ELIZABETH HOFMANNEXPERT STUDYCI 4300 – FALL 2015PROFESSOR LAURIE RAMIREZ

A. Identification of Artifact/Evidence:Artifact: Function Paper

Topic/Title: What defines a function?

Medium: Microsoft Word Document; Research Paper

Technology Used: Microsoft Word for word-processing, iPhone pictures, Appalachian State library online sources

B. Analyze your content knowledge as it was demonstrated at the time the artifact was completed or developed.Which standards and indicators were addressed in this artifact – and at what level? You might also have a content specialist analyze your artifact and provide feedback about your level of understanding. If the artifact was a summative assessment, you might have received feedback from your professor. You might incorporate that feedback into this analysis.

Standard 3: Mathematics teacher candidates possess the mathematical knowledge needed to enable students to understand patterns, relations, and functions. This includes the use of algebraic symbols to represent and analyze mathematical situations, the use of mathematical models to represent and understand quantitative relationships, and the analysis of “change” in various contexts. Patterns, relationships, and functions

Middle School

Understand the critical importance of the concept of variable and the use of variables in expressing functional relationships

Understand and use algebra as a symbolic language; as a problem-solving tool; as generalized arithmetic; as a study of functions, relations, and variation; and as a way of modeling physical situations.

Understand functions, including the abilities to read, interpret and create graphs, formulas (in closed and recursive forms), and tables for particular classes of functions.

Understand linearity and how linear functions can illustrate proportional relationships. Recognize patterns of change associated with linear, quadratic, and exponential

functions. Demonstrate algebraic skills and be able to provide rationales for common algebraic

procedures.

Rationale for Standard 3:Standard 3 explicitly states that the teacher candidate should “possess the mathematical knowledge needed to enable students to understand patterns, relations, and function.” It further reads that this includes, “…the analysis of ‘change’ in various contexts.” Below I have shared a screenshot from a response Dr. Searcy gave me after a rough draft submission of my function paper. Here is where we discuss “change.”

And here are two screenshots that come from the final submission, in which I revise this section of the paper:

Standard 5: Mathematics teacher candidates possess the mathematical knowledge needed to enable students to develop skills in problem solving, making connections between various branches of mathematics, reasoning and proof, and communication and representation of mathematical ideas. Mathematical process skills

Middle School

Use problem solving to build new mathematical knowledge, apply and adapt a variety of appropriate strategies to solve problems, and monitor and reflect on the process of mathematical problem solving.

Use reasoning and proof to make and investigate mathematical conjectures, develop and evaluate mathematical arguments and proofs, and select and use various types of reasoning and methods of proof

Communicate mathematical thinking coherently and clearly, analyze and evaluate mathematical thinking and strategies of others, and use the language of mathematics to express mathematical ideas precisely.

Make connections by understanding how mathematics ideas interconnect and by applying mathematics in context outside of mathematics.

Use representations to organize, record, and communicate mathematical ideas.

Rationale for Standard 4:This standard states that the teacher candidate should “…possess the mathematical knowledge needed to enable students to develop skills in problem solving, making connections between various branches of mathematics, reasoning and proof, and communication and representation of mathematical ideas.” To verify my knowledge of this, I took a screenshot of a section of a rough draft in which I prove the reasoning behind the multiple ways to explain slope:

And below is a screenshot from the final paper in which I edit this section:

Standard 6: Mathematics teacher candidates must be versed in the appropriate use of mathematical tools and manipulatives. Mathematical tools

Middle School

Understand appropriate use of technology (e.g. graphing calculators, computer algebra systems, dynamic drawing tools, spreadsheets, or statistical graphing software) to explore algebraic, geometric and data analysis concepts.

Use appropriate math manipulatives (e.g., algebra tiles, computer virtual manipulatives, or computer applets) to clarify and develop mathematical concepts.

Rationale for Standard 6:The function paper did not ask for an explanation of how to teach function via manipulatives. So, to rationalize this standard, I will share a screenshot from a rough draft in which I offer a real-world connection for functions:

And below are screenshots from the final paper in which I edit this section:

C. Analyze the development of your content knowledge since the artifact was completed or developed.How? Has it decreased? Why? What are the weaknesses and/or limitations related to understanding this content? What is the next level of understanding related to this content? What would it take for you to attain that level of understanding? What are the challenges, barriers, or obstacles that prevent you from becoming an expert in this area of study? How will you overcome them?

I believe that my content knowledge has both increased and decreased; both in different ways. What has decreased is some of my mathematical skills. There are times when my CI 4040 (middle grades mathematics) professor puts a worksheet on my desk and I cannot remember the “short cuts” and “tricks” to find the answer.However, one thing that has increased is my ability to present mathematical topics in one fluid motion. I attribute this to last semester in my CI 3900 internship at Johnson County Middle School. In an eighth grade mathematics class for adolescents with mixed abilities, I quickly learned that I would need to present an idea in multiple formats in order to reach every student in the classroom. Walking around the room and helping individual students helped me to understand what it is they missed during my instruction. Another aspect of my teaching math skills that has increased is my ability to work with manipulatives. In my internship at JCMS, I would use peg blocks and rubber bands to teach shapes and area. And then, in CI 4040, we would connect each aspect of the middle grades math curriculum with some form of manipulative. For a microteaching group project, I created a worksheet that students can use to further their understanding of statistics. Here, students would use “spinners” on the IPad to gather their data. The worksheet is below:

D. Create a plan for deepening your knowledge in this area of study.Possible Strategies:

Have a content specialist (e.g., content area professor, middle grades teacher) evaluate the artifact and provide feedback about how knowledge in the content area might be advanced or refined.

Discuss the challenges you anticipate in deepening your knowledge in this particular area of study and/or in this discipline, including how you will overcome these challenges.

Create a real-world example of how this knowledge might be applied/used.

I have attached at the end of this document three attachments:1. Draft I with comments from Dr. Searcy2. Draft II with comments from Dr. Searcy3. The final draft (Function Paper)

Challenges in Deepening KnowledgeThe content of functions is either in my brain or needing freshening up. This is the kind of content in which the basics must be covered in order to deepen knowledge. The edits that Dr. Searcy made to my drafts are not her correcting my content knowledge, but more so her helping to re-organize my content and make the paper flow using examples, graphs, tables, equations, etc. The main challenge in deepening my knowledge of functions is the question as to if I will teach math, ELA, or both. If I end up in only a mathematics classroom, I will be able to play with different methods of delivering this content, extending it, etc. But, if I am in an ELA classroom, I will have to refresh my knowledge of functions. This is done through professional development seminars, math education conferences, shadowing other classrooms, etc.

Real-World Example:There are screenshots of real-world applications to functions on pages 7, 8, and 9.E. Reflect on your depth of knowledge in this area of study.Possible Strategies

Use the characteristics of experts identified in the background section of this document. Comment on your level of expertise on this topic relative to the applicable expert characteristics.

Use another taxonomy (e.g., SOLO Taxonomy, revised Bloom, et al., Marzano) to explain your depth of knowledge.

Discuss ways to use or integrate this content knowledge into other areas of the discipline.

Discuss literacy connections: What does it mean to be literate in this content area? How does this artifact and analysis process demonstrate my literacy in this content area?

Discuss interactions with content specialists: You have likely discussed the content of the feedback of specialists previously. Here, discuss your reactions to the feedback.

During my process to complete the function paper, I would sit down and discuss my research with Dr. Searcy through a series of one-on-one conferences. Her professional guidance was excellent in both reflection and foreshadowing. Her help would aid me in both figuring out what kind of teacher I wanted to be and how to teach middle grades math. She has traveled in her past, so she and I would talk a lot about how I could integrate social studies into this content. Knowing that I was about to leave for Chile, she would plant the idea in my head to teach middle grades Spanish. She and I would discuss ways for me to expand my professional development into how to teach math in Spanish. In reflection of the function paper,

she would help me to write about math. Never before have I had an assignment in which I had to write so much about math. It was rather difficult for me. Her guidance in this paper was crucial to me completing it. If it weren’t for her responses to my drafts, I would not of been able to create an acceptable math essay.

F. Debrief your growth as a middle grades teacher through completion of this exert study. Describe your professional growth from completing this expert study. What did

you learn about yourself, your learning, and your future teaching? Explain how your understanding of this topic and/or your plan of action for

deepening your content knowledge will benefit your future middle grades students. How will your depth of content knowledge impact student learning? How does your content knowledge demonstrated in this artifact and study align with the content your students will be expected to know?

Discuss how the process of assessing, analyzing, and reflecting on your depth of content knowledge in this particular area will help you engage in this process with other content or in other concentration areas/disciplines.

In reflecting on my function paper, I find a sense of satisfaction knowing that I hold so much knowledge on function. However, I recall a lot of unnecessary information. I finished this paper a matter of months ago, and I cannot recall some chunks of information, such as the people who contributed to research on function. But, I feel confident in my knowledge of how to define a function and explain function using multiple methods. So, looking back on this paper makes me think how important it is to assign assignments with such excessive expectations. I remember writing this paper extremely stressed out, submitting what I thought was a decent paper, and receiving a grade short of my expectations. However, it was the one-on-one conferences that would set me back and allow me to de-stress. After I submitted rough draft I, I would create Draft II while waiting for a response to draft I. After receiving a response from draft II, I was able to construct the final paper into one completed function paper. One thing that I think will benefit my future middle grades students is the fact that I had to work so hard to write about math. Taking this into considering is important as a future math teacher, because when I ask students to write about math, I must reflect on the process that it took me to complete this paper at the age of twenty.

G. Rationale Reflection As part of your online portfolio, you will need to include a rationale reflection

that demonstrates your knowledge/understanding of and commitment to ASU Middle Grades Standard 4 – Content Knowledge.

A rationale is a persuasive argument. You are writing a convincing exposition giving reasons why and how the artifact demonstrates your competency of the standard.

A reflection is a thoughtful consideration of what you have learned about the standard by creating the artifact. Include language of the particular Middle Grades Program Standard in the narrative portion of your rationale-reflection.

This rationale reflection should include specific examples of how the Expert Study and other assignments from this course (i.e., the textbook critique or group assignments) have contributed to your content area expertise. Be sure that the narrative explicitly connects the Standard and indicators to the artifact. Do not use generalizations. Indicators for your particular content area can be found online and can be included in your rationale.

http://middlegrades.appstate.edu/undergraduate/state-and-program- standards

Standard 4: Content Knowledge

Middle grades teacher candidates understand and use the central concepts, tools of inquiry, standards, and structures of content in their chosen teaching fields and create meaningful learning experiences that develop young adolescents’ competence in subject matter and skills.

Middle grades majors at ASU complete a minimum of 24 semester hours in each of two concentrations (English Language Arts, Social Studies, Science, Math). For specific requirements of this standard, please consult the subject-specific standards for the licensure/concentration areas.

Rational:When I first started my education career during my sophomore year at Appalachian State, I would worry that I would never “master” my content areas. However, four semesters and four internships later, I am confident in my content areas. I have had the opportunity to tutor, teach, listen, watch and learn. With such an intense prompt, the function paper would pull together all of my teaching skills and highlight their importance. I know that this artifact demonstrates my competency of standard 4 because of this intensity. This paper not only demonstrates my knowledge of how to use, explain, and analyze functions, but it also gives insight to my knowledge of mathematics in general. The entire first half of the paper is on the history of math and the evolution of function. By giving insight to the evolution, it is easier for me to understand functions and how they came about. This is important to know, because without a brief history lesson on the topic, one can never truly

“master” it, as I feel I did.

Reflection:When I reflect on my development into a middle school teacher, I can tell that my professional development comes from a variety of resources. My content area classes would deepen my knowledge of mathematical concepts. CI 3910 and CI 3920 would expand my current knowledge and help me to master my knowledge. My in-class discussions with my professors and fellow classmates help me to deepen my knowledge of how to work with students and understand their developmental needs. My internships would benefit the way I communicate with young adolescents. I have evolved into a young professional who can serve as a mentor of mathematics for young adolescents. Especially with my five-week internship I feel I have evolved quickly into a better teacher. Even though I was in an ELA classroom, I would complete an instructional design sequence. This is a three-day lesson progression on box and whiskers plots. This really helped to broaden my horizons on teaching math, because I had to gather all information on box and whiskers plots in order to compress them into three days of lessons.

Draft I:

Draft II:

Final Function Paper:Elizabeth Hofmann

Dr. Mary Searcy

MAT 3920

Spring 2015

Depth of Knowledge

A function is a relationship between two quantities, one is the input and the

other is the output, in such for each input there is exactly one output. This concept is

one in which we discuss each class period. However, how did this definition come

about? That is—who coined the term function? Did more than one person

contribute to the theory, or concept of function?

The concept of function could go as far back as far as the Babylonians of c. 2000

B.C. who might be credited with a working definition of “function” because of their use

of tables like the one for n3 + n2, n = 1, 2,…,30, suggesting the definition that a function

is a table or correspondence. (Kennedy).

Amy Dahan-Dalmedico and Jeanne Peiffer found that The principal aim of

Descartes, displayed in his 1637 Geometrie, was to reduce the solution of all algebraic

problems that involve solutions of equations to some standard procedures for

construction of their real roots, which will be coordinates of the points of intersection of

appropriate planar curves, of degree as low as possible.

Descartes distinguished between geometric curves and mechanical curves and

restricted himself to “geometric curves”, those for which the two coordinates x and y are

connected by an algebraic equation P(x,y) = 0 (today called algebraic curves). To their

resolution Descartes wrote: ‘Taking successively infinitely many different magnitudes for

the line y, there will be found also on it values for the line x, and thus will be had an

infinity of points such that each one is marked C, by means of which the curvilinear line

asked for is described.’

Here, without a doubt for the first time also clearly, is displayed the idea that an

equation between x and y is a way to introduce a functional relation of dependence

between variable quantities, in the sense that one of them allows the determination of the

other. (Dahan-Dalmedico).

In 1692, Gottfried Wilhelm von Leibniz thought of a function as any quantity

associated with a curve, such as the coordinates of a point on a curve, the length of a

tangent to the curve, and so on.

In 1718, Johann Bernoulli defined a function to be any expression involving one

variable and any constants.

In 1750, Leonhard Euler called function in the sense of Bernoulli’s definition

“analytic functions” and sued also a second definition, according to which a function was

not required to have an analytic expression but could be represented by a curve, for

example. Euler also introduced the now standard notation f(x). (Kennedy).

According to Carl B. Boyer, Euler defined a function of a variable as “any

analytic expression whatsoever composed of that variable quantity and numbers or

constant quantities” in the opening pages of his Introductio in analysin infinitorum

(1748). He did not immediately make clear what an “analytic expression: is; but later he

stated, concerning a variable z, that “there will be no doubt but that every function of this

variable can be transformed into an infinite expression of the form Aza + Bzb + Czv +

Dzd, etc.” Here Euler did not quite reach the modern view that such a series defines a

function, but his treatise did make clear the key role that infinite series play in the study

of functions. (Boyer).

In a 1976 publication of Algebraic and Arithmetic Structures, the authors define a

function is a special kind of relation in which every element in the domain is assigned to

something in the range, and in which every element in the domain is assigned to only one

element in the range. Functions are classified according to whether they are one-to-one,

one-to-many, many-to-one, or many-to-many (Bell). This means how many range values

there are for each domain value. There could be more than way to write a value. For

example, sin(2x + pi) is the same as sin(4x + pi), so that would represent a “many-to-

one” relationship.

The mathematical concept of function emerged parallel to the development of the

calculus. The elements of calculus were put in the form in which they are discussed today

during the early nineteenth century, through the work of the Frenchman Augustin-Louis

Cauchy (1789-1867) and others. An excellent history of the calculus can be found in

Boyer’s book History of the Calculus (Durbin). It would not be accurate to leave the

impression that Cauchy was alone in the early nineteenth-century development of the

foundation of the calculus. At that time in Bohemia there was a priest, philosopher, and

mathematician, Bernhard Bolzano, who was seeking to arithmetize analysis; and his

definition of the derivative was the same as Cauchy’s. Moreover, Bolzano and Cauchy

independently reached the modern definition of continuity as a function. They defined a

function f(x) as continuous in an interval if for any value of x in this interval the

difference f(x + Dx) – f(x) becomes and remains less than any given quantity for Dx

sufficiently small (Boyer).

In 1800, Joseph Louis Lagrange restricted the meaning of function to a power

series representation. Twenty-two years later, Jean Joseph Fourier stated that an arbitrary

function can be represented by a trigonometric series. Seven years after that, P.G.

Lejuene Dirichlet said that y is a function of x if y possesses one or more definite values

for each of certain values that x may take in a given interval, x0 to x1.

The notion of function originated with the development of analytic geometry and

calculus in the seventeenth century, in the narrow sense of functions from the set of real

numbers (or some subset of real numbers) to the set of real numbers. In fact, the only

functions considered then were those that could be given by formulas. It was not until the

nineteenth century that it was realized that even for the purposes of calculus it is

necessary to take into account the existence of functions that cannot be given by

formulas. An example of such a function is the Dirichlet function: for x a real number, let

x = 0 if x is rational and x = 1 if x is irrational. Dirichlet 1805-1859 was responsible for

the spirit of the broad definition of function (Durbin).

To gain a better understanding of contemporary use of function, Joe Kennedy and

Ragan Esther provide four definitions for function using twenty elementary texts:

Definition 1: A function is a set of ordered pairs whose first elements are all different.

Definition 2: When the value of one variable depends on another, the first is said to be a

function of the second.

Definition 3: If to each permissible value of x there corresponds one or more values of y,

then y is a function of x.

Definition 4: If y is a function of x, then it is equal to an algebraic expression in x.

Eleven of these texts were published before 1959, nine after 1959. The older texts

used definitions 2, 3, 4, and others; six of the newer ones used definition 1.

Fifteen college algebra texts were examined, seven published before 1959 and eight after

1959. None of the older texts sued definition 1; four of the eight newer ones did.

This quite recent history of ‘function’ has additional significance in the context of

the earlier history of both the idea and word.

Kennedy and Esther go on to provide a more in-depth timeline of the emergence

of the concept of function, beginning in 1637 with Rene Descartes. Descartes may have

been the first to use the term; he defined a function to mean any positive integral power

of x, such as x2, x3, …

More recently, the study of point sets by Georg Cantor and other has led to a

definition of function in terms of ordered pairs of elements, not necessarily numbers.

Modern-day society has a much more in-depth understanding of function with

increased technology, more information, etc. According to the 2012 publication The Joy

of X: A Guided Tour of Math, form One to Infinity, “A mathematician needs functions for

the same reason that a builder needs hammers and drills. Tools transform things. So do

functions. In fact, mathematicians often refer to them as transformations because of this.

But instead of wood and steel, the materials that functions pound away on are numbers

and shapes and, sometimes, even other functions.” (Strogatz). He goes on to explain that,

in theory, exponential growth is also supposed to grace your bank account. He relates

exponential growth in bank accounts to the idea of a function (Strogatz). This is

important to note in the evolution of function. Published in 2012, this book uses the

modern-day example of bank accounts to compare to a function.

It is also important to note the evolution of the graph of a function. Nicole Oresme

(1323-1382), bishop of Lisieux. In studying, for example, the distance covered by an

object moving with variable velocity, Oresme associated the instants of time within the

interval with the points on the horizontal line segment (called a “line of longitudes”), and

at each of these points he erected (in a plane) a vertical line segment (“latitude”), the

length of which represented the speed of the object at the corresponding time. Upon

connecting the extremities of these perpendiculars or latitudes, he obtained a

representation of the functional variation in velocity with time—one of the earliest

instances in the history of mathematics of what now would be called “the graph of a

function.” It was then clear to him that the area under his graph would represent the

distance covered, for it is the sum of all the increments in distance corresponding to the

instantaneous velocities (Boyer).

With an understanding of the evolution of the mathematical concept of function,

one can understand its place in contemporary mathematics. To do this, one can find the

definition of function in a modern-day college textbook. According to the 2003

instructor’s edition of College Algebra: Third Edition, Mark Dugopolski states, “A

function is a set of ordered pairs in which no two ordered pairs have the same first

coordinate and different second coordinates.” He goes on to state “if the value of a

variable y is determined by the value of another variable x, then y is a function of x. The

phrase ‘is a function of’ means ‘is determined by.’ If there is more than one value for y

corresponding to a particular x-value then y is not determined by x and y is not a function

of x” (Dugopolski).

In comparison, the 2000 publication College Algebra: Through Modeling and

Visualization first explains the concept of an algorithm, then presents the idea of a

function.

ALGORITHM 1.3 Converting yards to feet

Step 1: Input x, the number of yards.

Step 2: Multiply x by 3. Let the result be y.

Step 3: Output y, the equivalent number of feet.

It goes on to read, “This algorithm establishes a relation between two sets of

numbers. Each valid input x in yards determines exactly one output y in feet, which can

be represented by the ordered pairs (x, y). If an algorithm produces exactly one output for

each valid input, then the algorithm computes a function” (Rockswold).

These two textbooks are interesting to compare, as they are only three years apart

in publication date. The 2003 book states a function in terms of a worded definition. The

2000 book uses an example to explain the concept of a function, more so than the

definition of a function. Both textbooks read on and define independent variables and

depended variables after their explanations of functions. However, they begin the process

of explaining function a little different. The 2003 book is more of a standard textbook,

i.e. it is filled with definitions, then short and sweet examples. The 2000 book is a visual

interpretation of mathematical concepts, so it uses simple examples to introduce the idea

of function, more than a definition.

Part II

A linear function takes the form of Y=mX+b in which m is the slope and b is the

y-intercept. There is a consistent slope between all of the points on the line. Any two

points can be input into the equation Y2-Y1/X2-X1 and get the same slope. In other

words, it creates a straight line. A table gives the values of X and Y, which makes it easy

for the student to take the values of X and Y and plug them into the equation Y2-Y1/X2-

X1 to determine the exact slope of the line. When looking at a graph, one can draw from

two different points on the graph that are at perfect whole numbers to see if the slope is

the same. This proves that there is a consistent slope in the line, so the rise over run is

increasing or decreasing the same amount, which would allot for a straight, linear line.

We model linear equations using the model y = mx + b with m being the slope and b

being the y-intercept. I will explain these in-depth when I get to slope and intercepts.

To better understand my depth of knowledge on linear functions, I will use a real-life

example:

A cab company has a boarding fee of four dollars then has a meter rate of two dollars for every mile driven. The equation of the line is as followed:

Y = 2x + 4

Below is the graph of this function:

The graph displays a straight, linear line. There are no “endpoints.” Instead, it

continues on forever both ways down the x-axis. The domain represents all of the X-

values in the function. If the function is continuous on the X-axis, then the X-values go

from negative infinity to positive infinity. If the function is discontinuous, then there is a

point on the function where the values do not go any further to the left or right. In this

case, the domain can be represented by an integer. So, for example, if the least X-value is

-10 and the greatest X-value is 10, then our domain is -10 > X < 10. However, here we

are dealing with linear functions, which are continuous. Therefore, the domain continues

on between negative infinity and positive infinity. It can be written as: (-∞,∞). The ∞

symbol is the symbol used to represent infinity. With the example of the cab company,

the graph is continuous between negative infinity and positive infinity. Even though the

boarding fee marks the beginning of the money owed, the model of the equation goes on

forever.

The range represents all of the Y-values in the function. It is determined similar to how

to find the domain. So, if the function is continuous on the Y-axis, then the range is from

negative infinity to positive infinity. However, if it stops, say, at -5 and 5, then the range

is -5 > X < 5. However, here we are dealing with linear functions, which are continuous.

Therefore, the range is from negative infinity to positive infinity if the linear function

has a slope. If the slope is zero, then the range would only be represented by y = c, some

constant. Typically, the range for a linear function with a slope that is not zero is (-

∞,∞).

A function is increasing when the y-value increases as the x-value increases. A

function is decreasing when the y-value decreases as the x-value increases. When a

function is increasing at a decreasing rate, it is the same as saying the X-values are

increasing with a negative slope. When a function is decreasing at an increasing rate, it is

when the X-values are decreasing with a positive slope. Linear functions can either be

increasing or decreasing. So, when the slope is positive it is increasing and when it is

negative it is decreasing.

One way to find if the functions is increasing or decreasing is by finding the rate of

change, or the slope. The slope is used to describe the measurement of the steepness of a

straight line. It can be found by dividing the rise (vertical change) by the run (horizontal

change). This is using the same formula, Y2-Y1 / X2-X1 that we use to find the slope of

a linear function. The term “rate of change” is used to describe the slope, because the

slope defines the rate in which the X and Y values change. The function of the cab model

exemplifies a positive slope for an increasing function. The slope, m, is 2/1 because the

change in Y, or dollars, is two dollars and it is over the change in X, or miles driven,

which is one. So, for every mile driven, the amount due is two dollars. Two divided by

one gives us two. Therefore, the slope, or rate of change, is two. Linear functions

demonstrate a slope that is consistent over the entire course of the model, because they

either increase or decrease at a constant rate using a common rate of change between

two points.

One way to better understand the slope is to look at a table of the function’s values.

Below is a table of coordinate points:

X Y = mx + bY = 2x + 4

F(x)

-2 2(-2) + 4 0-1 2(-1) + 4 20 2(0) + 4 41 2(1) + 4 62 2(2) + 4 8

The concavity of a function is found by looking at the graph. If the function is

non-linear, meaning the slope is not consistent, then we can look at the direction in which

it is going to determine the concavity. When the curve is “bent upward” the function is

concave up, and when it is curved downward it is concave down. Linear functions do not

have any sort of concavity. Concavity is found by looking at the slope. A function only

bends if there is not a consistent slope. Since linear functions maintain the same slope,

there is no concavity.

The point of inflection can be found when the function changes directions, which is

typically found by a change in concavity. Since there is no concept of concavity with

linear functions, there will be no point of inflection.

An asymptote is when there is a “halt” to the function. For example, an asymptote of

X=9 means that everywhere that X is 9 there is no value to the function. When there is an

asymptote, a line or curve can approach it arbitrarily closely, but it does not touch or

cross it. Asymptotes may be vertical or horizontal. So, there could be an asymptote that

does not cross a certain value of either X or Y. But, it is more accurate to describe

asymptotes by saying it as a point where the function gets close to or approaches but

does not touch. Linear functions do not have asymptotes. There is no change in slope, so

the function cannot “approach” a line on a certain X or Y value.

Intercepts are where a graph crosses either the x-axis or the y-axis. The X-axis is found

on a table when the Y value is 0. The Y-axis is found on a table when the X value is 0.

When looking at a graph, the X and Y intercepts can be found where the function literally

crosses either the X or Y-axis, or both. Not all functions cross the X or Y-axis. Some

function can cross the X-axis at more than one intersection, but if it crosses the Y-axis

more than once it will not be considered a function, as it would not cross the vertical line

test. All linear functions have an X and Y intercept. If a linear function has the

equation y = 2x for example, then we assume it begins at (0,0) and moves right at a rate

of 2/1 and left at a rate of 2/1. In the cab example, the boarding fee of four dollars is the

initial value. No matter how far the cab travels, there is a fee of four dollars for boarding.

Therefore, this linear function will only cross the y-axis at (0,4) and will continue to

increase with a slope of two. However, we know, since it is continuous, that there are

values to the left of the y-intercept. From (0,4), if we move down two and left one, we get

to (-1,2). Then again, we get to (-2,0). This is the x-intercept, as there is no value for y.

This does not mean that the cab drives in a negative direction for two miles. Instead, it

models where the function will go when it continues to travel towards negative infinity.

The absolute minimum is the lowest point over the entire domain of a function. The

absolute maximum is the highest point over the entire domain. So, the absolute

minimum/maximum can be found by finding either the highest or lowest value of the X-

values. The local maximum is the highest point in a particular section of a graph. The

term “local” refers to the fact that one is looking between certain ranges. Therefore, the

local minimum is the lowest point in a particular section of a graph. Linear functions that

are increasing or decreasing do not have maximum or minimum values, since they are

continuous and have a domain and range of (-∞,∞). However, consider the linear function

of y = c, in which c is a constant value the slope is zero, so there is technically no “mx” in

the equation. Every point on the graph would be (x, c). If, for example, we have Y = 2,

then we would have a horizontal, linear function in which the X values go from negative

to infinity and positive infinity. The Y values, or range, is simply y=2.

There are many other real-life applications that can be modeled by a linear function.

For example, a runner could graph her run using a linear function, if she runs at a

constant pace. So, she could start at zero and increase at a constant rate. The X-value

would determine how long she has been running (typically minutes) and the Y-value

could determine how far she has run, typically in feet. The rate would show how many

feet she runs per minute. The y-intercept would most likely start at zero, unless she starts

at a different position over her running course. This is just one example out of many.

An exponential function is a lot like what it sounds like. It takes the formula Y=ex in

which e is a fixed number, or constant, and X is the variable in the power. In class, we

discussed how exponential functions can take on the form Y=a(b)x in which A is an initial

amount, b is the rate of exponential growth and X is the time. In either form, the graph

will look like it is slowing increasing/decreasing, then as time goes on, it takes an intense

shift in the rate, allowing it to increasing/decrease way larger than it did at the beginning.

And, exponential functions may even begin to stat out exponential and get even smaller

as time goes on, depending on the value of b. To gain a better understanding of how this

works, I used two different exponential functions to compare to one another:

Example 1: Dr. Searcy has $2,000 in her bank account in the year 2000. Each year past 2000, the amount of money in her bank account increases by a factor of two. The amount of money in her bank account follows the exponential model:

Y = 2(2)x

Example 2: Miss Hofmann has $2,000 in her bank account in the year 2000. Each year past 2000, the amount of money in her bank account decreases by a factor of one-half. The amount of money in her bank account follows the exponential model:

Y = 2(0.5)x

In both functions, a is two, which represents the initial value, or initial amount of

money in their bank accounts in the year 2000. This will be further explained in my paper

when I discuss y-intercepts. In example 1, the b value is greater than 1, so b > 1. In

example 2, the b value is between 0 and 1, so 0 < b < 1. The graphs of the equations are

shown below:

Notice that the graph for example 1 and the graph for example 2 are very

different. As I explain the characteristics for these exponential functions, I will explain

how and why this occurs.

The domain of an exponential function is similar to the domain of a linear

function in the sense that it is a description of all of the X-values. The domain typically is

separated by the same amount, as it typically represents time. This means the function

will increase/decrease in the same interval. In example 1, the x-values come from

negative infinity of the x-axis, and increase as they travel in a positive direction on the x-

axis all the way to positive infinity. So, in this case, the domain is from (-∞,∞). Same

with example 2, the x-values go on from (-∞,∞).

As for the range, which is the description of all of the Y-values, the Y-values are

not typically separated by a common difference. Instead, they increase/decrease

exponentially, so there is a different rate of change between each value. Both examples 1

and 2, as the functions approach negative and positive infinity on the x-axis, they never

cross or touch the x-axis. Therefore, they never cross y = 0. So, the y values are only

positive. The range can be written as (0,∞).

Both the domain and range may be different depending on if there is a

transformation in the function. I will discuss this shorty after I discuss transformations.

With exponential functions, an increasing function can be characterized by a

function that is increasing along the graph. So, as the X value increases, so does the Y

value. It is decreasing if, as the X-value increases, the Y-value decreases. In example 1,

from negative infinity to the y intercept, (0,2), the function is increasing at an increasing

rate. Then, after the y-intercept, the function continues to increase at an increasing rate.

So the entire function is increasing at an increasing rate. In example 2, the entire function

is decreasing at a decreasing rate.

The rate of change is a little different in an exponential function versus a linear

function. As we previously discussed, a linear function has a constant rate of change.

This means that we can use the formula Y2-Y1 / X2-X1 between any two coordinates

and find the same slope, or rate of change. However, exponential functions grow

exponentially. This is why we use exponents when we write exponential functions. As

the X-values increase/decrease, the Y-values will build up slowly then shoot up/down at

an exponential rate. To better understand the rate of change, consider the equations of the

examples:

Y = 2(0.5)x

Y = 2(2)x

Notice that for each one, b is a different value. In example one, b >1. In example

two, 0 < b < 1. The value of b determines the way in which the function grows

exponentially. It makes sense that this occurs, because when the b value is smaller than

one, we are technically dividing by an amount, which means that as the x-values increase

the y-values will decrease. And, vise versa, when the b value is greater than one, we are

technically multiplying the input values. So, as the x-values increase, the output values

will also.

That leads us to concavity. The concavity of an exponential function is

determined by the way it bends. The function is concave up if it bends upward and it is

concave down if it bends downward. One difference between exponential and linear

functions is that with an exponential function, it can be increasing or decreasing, but not

both at the same time. In example one, where b > 1, the graph is concave up and example

two where 0 < b < 1, the graph is concave down. So, we see here that a b value between

0 and 1 creates a function that is concave down and a v value greater than 1 creates

a function that is concave up.

Asymptotes are very common with exponential functions. That is, a function may

grow exponentially, but never hit or cross a certain value of either X or Y. For example,

if the Function Y=7X is growing exponentially, and there is an asymptote of Y=20, it will

approach 19.999… and continue to approach it, but it will never actually go to 20. We

use the phrase “goes to” to visualize asymptotes. So, a function may “go to” a certain

value, either horizontally or vertically, but will never touch it. Exponential functions have

asymptotes. Look at the graph of example one. If you trace back the x-values toward

negative infinity, they will continue to divide and divide, but they will never go to a

negative value because both a and b are positive values. Therefore, there is a horizontal

asymptote at x = 0. Same goes for example two. Exponential functions only have

horizontal asymptotes. This is in part because we cannot transform the function in any

way so that the graph increases or decreases parallel to the y-axis.

The Y-intercept of an exponential function is where the graph of the function

crosses the Y-axis. We already stated that the y-intercept for both examples is at y = 2.

We got this because the initial value, or a, is two. Similar to linear functions, the initial

value marks where the intent of the y-value begins. Then, for exponential functions, they

either increase or decrease from there or on both sides. Typically, most exponential

functions have an asymptote at x = 0, so there is only a y-intercept. However, if the

function were to take on a vertical shift up or down, then the function could cross both

the x and y axis, or just one or the other. Below is an example of a graph of a function

that crosses both the x and y-axis:

We see here that the horizontal asymptote took a vertical shift down a couple of

units. It is concave up, so we can assume the b value is between 0 and 1.

The absolute maximum is the largest Y value on the function and the absolute

minima is the smallest Y-value on the function. Either looking at the table or the graph

finds it. With exponential functions, there are not always both. In some instances,

especially with asymptotes, the “largest” X or Y value can be infinity if the function

continues on infinitely and never “touches” a value. It is difficult to determine a local

maximum or minimum with an exponential function, as they typically take on the visual

of a Nike check. So, there is not more than one “hump”, or concavity, from which we can

choose more than one maximum/minimum. Examples one and two are both positive

exponential functions that do not touch or cross the x-axis. Therefore, their minimum

value is open to the x-axis. Their maximum values would technically be positive infinity.

If their output variables were reflected across the x-axis, then their maximums would be

open to the x-axis and their minimums would technically be negative infinity.

Part III

With linear functions, we find that there is a consistent slope, y-intercept, and

a continuous domain. With exponential functions, we find that the there is a

beginning “point” and a slope that increases or decreases exponentially. So, it starts

off slow then gets faster and faster either in a positive or negative direction. With

quadratic functions, we find that the function “changes its mind” and curves to

create what we call a parabola. A parabola is a curve that is formed by a quadratic

function. It is concave up if it makes a “U-shape” and concave down if it makes an

“upside-down U-shape.”

The most basic form of a quadratic function is y = x2. Squaring the values is

asking for repeated multiplication. So, the lower “x-values” will give us a smaller y-

value than the larger “x-values.” This explains the curve of the parabola.

We see here that the function crosses the y-intercept at four. This represents the

“start” of the ball toss. The ball reaches its max at (2,6), which represents the change

in slope.

So, we see that one of the values in the formula is squared. That is because

the most basic form, also known as the parent function, is y = x2. The graph of y = x2

looks like so:

Notice that the function reflects across the y-axis. To get a better understanding of

why that is, look at the table of the function’s values:

X Y = x2 F(x)-2 Y = (-2)2 4-1 Y = (-1)2 10 Y = (0)2 01 Y = (1)2 12 Y = (2)2 4

Note the value of f(x) when x is zero. Then, when we travel the x-values

towards a negative direction, we get the same f(x) value as when we travel the x-

values in a positive direction. That is because when you put a negative value in

parenthesis and square it, you are basically getting the absolute value, which is the

same as when it is positive. Therefore, we see a reflection on the y-axis, or zero.

With my explanation of quadratic functions, I will use the example of

throwing a ball up in the air and watching it come back down. So, say a tall man

begins by tossing a ball at six feet, where his hands are. Assume he throws it upward

at two feet per second and the earth’s gravity is reducing the ball’s speed at a rate of

half a foot per second squared. We will use “t” to represent time, in seconds. Let’s

look at what we just stated:

Height starts at four feet 6Ball travels up at twelve feet per second 2tGravity pulls it down, changing the speed by about five feet per second. -0.5t2

So, we have the equation f(x) = -0.5t2 + 2t + 6. I will further explain how I got

this equation when I get to rates of change.

When we graph this, we see:

We see that the graph and formula y = x2 is very similar to the given example.

Notice that the graph of y = x2 demonstrates one that reflects across the y-axis. But

what if this is not the case? That is where we find the general formula for quadratic

functions: y = ax2 + bx + c where a is the initial slope, b is the “changed” slope and c

is some constant at the y-intercept. When the absolute value of a is greater than one,

the parabola will be “skinny” because the function will grow faster and faster. When

the absolute value of a is less than one, the parabola will be “fat” because the

function will grow slower.

All quadratic functions will have a domain of negative infinity to positive

infinity. Regardless if it is concave up or concave down, the x-values will continue to

travel either in a negative or positive direction. We cannot turn the function so that

it is sideways because then it would not be a function, as there would be two output

values for every input value.

The range of quadratic functions depends on the maximum or minimum

value, similar to finding the range of an exponential function. So, looking at the ball

toss example, our maximum value is (2,6). No values go higher that when y = 6. If we

track the function from left to right, we go from negative infinity to six. Then, when

the direction changes, we go from six to positive infinity. Therefore, we can state the

range as: (-∞, 6) U (6,∞). The ∞ symbol is the symbol we use to represent infinity.

The U is a “union” symbol, which we use to represent the fact that the function does

not stop at 6. It simply changes direction at 6.

Again, if we follow the function from left to right we can examine where it is

increasing and decreasing. From negative infinity to 6, we can see that the slop goes

from really steep to not as steep. This shows that it is increasing at a decreasing rate.

So, even though the function is increasing, the rate at which it travels to 6 is slowing

down. Then, from 6 to positive infinity, the function is decreasing but the slope is

getting steeper and steeper. This shows that it is decreasing at an increasing rate.

The rate of change can be found in the equation. Look at the equation:

f(x) = -0.5t2 + 2t + 6

Let’s break it up by going back to the table I made when I wrote out the equation:

Height starts at four feet 6Ball travels up at twelve feet per second 2tGravity pulls it down, changing the speed by about five feet per second. -0.5t2

So, the ball toss begins when Y = 6, or (0,6). Here, the x-value is 0 because the

x-values represent t, time, and the ball has yet to travel when it is at “start”. The ball

travels up two feet per second, and that is where we find 2t. If we just graphed Y =

2t, we would get a linear graph. But, the addition/subtraction of “ax2” in the

quadratic formula gives us a change in slope with a “second” slope, which is why the

formula can increase at a decreasing rate or decreasing with an increasing rate. So,

when we add -0.5t2 into the equation, this changes the slope to negative, since it’s

concave down. It’s squared because we’re changing the speed (in feet per second)

and putting that in “per second.”

As previously stated, the ball toss example is of a concave down parabola. If

we were to graph an example in which an item decreases at a decreasing rate, hits a

minimum value, then turns to travel increasing at an increasing rate, then the

parabola would be concave up. There are no points of inflections with quadratic

functions that begin with a squared value. A point of inflection is one in which

there is a change in concavity. But, if we were to graph a cubic function, such as y =

x3, then there would be a point of inflection because there would be yet another

“change” in the graph, which would demonstrate a change in concavity.

It is also important to note that there are no asymptotes in quadratic

formulas. An asymptote is when a functions infinite domain or range value gets

really close to a certain value, but never touches it. With quadratic functions, there

are no y values above a maximum value, or none under a minimum value. But, the

function is not approaching a certain value. Instead, like we already discussed, the

domain values continue on to either negative or positive infinity.

Every quadratic function has a y-intercept at some point. We know this

because the domain is from negative to positive infinity, so it is going to cross the y-

axis at some point. However, not all quadratic functions have x-intercepts. If, the

maximum value is above the x-axis and it is concave up, then it will never cross the

x-axis. But, if the maximum value is above the x-axis and it is concave down, then it

will cross the x-axis at two different points. With the ball toss, the ball begins at four

feet, which is at the point (0,4), which gives us a y-intercept of four. This is where

time (t) is zero because it has yet to start traveling. The ball then travels up, then

takes a turn down and crosses the x-intercept at (6,0), which is where the ball lands

or reaches the ground. This is where the number of feet (y) is zero. The ball is not in

the air, so it is at zero feet. Notice that there is another x-intercept at (-2,0). This is

because the quadratic function is continuous, so when we graph it, we can assume

that it will continue on left and right.

All quadratic functions have either an absolute maximum or an absolute

minimum. If the parabola is concave up, it will have an absolute minimum because

no other y-values in the entire function will go below the minimum. Likewise, if the

parabola is concave down, it will have a maximum value at which no y-values will

exist above it. With the ball toss, the ball travels up then down, reaching a maximum

value of (2,6). This is where time is two, representing that the ball is two seconds

into its journey. And the ball is at height six feet, which is the maximum height it will

reach.

Through my process of explaining functions, I have learned about how to pay

attention to detail better than I did before this class. Before, I understood the

information at the same level, but I did not know how to correctly point out errors

and changes. Now, I feel confident in my ability to explain functions better, using

correct terminology and multiple methods (tables, graphs, equations, etc.). As I

learn more mathematics, the smaller concepts make better sense in my head. Even

though functions are a foundation to many other forms of mathematics, there is a lot

to learn about them. Once I think I mastered one aspect of function behavior, I learn

something about them that I did not previously know. Every characteristic of a

function links with another one to make for an entire culture of functions. For

example, it was not until I mastered the concept of “increasing at a decreasing rate”

or “decreasing at a decreasing rate” and so on that I completely understood why

quadratic functions are written y = ax2 + bx + c. With this knowledge I have attained,

I feel much more confident in my abilities to turn around and apply it to other

aspects of mathematics. I know that this class has made a huge impact on my career

as a middle school math teacher and I am overjoyed to be able to make an

impression on adolescent learners of mathematics in the future. I also hope that

with teaching mathematics, I will continue to expand my depth of knowledge in the

area of function.

Elizabeth Hofmann

Dr. Mary Searcy

MAT 3920

Spring 2015

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