Domain:
Horizontal asymptote:
Range:
B Vertical asymptote:
Domain:
Horizontal asymptote:
Range:
C Vertical asymptote:
Domain:
Horizontal asymptote:
Range:
D Vertical asymptote:
Domain:
Horizontal asymptote:
Range:
.
4
____
6. Identify the zeros and vertical asymptotes of
. Then graph.
A Zeros at and .
Vertical asymptote:
C Zeros at and .
Vertical asymptote:
y
y
20
20
10
10
–20
–20
–10
10
20
–10
10
20
x
10
20
x
x
–10
–10
–20
–20
B Zeros at
and .
Vertical asymptote:
D Zeros at
and .
Vertical asymptote:
y
–20
____
y
20
20
10
10
–10
10
20
x
–20
–10
–10
–10
–20
–20
7. Identify the zeros and asymptotes of
A Zeros:
and 3
Vertical asymptotes:
Horizontal asymptote:
,
. Then graph.
C Zeros:
and 4
Vertical asymptotes:
Horizontal asymptote:
,
y
y
12
12
10
10
8
8
6
6
4
4
2
2
–12 –10 –8 –6 –4 –2–2
2
4
6
x
8 10 12
–12 –10 –8 –6 –4 –2–2
–4
–4
–6
–6
–8
–8
–10
–10
–12
–12
B Zeros:
and 3
Vertical asymptotes:
Horizontal asymptote:
2
____
12
12
10
10
8
8
6
6
4
4
2
2
2
4
6
x
8 10 12
–12 –10 –8 –6 –4 –2–2
–4
–4
–6
–6
–8
–8
–10
–10
–12
–12
.
4
6
x
C There is a hole in the graph at
.
y
10
10
8
8
6
6
4
4
2
2
–2
–2
8 10 12
,
2
y
–4
x
. Then graph.
A There is a hole in the graph at
–6
8 10 12
y
8. Identify holes in the graph of
–10 –8
6
D Zeros:
and 4
Vertical asymptotes:
Horizontal asymptote:
,
y
–12 –10 –8 –6 –4 –2–2
4
2
4
6
8
10
x
–10 –8
–6
–4
–2
–2
–4
–4
–6
–6
–8
–8
–10
–10
B There are no holes in the graph.
2
4
6
D There is a hole in the graph at
8
x
10
.
y
–10 –8
–6
–4
y
10
10
8
8
6
6
4
4
2
2
–2
–2
2
4
6
8
10
x
–10 –8
–6
–4
–2
–2
–4
–4
–6
–6
–8
–8
–10
–10
2
4
6
8
10
x
Multiple Response
Identify one or more choices that best complete the statement or answer the question.
____
1. The graph of the function
is shown. The graph is symmetric with respect to which of the
following?
y
6
4
2
–6
–4
–2
2
4
6
x
–2
–4
–6
A
B
C
D
E
The origin
The -axis
The -axis
The line
The line
Short Answer
1. Alyssia is a pilot whose airplane travels 150 miles per hour when there is no wind. One day, it took Alyssia a
total of t hours to fly 200 miles from airport A to airport B with a headwind of w mi/h and then back from
airport B to airport A with a tailwind of w mi/h. Write an equation relating and .
2. A soup company wants to make a new soup can that has a volume of 450 mL. Write an equation that relates
the can’s height to its volume and its radius where h and r are measured in centimeters (remember that
). Then graph this relationship with the appropriate axis labels and scales.
3. Elena, Brian, Carmen, and Drew carpool to work every morning. The fuel efficiency of Brian’s car while
driving in the city is 4 miles per gallon less than while driving on the highway. Their commute to work is 20
miles on the highway and 8 miles in the city. Brian wants to determine how much gas is consumed driving to
and from work.
a. Choose variables to represent the unknown quantities in this problem. State which variable is the
independent variable and which is the dependent variable, and give the unit of measurement for each.
b. Using unit analysis as a guide, write a function that models the total amount of gas consumed on a round
trip to and from work. Explain your reasoning.
4. Consider the functions
and
. The function
Explain how this graph can be used to approximate the solutions of the equation
approximate the solutions to the nearest 0.5 unit.
y
6
4
h(x)
2
–6
–4
–2
2
–2
–4
–6
4
6
x
is graphed below.
. Then
5. A new computer game costs $97,500 to develop. Once completed, individual games can be produced for
$0.20 each. The first 225 are given away as samples. Write and graph a function C(x) for the average cost of
each game that is sold. How many games must be sold for the average cost to be less than $3? (Hint: when
graphing, use large values on your x-axis)
6. You are selling T-shirts for a fundraiser. The cost of making the designs and buying blank T-shirts is $425. In
addition to these one time charges, the cost of printing each T-shirt is $1.75. Let x represent the number of
T-shirts that are printed. Write a model that represents the average cost per T-shirt. Then graph the model.
7. The graph below has a vertical asymptote at
0). What could be the function of this graph?
, a horizontal asymptote at
8. The graph below has a vertical asymptote at
, a horizontal asymptote at
and a hole at (6, 4). What could be the function of this graph?
Essay
, and an x-intercept at (
, an x-intercept at (
, 0),
,
1. A rational function, R(x) has the following characteristics: a vertical asymptote at
, a horizontal
asymptote at
, and a hole at (2, ). Sketch the function. The follow the steps below to determine what
it could be.
Part a: Put in the factor that would account for the vertical asymptote at
Part b: Insert factors that would account for a hole at
.
.
Part c: Determine what must be true about the numerator and denominator for there to be a horizontal
asymptote at
.
Part d: Insert factors that would account for the horizontal asymptote at
Part e: Describe what you must do in order for the hole to appear at (2,
Part f: Solve your equation for a.
Part g: Write the complete function.
.
).
Algebra 2 Module 8 Study Guide
Answer Section
MULTIPLE CHOICE
1. ANS:
STA:
KEY:
2. ANS:
MSC:
3. ANS:
MSC:
4. ANS:
B
PTS: 1
DIF: DOK 2
A-APR.6
TOP: Rewrite Rational Expressions
rational function | graph rational functions
B
PTS: 1
DIF: Average
DOK 2
A
PTS: 1
DIF: Average
DOK 2
B
Write
in the form
NAT: A-APR.D.6
TOP: Section 5A Quiz
TOP: Section 5A Quiz
where h is the horizontal translation and k is the vertical
translation.
and
. Translate
up 3 units.
Feedback
A
B
C
D
(1/x) + c represents a vertical translation of f(x).
Correct!
The sign of c determines whether (1/x) + c represents a vertical translation of f(x) |c|
units up or down.
(1/x) + c represents a vertical translation of f(x).
PTS:
OBJ:
TOP:
5. ANS:
1
DIF: Average
REF: 16afe3fa-4683-11df-9c7d-001185f0d2ea
5-4.1 Transforming Rational Functions
NAT: NT.CCSS.MTH.10.9-12.F.BF.3
5-4 Rational Functions
MSC: DOK 2
D
Write the function in the form
where x = h is the vertical asymptote and helps find the
domain, and y = k is the horizontal asymptote and helps find the range.
, so
Vertical asymptote:
Domain:
and
.
Horizontal asymptote:
Range:
Feedback
A
B
C
D
The horizontal asymptote is equal to the vertical translation of the parent function.
The vertical asymptote is at the value of x that makes the denominator equal 0.
The vertical asymptote is at the value of x that makes the denominator equal 0. The
horizontal asymptote is equal to the vertical translation of the parent function.
Correct!
PTS: 1
DIF: Average
REF: 16b21f46-4683-11df-9c7d-001185f0d2ea
OBJ: 5-4.2 Determining Properties of Hyperbolas
NAT: NT.CCSS.MTH.10.9-12.F.IF.7.d
TOP: 5-4 Rational Functions
MSC: DOK 2
6. ANS: A
Factor the numerator.
The zeros are the values that make the numerator zero, x = and x = .
The vertical asymptote is where the denominator is zero, x =
Plot the zeros and draw the asymptote, then make a table of values to fill in missing points.
Feedback
A
B
C
D
Correct!
Factor the numerator. Zeros are the x-values that make the numerator zero. The
asymptote is where the denominator is zero.
Factor the numerator. Zeros are the x-values that make the numerator zero. The
asymptote is where the denominator is zero.
Factor the numerator. Zeros are the x-values that make the numerator zero. The
asymptote is where the denominator is zero.
PTS:
OBJ:
NAT:
MSC:
7. ANS:
1
DIF: Average
REF: 16b481a2-4683-11df-9c7d-001185f0d2ea
5-4.3 Graphing Rational Functions with Vertical Asymptotes
NT.CCSS.MTH.10.9-12.F.IF.7.d
TOP: 5-4 Rational Functions
DOK 2
B
Factor the numerator and denominator.
Zeros:
and 3
Vertical asymptotes:
Horizontal asymptote:
,
The numerator is 0 when
or
.
The denominator is 0 when
or
.
Both p and q have the same degree: 2. The
horizontal asymptote is
.
Feedback
A
B
C
D
To find the horizontal asymptote, divide the leading coefficient of p by the leading
coefficient of q.
Correct!
Find the zeros by checking when the nominator is 0. Then find the vertical asymptotes
by checking when the denominator is 0. To find the horizontal asymptote, divide the
leading coefficient of p by the leading coefficient of q.
You reversed the values of the zeros and the vertical asymptotes.
PTS: 1
DIF: Average
REF: 16b4a8b2-4683-11df-9c7d-001185f0d2ea
OBJ: 5-4.4 Graphing Rational Functions with Vertical and Horizontal Asymptotes
NAT: NT.CCSS.MTH.10.9-12.F.IF.7.d
MSC: DOK 2
8. ANS: A
Except for the hole at
TOP: 5-4 Rational Functions
Factor the numerator.
is a factor in both the numerator and the
denominator, so there is a hole at
.
Divide out common factors.
, the graph of f is the same as
. On the graph, indicate the hole with an
open circle. The domain of f is
.
Feedback
A
B
C
D
Correct!
For what value(s) of x are the numerator and denominator of f(x) equal to zero?
For what value(s) of x are the numerator and denominator of f(x) equal to zero?
For what value(s) of x are the numerator and denominator of f(x) equal to zero?
PTS: 1
DIF: Average
REF: 16b6e3fe-4683-11df-9c7d-001185f0d2ea
OBJ: 5-4.5 Graphing Rational Functions with Holes
NAT: NT.CCSS.MTH.10.9-12.F.IF.7.d
TOP: 5-4 Rational Functions
MSC: DOK 2
MULTIPLE RESPONSE
1. ANS: A, D, E
A. For every point
on the graph, the point
is also on the graph. So, the graph is symmetric
with respect to the origin.
B. The graph of a function can never be symmetric with respect to the -axis because the points
and
would lie on a vertical line.
C. Since for any point
on the graph, the point
is not on the graph, the graph is not symmetric
with respect to the y-axis.
D: For every point
on the graph, the point
is also on the graph. So, the graph is symmetric with
respect to the line
.
E: For every point
on the graph, the point
is also on the graph. So, the graph is symmetric
with respect to the line
.
Feedback
Correct
Incorrect
That’s correct!
Examine the graph and determine the symmetries.
PTS: 1
DIF: DOK 1
NAT: F-IF.B.4*
KEY: interpreting graphs | rational functions | symmetry
SHORT ANSWER
1. ANS:
STA: F-IF.4*
Rubric
1 point for each rational expression in the equation
PTS: 2
DIF: DOK 2
STA: A-CED.2* | MP.4
2. ANS:
100
NAT: A-CED.A.2* | MP.4
KEY: rational function | modeling
h
Height (cm)
80
60
40
20
1
2 3 4 5 6
Radius (cm)
7
8
r
Rubric
1 point for equation; 1 point for curve; 1 point for appropriate axis labels and scales
PTS: 3
DIF: DOK 2
NAT: A-CED.A.2* | MP.4
STA: A-CED.2* | MP.4
KEY: rational function | volume | modeling
3. ANS:
a. The independent variable is f, the fuel efficiency of Brian’s car while driving on the highway, and it is
measured in miles per gallon. The dependent variable is , the amount of gas consumed, which is
measured in gallons.
b. Unit analysis shows that dividing miles by miles per gallon gives gallons. So, the amount of gas consumed
driving on the highway is
, and the amount of gas consumed driving in the city is
. The sum of
these quantities gives the amount of gas consumed driving one way, so this sum must be doubled for a
round trip. Therefore, the function that models the gas consumed on a round trip is
Rubric
a. 1 point for appropriate variable names; 1 point for identifying the independent and dependent variables; 1
point for correct units of measurement
b. 1 point for correct use of unit analysis; 2 points for correct reasoning in developing the model
PTS: 6
DIF: DOK 3
NAT: N-Q.A.2* | N-Q.A.1* | MP.2 | MP.4
.
STA: N-Q.2* | N-Q.1* | MP.2 | MP.4
KEY: modeling | rational functions
4. ANS:
The equation
can be transformed by subtracting
from both sides. This results in the
equation
, or
. The solutions to the equation
are the values of for
which
. So, the -intercepts of the graph of h(x) must be found.
The -intercepts are approximately
and
.
and
. So, the solutions of the equation
Rubric
2 points for a reasonable explanation of how the graph of
; 1 point for finding the solutions of the equation
can be used to find the solutions of
PTS: 3
DIF: DOK 3
NAT: A-REI.D.11* | MP.3
STA: A-REI.11* | MP.3
KEY: solving equations graphically | exponential functions | rational functions
5. ANS:
; 35,000 games
PTS: 1
DIF: DOK 2
TOP: Rewrite Rational Expressions
6. ANS:
NAT: A-CED.A.1 STA: A-CED.1
KEY: graph | equation | word | rational function
A=
PTS: 1
DIF: DOK 2
NAT: A-CED.A.1 STA: A-CED.1
LOC: NCTM.PSSM.00.MTH.9-12.GEO.4.e | NCTM.PSSM.00.MTH.9-12.PRS.3 |
NCTM.PSSM.00.MTH.9-12.REP.2
TOP: Graph Simple Rational Functions
KEY: word | function | graph | rational
are
7. ANS:
PTS: 1
MSC: DOK 2
8. ANS:
PTS: 1
MSC: DOK 2
DIF: Average
TOP: Free Response Test, Chapter 5, Form B
DIF: Advanced
TOP: Free Response Test, Chapter 5, Form C
ESSAY
1. ANS:
y
10
8
6
4
2
–10 –8
–6
–4
–2
–2
2
4
6
8
10
x
–4
–6
–8
–10
Part a:
Part b:
Part c: The numerator and denominator must have the same degree, and the leading coefficient of the
numerator must be two times the leading coefficient of the denominator.
Part d:
Part e: For the function
,
Part f:
;
;
;
.
Part g:
PTS: 1
MSC: DOK 3
DIF: Average
TOP: Performance Assessment, Chapter 5