Notes 5.1 & 5.2 honors
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This document provides examples and exercises for understanding functions using ordered pairs, function rules, and graphs. It includes examples of writing real-life data as ordered pairs and identifying the domain and range, using equations to represent functions, determining whether a relation is a function, writing function rules from patterns or tables, representing word problems as functions, and analyzing trends in functional data from a graph. The goal is for students to understand different ways to represent and analyze functional relationships.
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Instructions:
This is an open-book exam. You may refer to your text and other course materials as you work on the exam, and you may use a calculator.
Record your answers and work in this document. There are 30 problems.
Problems #1-12 are multiple choice. Record your choice for each problem. Problems #13-21 are short answer. Record your answer for each problem.
Problems #22-30 are short answer with work required. When requested, show all work and write all answers in the spaces allotted on the following pages. You may type your work using plain- text formatting or an equation editor, or you may hand-write your work and scan it. In either case, show work neatly and correctly, following standard mathematical conventions. Each step should follow clearly and completely from the previous step. If necessary, you may attach extra pages.You must complete the exam individually. Neither collaboration nor consultation with others is allowed. Your exam will receive a zero grade unless you complete the following honor statement.
Please sign (or type) your name below the following honor statement:
I have completed this final examination myself, working independently and not consulting anyone except the instructor. I have neither given nor received help on this final examination.
NameDate
Math 107 Final Examination
Fall, 2019
1
MULTIPLE CHOICE. Record your answer choices.
1.7.
2.8.
3.9.
4.10.
5.11.
6.12.
SHORT ANSWER. Record your answers below.
13.
14.
15.
16.
17.
18.
19. (a) (b) (c)
20.(a)
(b)
(c)
(d)
21. (a) (b) (c) (d)
SHORT ANSWER with Work Shown. Record your answers and work.
Problem Number
Solution
22
Answers:
(a)
(b)
Work/for part (a) and explanation for part (b):
23
Answers:
(a)
(b)
(c)
Work for part (a):
24
Answer:
Work:
25
Answer:
Work:
26
Answers:
(a)
(b)
Work for part (a) and for part (b):
27
Answer:
Work:
28
Answer:
Work:
29
Answers: (a)
(b)
Work for (b):
30
Answer:
Work:
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Show all of your work details for these calculations. Please review this Web site to see how to type mathematics using the keyboard symbols.
Problem 1: Modeling Profit for a Business
IMPORTANT: See Question 3 below for special IP instructions. This is mandatory.
Remember that the standard form for the quadratic function equation is
y
=
f
(
x
) =
ax
2
+
bx
+
c
and the vertex form is
y
=
f
(
x
) =
a
(
x
–
h
)
2
+
k
, where (
h
,
k
) are the coordinates of the vertex of this quadratic function’s graph.
You will use
P
(
x
) = −0.2
x
2
+
bx
–
c
where (−0.2
x
2
+
bx
) represents the business’ variable profit and
c
is the business’s fixed costs.
So,
P
(
x
) is the store’s total annual profit (in $1,000) based on the number of items sold,
x
.
1. Choose a value between 100 and 200 for
b
. That value does not have to be a whole number.
2. Think about and list what the fixed costs might represent for your fictitious business (be creative). Start by choosing a fixed cost,
c
, between $5,000 and $10,000, according to the first letter of your last name from the values listed in the following chart:
If your last name begins with the letter
Choose a fixed cost between
A–E
$5,000–$5,700
F–I
$5,800–$6,400
J–L
$6,500–$7,100
M–O
$7,200–$7,800
P–R
$7,800–$8,500
S–T
$8,600–$9,200
U–Z
$9,300–$10,000
4.Replace
b
and
c
with your chosen values in Parts 1 and 2 in
P
(
x
) = −0.2
x
2
+
bx
−
c
. This is your quadratic profit model function. State that quadratic profit model functions equation.
5. Next, choose 5 values of
x
(number of items sold) between 500 and 1,000. Think about the general characteristics of quadratic function graphs (parabolas) to help you with choosing these 5 values of
x
.
6. Plug these 5 values into your model for
P
(
x
), and evaluate the annual business profit given those sales volumes. (Be sure to show all of your work for these calculations.)
7. Use the 5 ordered pairs of numbers from 5 and 6 and Excel or another graphing utility to graph your quadratic profit model, and insert the graph into your Word answer document. The graph of the quadratic function is called a
parabola
.
8. What is the vertex of the quadratic function graph? (Show your work details, or explain how you found the vertex.)
9. What is the equation of the line of symmetry? Explain how you found this equation.
10. Write the vertex form for your quadratic profit function.
11. Is there a maximum profit for your business? If so, how many items must be sold to produce the maximum profit, and what is that maximum profit? If your quadratic profit function has a maximum, show your work or explain how the maximum profit figure was obtained.
12. How would knowing the number of items sold that produces the maximum profit help you to run your business more effectively.
13. Analyze the results of these profit cal.
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Functions D1 Notes
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