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A large part of science may be described as the study of the dependence of one measured quantity on another measured quantity. The word *function *is used in this context in a special way. In previous examples, the word *function *may have been used as follows:

- The density of
*V. natriegens*is a*function*of time. - Light intensity is a
*function*of depth below the surface in an ocean. - Light intensity is a
*function*of distance from the light source. - The frequency of cricket chirps is a
*function*of the ambient temperature. - The percentage of females turtles from a clutch of eggs is a
*function*of the incubation temperature.

## 2.2.1 Three definitions of “Function”.

Because of its prevalence and importance in science and mathematics, the word *function *has been defined several ways over the past three hundred years, and now is usually given a very precise formal meaning. More intuitive meanings are also helpful and we give three definitions of function, all of which will be useful to us.

The word *variable *means a symbol that represents any member of a given set, most often a set of numbers, and usually denotes a value of a measured quantity. Thus density of *V. natriegens*, time, percentage of females, incubation temperature, light intensity, depth and distance are all variables.

The terms *dependent *variable and *independent *variable are useful in the description of an experiment and the resulting functional relationship. The density of *V. natriegens* (dependent variable) is a function of time (independent variable). The percentage of females turtles from a clutch of eggs (dependent variable) is a *function *of the incubation temperature (independent variable).

Using the notion of variable, a function may be defined:

**Definition 2.2.1 Function I** *Given two variables, x and y, a function is a rule that assigns to each value of x a unique value of y.*

In this context, x is the *independent variable*, and y is the *dependent variable*. In some cases there is an equation that nicely describes the ‘rule’; in the percentage of females in a clutch of turtle eggs examples of the preceding section, there was not a simple equation that described the rule, but the rule met the definition of function, nevertheless.

The use of the words dependent and independent in describing variables may change with the context of the experiment and resulting function. For example, the data on the incubation of turtles implicitly assumed that the temperature was held constant during incubation. For turtles in the wild, however, temperature is not held constant and one might measure the temperature of a clutch of eggs as a function of time. Then, temperature becomes the dependent variable and time is the independent variable.

In Definition 2.2.1, the word ‘variable’ is a bit vague, and ‘a function is a rule’ leaves a question as to ‘What is a rule?’. A ‘set of objects’ or, equivalently, a ‘collection of objects’, is considered to be easier to understand than ‘variable’ and has broader concurrence as to its meaning. Your previous experience with the word function may have been that

**Definition 2.2.2 Function II** *A function is a rule that assigns to each number in a set called the domain a unique number in a set called the range of the function.*

Definition 2.2.2 is similar to Definition 2.2.1, except that ‘a number in a set called the domain’ has given meaning to *independent variable* and ‘a unique number in a set called the range’ has given meaning to *dependent variable*.

The word ‘rule’ is at the core of both definitions 2.2.1 **Function I**. and 2.2.2 **Function II**. and is still a bit vague. The definition of function currently considered to be the most concise is:

**Definition 2.2.3 Function III** *A function is a collection of ordered number pairs no two of which have the same first number.*

A little reflection will reveal that ‘a table of data’ is the motivation for Definition 2.2.3. A data point is actually a number pair. Consider the tables of data shown in Table 2.1 from *V. natriegens* growth and human population records. (16,0.036) is a data point. (64,0.169) is a data point. (1950,2.52) and ( 1980,4.45) are data points. These are basic bits of information for the functions. On the other hand, examine the data for cricket chirps in the same table, from Chapter 1. That also is a collection of ordered pairs, but the collection does not satisfy Definition 2.2.3. There are two ordered pairs in the table with the same first term – (66,102) and (66,103). Therefore the collection contains important information about the dependence of chirp frequency on temperature, although the collection does not constitute a function.

V. natriegens Growth pH 6.25 | |
---|---|

Time (min) | Population Density |

0 | 0.022 |

16 | 0.036 |

32 | 0.060 |

48 | 0.101 |

64 | 0.169 |

80 | 0.266 |

World Population | |
---|---|

Year | Population (billions) |

1940 | 2.30 |

1950 | 2.52 |

1960 | 3.02 |

1970 | 3.70 |

1980 | 4.45 |

1990 | 5.30 |

2000 | 6.06 |

Cricket Chirps | |
---|---|

Temperature ((^{circ} F)) | Chirps per Minute |

67 | 109 |

73 | 136 |

78 | 160 |

61 | 87 |

66 | 103 |

66 | 102 |

67 | 108 |

77 | 154 |

74 | 144 |

76 | 150 |

In a function that is a collection of ordered number pairs, the first number of a number pair is always a value of the independent variable and a member of the domain and the second number is always a value of the dependent variable and a member of the range. Almost always in recording the results of an experiment, the numbers in the domain are listed in the column on the left and the numbers in the range are listed in the column on the right. Formally,

**Definition 2.2.4 Domain and Range** *For Definition 2.2.3 of function, the domain is defined as the set of all numbers that occur as the first number in an ordered pair of the function and the range of the function is the set of all numbers that occur as a second number in an ordered pair of the function.*

**Example 2.2.1** Data for the percentage of U.S. population in 1955 that had antibodies to the polio virus as a function of age is shown in Table 2.2.1.1. The data show an interesting fact that a high percentage of the population in 1955 had been infected with polio. A much smaller percentage were crippled or killed by the disease.

Although Table 2.2.1.1 is a function, it is only an approximation to a **perhaps **real underlying function. The order pair, (17.5, 72), signals that 72 percent of the people of age 17.5 years had antibodies to the polio virus. More accurately, (17.5,72) signals that of a **sample **of people who had ages in the interval from age 15 to less than 20, the percentage who tested positive to antibodies to the polio virus was greater than or equal to 71.5 and less than 72.5.

Table 2.2.1.1 is a useful representation of an enormous table of data that lists **for a certain instance of time during 1955**, for each U.S. citizen, their age (measured perhaps in hours (minutes?, seconds?)), and whether they had HIV antibodies, ’yes’ or ’no,’ This table would not be a function, but for each age, the percent of people of that age who were HIV positive would be a number and those age-percent pairs would form a function. The domain of that function would be the finite set of ages in the U.S. population.

Age | 0.8 | 1.5 | 2.5 | 3.5 | 4.5 | 5.5 | 6.5 | 7.5 | 8.5 | 9.5 | 12.5 | 17.5 | 22.5 | 27.5 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

% | 3 | 6 | 13 | 19 | 27 | 35 | 40 | 43 | 46 | 49 | 64 | 72 | 78 | 87 |

Remember that only a very few data ‘points’ were listed from the large number of possible points in each of the experiments we considered. There is a larger function in the background of each experiment.

Because many biological quantities change with time, the domain of a function of interest is often an interval of time. In some cases a biological reaction depends on temperature (percentage of females in a clutch of turtle eggs for example) so that the domain of a function may be an interval of temperatures. In cases of spatial distribution of a disease or light intensity below the surface of a lake, the domain may be an interval of distances.

It is implicit in the bacteria growth data that at any specific time, there is only one value of the bacterial density^{2} associated with that time. It may be incorrectly or inaccurately read, but a fundamental assumption is that there is only one correct density for that specific time. The condition that no two of the ordered number pairs have the same first term is a way of saying that each number in the domain has a unique number in the range associated with it.

All three of the definitions of function are helpful, as are brief verbal descriptions, and we will rely on all of them. Our basic definition, however, is the ordered pair definition, Definition 2.2.3.

## 2.2.2 Simple graphs.

Coordinate geometry associates ordered number pairs with points of the plane so that by Definition 2.2.3 a function is automatically identified with a point set in the plane called a *simple graph:*

Definition 2.2.5 Simple Graph

*A simple graph is a point set, G, in the plane such that no vertical line contains two points of G.*

*The domain of G is the set of x-coordinates of points of G and the range of G is the set of all y-coordinates of points of G.*

Note: For use in this book, every set contains at least one element.

The domain of a simple graph G is sometimes called the x-projection of G, meaning the vertical projection of G onto the x-axis and the range of G is sometimes called the y-projection of G meaning the horizontal projection of G onto the y-axis.

A review of the graphs of incubation temperature - percentage of females turtles in Figure 2.1.1 and Exercise Fig. 2.1.1 will show that in each graph at least one vertical line contains two points of the graph. Neither of these graphs is a simple graph, but the graphs convey useful information.

A circle is not a simple graph. As shown in Figure 2.2.1A there is a vertical line that contains two points of it. There are a lot of such vertical lines. The circle does contain a simple graph, and contains one that is ‘as large as possible’. The upper semicircle shown in Figure 2.2.1B is a simple graph. The points (-1,0) and (1,0) are filled to show that they belong to it. It is impossible to add any other points of the circle to this simple graph and still have a simple graph — thus it is ‘as large as possible’. An equation of the upper semicircle is

[y=sqrt{1-x^{2}}, quad-1 leq x leq 1]

The domain of this simple graph is [-1,1], and the range is [0,1]. Obviously the lower semicircle is a maximal simple graph also, and it has the equation

[y=-sqrt{1-x^{2}}, quad-1 leq x leq 1]

The domain is again [-1,1], and the range is [-1,0].

**Figure (PageIndex{1}):** *A. A circle; a vertical line contains two point of the circle so that it is not a simple graph. B. A subset of the circle that is a simple graph. C. Another subset of the circle that is a simple graph. The simple graphs in (b) and (c) are maximal in the sense that any point from the circle added to the graphs would create a set that is not a simple graph — the vertical line containing that point would also contain a point from the original simple graph.*

There is yet a third simple graph contained in the circle, shown in Figure 2.2.1C, and it is ‘as large as possible’. An equation for that simple graph is

[y=left{egin{aligned}

sqrt{1-x^{2}} & ext { if } &-1 leq x leq 0

-sqrt{1-x^{2}} & ext { if } & 0

The domain is [-1,1] and range is [-1,1] Because of the intuitive advantage of geometry, it is often useful to use simple graphs instead of equations or tables to describe functions, but again, we will use any of these as needed.

### Exercises for Section 2.2 Functions and Simple Graphs.

**Exercise 2.2.1** Which of the tables shown in Table Ex. 2.2.1 reported as data describing the growth of *V. natriegens* are functions?

Time | Abs | Time | Abs | Time | Abs |
---|---|---|---|---|---|

0 | 0.018 | 0 | 0.018 | 0 | 0.018 |

12 | 0.023 | 12 | 0.023 | 12 | 0.023 |

24 | 0.030 | 24 | 0.030 | 24 | 0.030 |

36 | 0.039 | 36 | 0.039 | 48 | 0.049 |

48 | 0.049 | 48 | 0.049 | 48 | 0.049 |

48 | 0.065 | 60 | 0.065 | 48 | 0.049 |

60 | 0.085 | 72 | 0.065 | 72 | 0.065 |

78 | 0.120 | 87 | 0.065 | 87 | 0.065 |

96 | 0.145 | 96 | 0.080 | 96 | 0.080 |

110 | 0.195 | 110 | 0.095 | 110 | 0.095 |

120 | 0.240 | 120 | 0.120 | 120 | 0.120 |

**Exercise 2.2.2** For the following experiments, determine the independent variable and the dependent variable, and draw a simple graph or give a brief verbal description (your best guess) of the function relating the two.

- A rabbit population size is a function of the number of coyotes in the region.
- An agronomist, interested in the most economical rate of nitrogen application to corn, measures the corn yield in test plots using eight different levels of nitrogen application.
- An enzyme, E, catalyzes a reaction converting a substrate, S, to a product P according to [mathrm{E}+mathrm{S} ightleftharpoons mathrm{ES} ightleftharpoons mathrm{E}+mathrm{P}] Assume enzyme concentration, [E], is fixed. A scientist measures the rate at which the product P accumulates at different concentrations, [S], of substrate.
- A scientist titrates a 0.1 M solution of HCl into 5 ml of an unknown basic solution containing litmus (litmus causes the color of the solution to change as the pH changes).

**Exercise 2.2.3** A table for bacterial density for growth of *V. natriegens* is repeated in Exercise Table 2.2.3. There are two functions that relate population density to time in this table, one that relates population density to time and another that relates population to time index.

- Identify an ordered pair that belongs to both functions.
- One of the functions is implicitly only a partial list of the order pairs that belong to it. You may be of the opinion that both functions have that property, but some people may think one is more obviously only a sample of the data. Which one?
- What is the domain of the other function?

pH 6.25 | ||
---|---|---|

Time (min) | Time Index (t) | Population Density (B_t) |

0 | 0 | 0.022 |

16 | 1 | 0.036 |

32 | 2 | 0.060 |

48 | 3 | 0.101 |

64 | 4 | 0.169 |

80 | 5 | 0.266 |

**Exercise 2.2.4** Refer to the graphs in Figure Ex. 2.2.4.

- Which of the graphs are simple graphs?
- For those that are not simple graphs,
- Draw, using only the points of the graph, a simple graph that is ‘as large as possible’, meaning that no other points can be added and still have a simple graph.
- Draw a second such simple graph.
- Identify the domains and ranges of the two simple graphs you have drawn.
- How many such simple graphs may be drawn?

**Figure for Exercise 2.2.4** Graphs for Exercise 2.2.4. Some are simple graphs; some are not simple graphs.

**Exercise 2.2.5** Make a table showing the ordered pairs of a simple graph contained in the graph in Figure Ex. 2.2.5 and that has domain

[{-1.5,-1.0,-0.5,0.0,0.5,1.0,1.5}]

How many such simple graphs are contained in the graph of Figure Ex. 2.2.5 and that have this domain?

**Figure for Exercise 2.2.5** *Graph for Exercise 2.2.5.*

**Exercise 2.2.6**

- Does every subset of the plane contain a simple graph?
- Does every subset of the plane contain two simple graphs?
- Is there a subset of the plane that contains two and only two simple graphs?
- Is there a line in the plane that is not the graph of a function?
- Is there a function whose graph is a circle?
- Is there a simple graph in the plane whose domain is the interval [0,1] (including 0 and 1) and whose range is the interval [0,3]?
- Is there a simple graph in the plane whose domain is [0,1] and whose range is the y-axis?

**Exercise 2.2.7 A bit of a difficult exercise.** For any location, (lambda) on Earth, let Annual Daytime at (lambda) , (AD(lambda)), be the sum of the lengths of time between sunrise and sunset at (lambda) for all of the days of the year. Find a reasonable formula for (AD(lambda)). You may guess or find data to suggest a reasonable formula, but we found proof of the validity of our formula a bit arduous. As often happens in mathematics, instead of solving the actual problem posed, we found it best to solve a ’nearby’ problem that was more tractable. The 365.24... days in a year is a distraction, the elliptical orbit of Earth is a downright hinderance, and the wobble of Earth on its axis can be overlooked. Specifically, we find it helpful to assume that there are precisely 366 days in the year (after all this was true about 7 or 8 million years ago), the Earth’s orbit about the sun is a circle, the Earth’s axis makes a constant angle with the plane of the orbit, and that the rays from the sun to Earth are parallel. We hope you enjoy the question.

## 2.2.3 Functions in other settings.

There are extensions of the function concept to settings where the ordered pairs are not ordered number pairs. A prime example of this is the genetic code shown in Figure 2.3. The relation is a true function (no two ordered pairs have the same first term), and during the translation of proteins, the ribosome and the transfer RNA’s use this function reliably.

**Figure (PageIndex{2}):** *The genetic code (for human nuclear RNA). Sets of three nucleotides in RNA (codons) are translated into amino acids in the course of proteins synthesis. CAA codes for Gln (glutamine). ∗AUG codes for Met (methionine) and is also the START codon.*

**Explore 2.2.1** List three ordered pairs of the genetic code. What is the domain of the genetic code? What is the range of the genetic code?

The ordered pair concept is retained in the preceding example; the only change has been in the types of objects that are in the domain and range. When the objects get too far afield from simple numbers, the word transformation is sometimes used in place of function. The genetic code is a transformation of the codons into amino acids and start and stop signals.

Another commonly encountered extension of the kinds of objects in the domain of a function occurs when one physical or biological quantity is dependent on two others. For example, the widely known Charles’ Law in Chemistry can be stated as

[P=frac{n R T}{V}]

where (P) = pressure in atmospheres, n = number of molecular weights of the gas, (R) = 0.0820 Atmospheres/degree Kelvin-mol = 8.3 /degree Kelvin-mol (the gas constant), (T) = temperature in kelvins, and (V) = volume in liters. For a fixed sample of gas, the pressure is dependent on two quantities, temperature and volume. The domain is the set of all feasible temperature-volume pairs, the range is the set of all feasible pressures. The function in this case is said to be a function of two variables. The ordered pairs in the function are of the form

[((x, y), z), quad ext{or} quad( ext { (temperature, volume) }, ext { pressure })]

There may also be multivalued transformations. For example, doctors prescribe antibiotics. For each bacterial infection, there may be more than one antibiotic effective against that bacterium; there may be a list of such antibiotics. The domain would be a set of bacteria, and the range would be a set of lists of antibiotics.

### Exercises for Section 2.2.3 Functions in other settings.

**Exercise 2.2.8** Describe the domain and range for each of the following transformations.

- Bird identification guide book.
- A judge sentences defendants to jail terms.
- The time between sunrise and sunset.
- Antibiotic side effects.

^{2} As measured, for example, by light absorbance in a spectrophotometer as discussed on page 4.

## Graphs as adjacency information.

can be represented as the function. This mechansim can be extended to a wide variety of graphs types by slightly altering or enhancing the kind of function that represents the graph. Here are a few examples.

### Directed graph.

We could represent this as a Dgraph as follows:

### Vertex labeled graph.

### Edge labeled graph.

### Advantages of representing graphs as functions

### Disadvantages of using graphs as functions

- Cannot be extended to accomodate queries about the set of Vertices or the set of Edges.
- Depending upon the compiler that compiles the functions may not be very efficient. In fact the worst case time could be proportional to the number of vertices.
- The graph must be known statically at compile time.

## Graphs as arrays of adjacent vertexes.

In the rest of this note we will assume that Vertices are of type Int , and that the Vertices set is a finite range of the type Int . Thus a graph can be represented as follows:

## Graphs and Functions

This lesson is designed to introduce students to graphing functions. These activities can be done individually or in teams of as many as four students. Allow for 2-3 hours of class time for the entire lesson if all portions are done in class.

### Objectives

- have been introduced to plotting functions on the Cartesian coordinate plane
- have seen several categories of functions, including lines and parabolas

### Standards Addressed:

- Functions and Relationships
- The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
- The student demonstrates algebraic thinking.

- Functions and Relationships
- The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
- The student demonstrates algebraic thinking.

- Expressions and Equations
- Analyze and solve linear equations and pairs of simultaneous linear equations.

- Define, evaluate, and compare functions.
- Use functions to model relationships between quantities.

- Building Functions
- Build a function that models a relationship between two quantities
- Build new functions from existing functions

- Construct and compare linear, quadratic, and exponential models and solve problems

- Algebra
- Represent and analyze mathematical situations and structures using algebraic symbols

- Algebra
- Represent and analyze mathematical situations and structures using algebraic symbols
- Understand patterns, relations, and functions
- Use mathematical models to represent and understand quantitative relationships

- Algebra
- Competency Goal 4: The learner will use relations and functions to solve problems.

- Algebra
- Competency Goal 4: The learner will use relations and functions to solve problems.

- Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra
- COMPETENCY GOAL 5: The learner will understand and use linear relations and functions.

- Algebra
- COMPETENCY GOAL 4: The learner will understand and use linear relations and functions.
- COMPETENCY GOAL 5: The learner will understand and use linear relations and functions.

- Algebra
- The student will demonstrate through the mathematical processes an understanding of numeric patterns, symbols as representations of unknown quantity, and situations showing increase over time.

- Algebra
- Standard 4-3: The student will demonstrate through the mathematical processes an understanding of numeric and nonnumeric patterns, the representation of simple mathematical relationships, and the application of procedures to find the value of an unknown.

- Standard 4-6: The student will demonstrate through the mathematical processes an understanding of the impact of data-collection methods, the appropriate graph for categorical or numerical data, and the analysis of possible outcomes for a simple event.

- Standard 4-4: The student will demonstrate through the mathematical processes an understanding of the relationship between two- and three-dimensional shapes, the use of transformations to determine congruency, and the representation of location and movement within the first quadrant of a coordinate system.
- Standard 4-4: The student will demonstrate through the mathematical processes an understanding of the relationship between two- and three-dimensional shapes, the use of transformations to determine congruency, and the representation of location and moveme

- Algebra
- The student will demonstrate through the mathematical processes an understanding of proportional relationships.

- Algebra
- The student will demonstrate through the mathematical processes an understanding of equations, inequalities, and linear functions.

- The student will demonstrate through the mathematical processes an understanding of the relationships between two variables within one population or sample.

- The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane and the effect of a dilation in a coordinate plane.
- The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane and the effect of a dilation

- Elementary Algebra
- Standard EA-1: The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.
- Standard EA-3: The student will demonstrate through the mathematical processes an understanding of relationships and functions.
- Standard EA-4: The student will demonstrate through the mathematical processes an understanding of the procedures for writing and solving linear equations and inequalities.
- Standard EA-5: The student will demonstrate through the mathematical processes an understanding of the graphs and characteristics of linear equations and inequalities.
- Standard EA-6: The student will demonstrate through the mathematical processes an understanding of quadratic relationships and functions.

- Algebra
- The student will demonstrate through the mathematical processes an understanding of functions, systems of equations, and systems of linear inequalities.
- The student will demonstrate through the mathematical processes an understanding of quadratic equations and the complex number system.
- The student will demonstrate through the mathematical processes an understanding of algebraic expressions and nonlinear functions.

- Geometry
- 4.15.b The student will describe the path of shortest distance between two points on a flat surface.
- 4.16 The student will identify and draw representations of lines that illustrate intersection, parallelism, and perpendicularity.

- 4.15.b
- 4.16

- Probability and Statistics
- 7.17 The student, given a problem situation, will collect, analyze, display, and interpret data, using a variety of graphical methods, including frequency distributions line plots histograms stem-and-leaf plots box-and-whisker plots and scattergrams.
- 7.17 The student, given a problem situation, will collect, analyze, display, and interpret data, using a variety of graphical methods, including

- Patterns, Functions, and Algebra
- 8.14a The student will describe and represent relations and functions, using tables, graphs, and rules and
- 8.16 The student will graph a linear equation in two variables, in the coordinate plane, using a table of ordered pairs.
- 8.14 The student will
- 8.16 The student will graph a linear equation in two variables, in the coordinate plane, using a

- Algebra II
- AII.10 The student will investigate and describe through the use of graphs the relationships between the solution of an equation, zero of a function, x-intercept of a graph, and factors of a polynomial expression.
- AII.18 The student will identify conic sections (circle, ellipse, parabola, and hyperbola) from his/her equations. Given the equations in (h, k) form, the student will sketch graphs of conic sections, using transformations.
- AII.20 The student will identify, create, and solve practical problems involving inverse variation and a combination of direct and inverse variations.
- AII.10
- AII.18
- AII.20

### Textbooks Aligned:

- 7th
- [ Module 1 - Search and Rescue ] Section 4: Function Models
**Reason for Alignment:**The Graphs and Functions lesson is a good follow up to the Introduction to Functions lesson, also aligned with this section of the text, by building on the graphing of functions. This one goes deeper into the vocabulary and algebra of functions. This lesson may take a while if completed together in class, but some students could move through it independently in a shorter time.

- [ Module 3 - The Mystery of Blacktail Canyon ] Section 2: Equations and Graphs
**Reason for Alignment:**This is a detailed lesson on graphing functions. There are discussion suggestions, vocabulary and a Graph Sketcher Activity Worksheet already made up for practice. This lesson fits with the Graphit activity.

### Student Prerequisites

*Arithmetic:*Student must be able to:- perform integer and fractional arithmetic
- plot points on the Cartesian coordinate system
- read the coordinates of a point from a graph

- work with simple algebraic expressions

- perform basic mouse manipulations such as point, click and drag
- use a browser for experimenting with the activities

### Teacher Preparation

- Access to a browser
- Pencil and graph paper
- Copies of supplemental materials for the activities:
- Graph Sketcher Activity Worksheet

### Key Terms

**constant functions**Functions that stay the same no matter what the variable does are called constant functions **constants**In math, things that do not change are called constants. The things that do change are called variables. **coordinate plane**A plane with a point selected as an origin, some length selected as a unit of distance, and two perpendicular lines that intersect at the origin, with positive and negative direction selected on each line. Traditionally, the lines are called x (drawn from left to right, with positive direction to the right of the origin) and y (drawn from bottom to top, with positive direction upward of the origin). Coordinates of a point are determined by the distance of this point from the lines, and the signs of the coordinates are determined by whether the point is in the positive or in the negative direction from the origin **coordinates**A unique ordered pair of numbers that identifies a point on the coordinate plane. The first number in the ordered pair identifies the position with regard to the x-axis while the second number identifies the position on the y-axis **function**A function f of a variable x is a rule that assigns to each number x in the function's domain a single number f(x). The word "single" in this definition is very important **graph**A visual representation of data that displays the relationship among variables, usually cast along x and y axes. **negative numbers**Numbers less than zero. In graphing, numbers to the left of zero. Negative numbers are represented by placing a minus sign (-) in front of the number ### Lesson Outline

Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson. You may ask the following questions:

- Can someone tell me what a function is?
- Will someone give me an example of a function?
- Will someone give me an example of something that is not a function?

Let the students know what it is they will be doing and learning today. Say something like this:

- Today, class, we are going to learn more about functions.
- We are going to use the computers to learn more about functions, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first.

- Have the students try plotting points for several simple functions to ensure that they have some skill at plotting by hand. Even if graphing calculators are available, have the students plot points on graph paper - this is a skill that is important to practice by hand. Here are a few functions that might be assigned:
- Practice the students' function plotting skills by having them check their work from the previous activity by plotting the same functions using the Graph Sketcher Tool.
- Have the students investigate functions of the form y = _____ x + ____ using the Graph Sketcher Tool to determine what kinds of functions come from this form, and what changing each constant does to the function. Be sure to have them keep track of what they try and record their hypotheses and observations.
- Relate these graphs to the lesson on Linear Functions to demonstrate the rationale for the terms m = slope and b = intercept in the formula .

- You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

### Alternate Outline

- Replace all Graph Sketcher activities with graphing calculator activities. Note: Depending on the graphing calculator, you might have to spend some additional time discussing setting the window ranges.
- Replace all Graph Sketcher activities with Simple Plot activities. Simple Plot is a point plotting activity, which requires that the students create tables of values for the functions before plotting.
- Limit investigations to functions with one operation as in the Function Machine lesson and/or to linear functions as in the Linear Functions lesson .

### Suggested Follow-Up

After these discussions and activities, students will have more experience with functions and graphing. The next lesson, Reading Graphs , shows the students that graphs can be used to convey lots of information about a given situation.

## 3.6 Graphs of Functions

In the last section we learned how to determine if a relation is a function. The relations we looked at were expressed as a set of ordered pairs, a mapping or an equation. We will now look at how to tell if a graph is that of a function.

The graph of a linear equation is a straight line where every point on the line is a solution of the equation and every solution of this equation is a point on this line.

A relation is a function if every element of the domain has exactly one value in the range. So the relation defined by the equation y = 2 x − 3 y = 2 x − 3 is a function.

If we look at the graph, each vertical dashed line only intersects the line at one point. This makes sense as in a function, for every

*x*-value there is only one*y*-value.If the vertical line hit the graph twice, the

*x*-value would be mapped to two*y*-values, and so the graph would not represent a function.This leads us to the vertical line test. A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. If any vertical line intersects the graph in more than one point, the graph does not represent a function.

### Vertical Line Test

A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.

If any vertical line intersects the graph in more than one point, the graph does not represent a function.

### Example 3.51

Determine whether each graph is the graph of a function.

#### Solution

ⓐ Since any vertical line intersects the graph in at most one point, the graph is the graph of a function.

ⓑ One of the vertical lines shown on the graph, intersects it in two points. This graph does not represent a function.Determine whether each graph is the graph of a function.

Determine whether each graph is the graph of a function.

### Identify Graphs of Basic Functions

Compare the graph of y = 2 x − 3 y = 2 x − 3 previously shown in Figure 3.14 with the graph of f ( x ) = 2 x − 3 f ( x ) = 2 x − 3 shown in Figure 3.15. Nothing has changed but the notation.

### Graph of a Function

The graph of a function is the graph of all its ordered pairs, ( x , y ) ( x , y ) or using function notation, ( x , f ( x ) ) ( x , f ( x ) ) where y = f ( x ) . y = f ( x ) .

As we move forward in our study, it is helpful to be familiar with the graphs of several basic functions and be able to identify them.

We wrote linear equations in several forms, but it will be most helpful for us here to use the slope-intercept form of the linear equation. The slope-intercept form of a linear equation is y = m x + b . y = m x + b . In function notation, this linear function becomes f ( x ) = m x + b f ( x ) = m x + b where

*m*is the slope of the line and*b*is the*y*-intercept.The domain is the set of all real numbers, and the range is also the set of all real numbers.

### Linear Function

We will use the graphing techniques we used earlier, to graph the basic functions.

### Example 3.52

#### Solution

Notice that for any real number we put in the function, the function value will be

*b*. This tells us the range has only one value,*b*.### Constant Function

### Example 3.53

#### Solution

### Identity Function

The next function we will look at is not a linear function. So the graph will not be a line. The only method we have to graph this function is point plotting. Because this is an unfamiliar function, we make sure to choose several positive and negative values as well as 0 for our x-values.

### Example 3.54

#### Solution

We choose

*x*-values. We substitute them in and then create a chart as shown.Looking at the result in Example 3.54, we can summarize the features of the square function. We call this graph a parabola. As we consider the domain, notice any real number can be used as an

*x*-value. The domain is all real numbers.The range is not all real numbers. Notice the graph consists of values of

*y*never go below zero. This makes sense as the square of any number cannot be negative. So, the range of the square function is all non-negative real numbers.### Square Function

The next function we will look at is also not a linear function so the graph will not be a line. Again we will use point plotting, and make sure to choose several positive and negative values as well as 0 for our

*x*-values.### Example 3.55

#### Solution

We choose

*x*-values. We substitute them in and then create a chart.Looking at the result in Example 3.55, we can summarize the features of the cube function. As we consider the domain, notice any real number can be used as an

*x*-value. The domain is all real numbers.The range is all real numbers. This makes sense as the cube of any non-zero number can be positive or negative. So, the range of the cube function is all real numbers.

### Cube Function

The next function we will look at does not square or cube the input values, but rather takes the square root of those values.

### Example 3.56

#### Solution

We choose

*x*-values. Since we will be taking the square root, we choose numbers that are perfect squares, to make our work easier. We substitute them in and then create a chart.### Square Root Function

### Example 3.57

#### Solution

We choose

*x*-values. We substitute them in and then create a chart.### Absolute Value Function

### Read Information from a Graph of a Function

In the sciences and business, data is often collected and then graphed. The graph is analyzed, information is obtained from the graph and then often predictions are made from the data.

We will start by reading the domain and range of a function from its graph.

Remember the domain is the set of all the

*x*-values in the ordered pairs in the function. To find the domain we look at the graph and find all the values of*x*that have a corresponding value on the graph. Follow the value*x*up or down vertically. If you hit the graph of the function then*x*is in the domain.Remember the range is the set of all the

*y*-values in the ordered pairs in the function. To find the range we look at the graph and find all the values of*y*that have a corresponding value on the graph. Follow the value*y*left or right horizontally. If you hit the graph of the function then*y*is in the range.### Example 3.58

Use the graph of the function to find its domain and range. Write the domain and range in interval notation.

#### Solution

To find the domain we look at the graph and find all the values of

*x*that correspond to a point on the graph. The domain is highlighted in red on the graph. The domain is [ −3 , 3 ] . [ −3 , 3 ] .To find the range we look at the graph and find all the values of

*y*that correspond to a point on the graph. The range is highlighted in blue on the graph. The range is [ −1 , 3 ] . [ −1 , 3 ] .Use the graph of the function to find its domain and range. Write the domain and range in interval notation.

Use the graph of the function to find its domain and range. Write the domain and range in interval notation.

We are now going to read information from the graph that you may see in future math classes.

## Graphs of Functions Contents: This page corresponds to § 1.4 (p. 116) of the text. Suggested Problems from Text p. 124 #1, 2, 5, 8, 9, 11, 16, 17, 21, 25, 27, 29, 31, 39, 40, 47, 50, 51, 52, 54, 57, 64, 65, 66 Defining the Graph of a Function The graph of a function f is the set of all points in the plane of the form (x, f(x)). We could also define the graph of f to be the graph of the equation y = f(x). So, the graph of a function if a special case of the graph of an equation. Let f(x) = x 2 - 3 . Recall that when we introduced graphs of equations we noted that if we can solve the equation for y, then it is easy to find points that are on the graph. We simply choose a number for x, then compute the corresponding value of y. Graphs of functions are graphs of equations that have been solved for y! The graph of f(x) in this example is the graph of y = x 2 - 3. It is easy to generate points on the graph. Choose a value for the first coordinate, then evaluate f at that number to find the second coordinate. The following table shows several values for x and the function f evaluated at those numbers. x -2 -1 0 1 2 f(x) 1 -2 -3 -2 1 Each column of numbers in the table holds the coordinates of a point on the graph of f. (a) Plot the five points on the graph of f from the table above, and based on these points, sketch the graph of f. (b) Verify that your sketch is correct by using the Java Grapher to graph f. Simply enter the formula x^2 - 3 in the f text box and click graph. Let f be the piecewise-defined function To express f in a single formula for the Java Grapher or Java Calculator we write (5 - x^2)*(xLE2) + (x - 1)*(2Lx). The factor (xLE2) has the value 1 for x <= 2 and 0 for x > 2. Similarly, (2Lx) is 1 for 2 < x and 0 otherwise. If we evaluate the sum above at x = 3, the first product is 0 because (xLE2) is 0 and the second product is (3 - 1)*1=2. In other words, for x > 2, the formula evaluates to x - 1. If x <= 2, then the formula above is equal to 5 - x^2, which is exactly what we want! The graph of f is shown below. We have seen that some equations in x and y do not describe y as a function of x. The algebraic way see if an equation determines y as a function of x is to solve for y. If there is not a unique solution, then y is not a function of x. Suppose that we are given the graph of the equation. There is an easy way to see if this equation describes y as a function of x. Vertical Line Test A set of points in the plane is the graph of a function if and only if no vertical line intersects the graph in more than one point. The graph of the equation y 2 = x + 5 is shown below. By the vertical line test, this graph is not the graph of a function, because there are many vertical lines that hit it more than once. Think of the vertical line test this way. The points on the graph of a function f have the form (x, f(x)), so once you know the first coordinate, the second is determined. Therefore, there cannot be two points on the graph of a function with the same first coordinate. All the points on a vertical line have the same first coordinate, so if a vertical line hits a graph twice, then there are two points on the graph with the same first coordinate. If that happens, the graph is not the graph of a function. Videos: Animated Gif, MS Avi File, or Real Video File Characteristics of Graphs Consider the function f(x) = 2 x + 1. We recognize the equation y = 2 x + 1 as the Slope-Intercept form of the equation of a line with slope 2 and y-intercept (0,1). Think of a point moving on the graph of f. As the point moves toward the right it rises. This is what it means for a function to be increasing. Your text has a more precise definition, but this is the basic idea. The function f above is increasing everywhere. In general, there are intervals where a function is increasing and intervals where it is decreasing. Using interval notation, we say that the function is decreasing on the interval (-3, 2) increasing on (-infinity, -3) and (2, infinity) Graph the function f(x) = x 2 - 6x + 7 and find the intervals where it is increasing and where it is decreasing. Answer Some of the most characteristics of a function are its Relative Extreme Values . Points on the functions graph corresponding to relative extreme values are turning points, or points where the function changes from decreasing to increasing or vice versa. Let f be the function whose graph is drawn below. f is decreasing on (-infinity, a) and increasing on (a, b), so the point (a, f(a)) is a turning point of the graph. f(a) is called a relative minimum of f. Note that f(a) is not the smallest function value, f(c) is. However, if we consider only the portion of the graph in the circle above a, then f(a) is the smallest second coordinate. Look at the circle on the graph above b. While f(b) is not the largest function value (this function does not have a largest value), if we look only at the portion of the graph in the circle, then the point (b, f(b)) is above all the other points. So, f(b) is a relative maximum of f. f(c) is another relative minimum of f. Indeed, f(c) is the absolute minimum of f, but it is also one of the relative minima. Here again we are giving definitions that appeal to your geometric intuition. The precise definitions are given in your text. Approximating Relative Extrema Finding the exact location of a function's relative extrema generally requires calculus. However, graphing utilities such as the Java Grapher may be used to approximate these numbers. Suppose a is a number such that f(a) is a relative minimum. In applications, it is often more important to know where the function attains its relative minimum than it is to know what the relative minimum is. For example, f(x) = x 3 - 4x 2 + 4x has a relative minimum of 0. It attains this relative minimum at x = 2, so (2,0) is a turning point of the graph of f. We will call the point (2,0) a relative minimum point. In general, a relative extreme point is a point on the graph of f whose second coordinate is a relative extreme value of f. Approximate the relative extremes points of f(x) = x 3 - x 2 - 6x. When you display the graph of f in the default viewing rectangle you see that f has one relative maximum point near (-1,4) and one relative minimum point near (2,-8). The approximations (-1,4) and (2,-8) are not very close to the real relative extreme points, so we will use the zoom and trace features to improve the approximations. When you click the Trace button, a point on the graph of f is indicated with a small circle. The coordinates of that point are reported in the two text boxes near the Trace button. As you click the box with the right arrow >, the trace point moves to the right, staying on the graph of f. If you select a larger Step Size from the pull down menu, then the trace point moves farther with each click. Also notice that once you have clicked the > button, then you can use the enter key to move further right. It is possible to move faster with the enter key than with the mouse. Using the default view, the lowest point found while tracing near the minimum point is (1.8, -8.208). Note that this is not the exact location of the minimum point. We need to look at the trace points on either side of this point to get an idea of how close we are. Find this trace point, make sure that the Step Size is set to 1, and then find the points on either side of this point. The table below lists the coordinates of these points. x y 1.7333333 -8.196741 1.8 -8.208 1.8666667 -8.180148 Note that different Java implementations will compute and report different accuracies, so the values you find may may be slightly different from those above. The points in the table show that the real minimum point has an x coordinate somewhere between 1.7333333 and 1.8666667. Note that we do not yet have enough information to report the x value with even one decimal place accuracy, because if the second decimal place were a 4, then the value would round to 1.7. If the second decimal place were a 7, then the value would round to 1.8. So we need to improve our estimate by zooming in on the minimum point. There are several ways to do this with the Grapher : Zoom In , Zoom Box , or set the coordinates of the viewing rectangle. In this example we will set the view coordinates. Type in Xmi n = 1.74, Xmax = 1.86, Ymin = -8.21, Ymax = -8.19, and set the viewing rectangle to these coordinates by clicking the Reset button. Changing the viewing rectangle removes the trace point. To get it back, click the Trace button again. Now you need to move left to get to the minimum point. There are two different x values that correspond to the lowest y value. x y 1.786 -8.20882 1.7864 -8.20882 We still do not know the exact location of the minimum point, but we know that its x coordinate is between 1.786 and 1.7864. That means the x coordinate will be 1.786 after rounding to three decimal places, . Since we are only reporting three decimal places for the x coordinate, we will also round the y coordinate to three decimal places, so our approximation is (1.786, -8.209). Find the relative maximum point to two decimal place accuracy. Answer Even and Odd Functions A function f is even if its graph is symmetric with respect to the y-axis. This criterion can be stated algebraically as follows: f is even if f(-x) = f(x) for all x in the domain of f. For example, if you evaluate f at 3 and at -3, then you will get the same value if f is even. This condition is very easy to check with the Java Grapher. Open the Grapher and type (x - 2)^2 into the f text box and click the Graph button. This function is not even, so when we graph its reflection about the y-axis, we will get a new graph. In the g text box type f(-x), and click the Graph button The graph of g is the reflection about the y-axis of the graph of f. Since we see two distinct graphs, we know that f is not even. Now replace the text in the f box with x^2 - 3 clear the text from the g box and graph the function. This graph is symmetric with respect to the y-axis, so when you enter f(-x) in the g box and graph again, you do not see anything new. This is because the graph of g is the same as the graph of f. A function f is odd if its graph is symmetric with respect to the origin. This criterion can be stated algebraically as follows: f is odd if f(-x) = -f(x) for all x in the domain of f. For example, if you evaluate f at 3, you get the negative of f(-3) when f is odd. If you enter any function in the f box of the Grapher and enter -f(-x) in the g box, then the graph of g is the reflection through the origin of the graph of f. So, if f is not odd, then you see two distinct graphs. If f is odd, you see only one graph. Graph f(x) = (x - 2)^2 and g(x) = -f(-x). Because you see two distinct graphs, f is not odd. Now enter f(x) = x*(x^2 - 1) and "turn off" the graph of g by unchecking the box to the right of the text box and click Graph again. With this function f, the graph of g is the same as the graph of f. So, when you turn on the graph of g and click Graph again, you see nothing new. There are four operations on functions: For the purposes of the following examples, I’ll use functions f(x) and g(x). You don’t have to use “f” and “g”. That notation is somewhat arbitrary. The functions could be represented by any letters The choice depends largely on the preference of a particular author or professor. For example: j(t), s(t) or h(t). You might also see time(t) instead of “x”, especially in economics and physics applications. INTRODUCTION TO GRAPHS

First, a coördinate . A coördinate is a number. It labels a point on a line.

The coördinates 0, 1, 2, 3, etc. label those points. They are their "addresses."

A coördinate axis is a line with coördinates.

To label the points in a plane, we will need more than one coördinate axis, and we place them at right angles. Hence, they are called rectangular coördinate axes . And the coördinates on them are called rectangular coördinates . They are also called Cartesian coördinates , after the 17th century philosopher and mathematician René Descartes for he was one of the first to exploit the geometrical possibilities of coördinates.

Finally, the rectangular coördinates of a point are an ordered pair , ( x , y ). (2, 3) labels a point different from (3, 2). The x -coördinate is always entered first the y -coördinate, second.

Example 1. The graph of a function .

In which graph are the values of y a function of the values of x ?

To see the answer, pass your mouse over the colored area.

To cover the answer again, click "Refresh" ("Reload").In graph a). To each value of x there is one and only one value of y . Any straight line parallel to the y -axis will cut that graph only once.

The coördinate pairs of a function

Consider the function y = x 2 . The variable y now signifies the y -coördinate, and x , the x -coördinate. Therefore, every coördinate pair on the graph of that function is

In other words, the graph of a function

is that geometrical figure such that every coördinate pair on it is

then every coördinate pair on its graph has the form

Problem 1. The coördinate pairs on the graph of the following functions have what form?

To see the answer, pass your mouse over the colored area.

To cover the answer again, click "Refresh" ("Reload").a) y = 1 &minus x ( x , 1 &minus x ) b) y = ax + b ( x , ax + b ) c) y = 3 ( x , 3) d) y = f ( x ) ( x , f ( x )) Problem 2. Let y = f ( x ). Write the second member of each coördinate pair.

a) (1, ?) (1, f (1)) b) (&minus1, ?) (&minus1, f (&minus1)) c) ( a , ?) ( a , f ( a )) d) ( t &minus 4, ?) ( t &minus 4, f ( t &minus 4)) Problem 3. Which of these points are on the graph of the function

a) (1, 4) No, because 4 is not equal to 2 · 1 2 . b) (&minus1, 2) Yes. c) ( a , 2 a 2 ) Yes. d) ( b /2, b 2 /2) Yes. a) The coördinate pair ( p , q ) is on the graph of the function

What is the relationship between the coördinates p , q ?

b) The coördinate pair ( r , s ) is on the graph of the function

What is the relationship between the coördinates r , s ?

Let y = f ( x ). Then the coördinates of every point on its graph are

( x , f ( x )). (Look at the first quadrant.)f ( x ) -- or y -- is the height of the graph at x . It is the length of that vertical line.

Now, if f (&minus x ) = f ( x ) -- that is, if the height of the graph at &minus x is equal to the height of the graph at x -- then that point is shown at a ).

While if f (&minus x ) = &minus f ( x ) -- if the height of the graph at &minus x is the negative of the height of the graph at x -- then that point is shown at b ).

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## Modules

*Data**Graph*- Data.Graph.AdjacencyList
- Data.Graph.AdjacencyMatrix
- Data.Graph.Algorithm
- Data.Graph.Algorithm.BreadthFirstSearch
- Data.Graph.Algorithm.DepthFirstSearch

- Data.Graph.Class.AdjacencyList
- Data.Graph.Class.AdjacencyMatrix
- Data.Graph.Class.Bidirectional
- Data.Graph.Class.EdgeEnumerable
- Data.Graph.Class.VertexEnumerable

Graph each function in a viewing window [-2, 2] by [-1, 6].

(i) Which point is common to all four graphs ?

(ii) Analyze the functions for domain, range, continuity, increasing or decreasing behavior, symmetry, extrema, asymptotes and end behaviour.

(i) Every graph is passing through the point (0, 1).

(ii) Analyzing the function :

Domain is the defined value of x. For this function, the domain is all real numbers.

Every exponential functions are defined and continuous for all real numbers.

**Increasing / decreasing :**Since the base is integer, the graph is increasing.

It is symmetric about none.

The graph is asymptotic to the x-axis as x approaches negative infinity

The graph increases without bound as x approaches positive infinity. So there is no extreama.

When x approaches x to ∞, f(x) = ∞

When x approaches x to - ∞, f(x) = 0

Graph each function in viewing windows [-2, 2] by [-1, 6]

(i) Which point is common to all four graphs ?

(ii) Analyze the functions for domain, range, continuity, increasing or decreasing behavior, symmetry, extrema, asymptotes and end behaviour.

(i) Every graph is passing through the point (0, 1).

(ii) Analyzing the function :

Domain is the defined value of x. For this function, the domain is all real numbers.

Every exponential functions are defined and continuous for all real numbers.

**Increasing / decreasing :**Since the base is integer, the graph is increasing.

It is symmetric about none.

The graph is asymptotic to the x-axis as x approaches negative infinity

The graph increases without bound as x approaches positive infinity. So there is no extreama.

When x approaches x to -∞, f(x) = ∞

When x approaches x to ∞, f(x) = 0

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## Exponential Functions and their Graphs - Concept

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

There are certain functions, such as exponential functions, that have many applications to the real world and have useful inverse functions.

**Graphing exponential functions**is used frequently, we often hear of situations that have exponential growth or exponential decay. The inverses of exponential functions are logarithmic functions. The graphs of exponential functions are used to analyze and interpret data.Exponential or power function are a new type of function. What they are are a function where instead of having x in the base of the problem where you say like x squared, x cubed and things like that. What we actually have is our variable moves to the exponent, moves to the top, okay? So a exponential power function is anything of the form a to the x. And there is a restriction on a and that it has to be greater than zero and it can't be 1. Reason it can't be 1 is if you have 1 to any power, it's just always going to remain 1. So it doesn't really make sense to have our base be 1 anyway.

Some language that goes along with this, okay? We call a the base and we call x the exponent, okay? So exponential power functions. Anything that is of the form a to the x with a being a positive number not equal to 1.

- [ Module 1 - Search and Rescue ] Section 4: Function Models