4.5: Atomic Structure
12
Harper College Chemistry Department
[latexpage]
Composition of an Atom
The development of the modern atomic theory revealed much about the inner structure of atoms. It was learned that an atom contains a very small nucleus composed of positively charged protons and uncharged neutrons, surrounded by a much larger volume of space containing negatively charged electrons. The nucleus contains the majority of an atom’s mass because protons and neutrons are much heavier than electrons, whereas electrons occupy almost all of an atom’s volume. The diameter of an atom is on the order of 10 −10 m, whereas the diameter of the nucleus is roughly 10 −15 m—about 100,000 times smaller. For a perspective about their relative sizes, consider this: If the nucleus were the size of a blueberry, the atom would be about the size of a football stadium ( Figure 1 ).
Atoms—and the protons, neutrons and electrons that compose them—are extremely small. For example, a carbon atom weighs less than 2 × 10 −23 g and an electron has a charge of less than 2 × 10 −19 C (coulomb). When describing the properties of tiny objects such as atoms, we use appropriately small units of measure, such as the atomic mass unit (amu) and the fundamental unit of charge (e). The amu was originally defined based on hydrogen, the lightest element, then later in terms of oxygen. Since 1961, it has been defined with regard to the most abundant isotope of carbon, atoms of which are assigned masses of exactly 12 amu. This isotope is known as “carbon-12” as will be discussed later in this module. Thus, one amu is exactly $\frac{1}{12}$ of the mass of one carbon-12 atom: 1 amu = 1.6605 × 10 −24 g. The fundamental unit of charge (also called the elementary charge) equals the magnitude of the charge of an electron (e) with e = 1.602 × 10 −19 C.
A proton has a mass of 1.0073 amu and a charge of 1+. A neutron is a slightly heavier particle with a mass 1.0087 amu and a charge of zero; as its name suggests, it is neutral. The electron has a charge of 1− and is a much lighter particle with a mass of about 0.00055 amu. It would take about 1800 electrons to equal the mass of one proton. The properties of these fundamental particles are summarized in Table 1 . An observant student might notice that the sum of an atom’s subatomic particles does not equal the atom’s actual mass: The total mass of six protons, six neutrons and six electrons is 12.0993 amu, slightly larger than 12.00 amu. This “missing” mass is known as the mass defect, and you will learn about it in the chapter on nuclear chemistry.
| Properties of Subatomic Particles | |||||
|---|---|---|---|---|---|
| Name | Location | Charge (C) | Unit Charge | Mass (amu) | Mass (g) |
| electron | outside nucleus | −1.602 × 10 −19 | 1− | 0.00055 | 0.00091 × 10 −24 |
| proton | nucleus | 1.602 × 10 −19 | 1+ | 1.00727 | 1.67262 × 10 −24 |
| neutron | nucleus | 0 | 0 | 1.00866 | 1.67493 × 10 −24 |
The number of protons in the nucleus of an atom is its atomic number (Z). This is the defining trait of an element: Its value determines the identity of the atom. For example, any atom that contains six protons is the element carbon and has the atomic number 6, regardless of how many neutrons or electrons it may have. A neutral atom must contain the same number of positive and negative charges, so the number of protons equals the number of electrons. The total number of protons and neutrons in an atom is called its mass number (A). The number of neutrons is therefore the difference between the mass number and the atomic number: A – Z = number of neutrons.
\[\begin{equation*} \begin{align} \text{atomic number (Z)} &= \text{number of protons} \\ \text{mass number (A)} &= \text{number of protons} + \text{number of neutrons} \\ \text{A} - \text{Z} &= \text{number of neutrons} \\ \end{align} \end{equation*}\]
Atomic charge = number of protons − number of electrons
As will be discussed in more detail later in this chapter, atoms (and molecules) typically acquire a charge by gaining or losing electrons. An atom that gains one or more electrons will exhibit a negative charge and is called an anion. Positively charged atoms called cations are formed when an atom loses one or more electrons. For example, a neutral sodium atom (Z = 11) has 11 protons and 11 electrons. If this atom loses one electron, it will become a cation with a 1+ charge (11 − 10 = 1+). A neutral oxygen atom (Z = 8) has eight protons and electrons, if it gains two electrons it will become an anion with a 2− charge (8 − 10 = 2−).
Iodine is an essential trace element in our diet; it is needed to produce thyroid hormone. Insufficient iodine in the diet can lead to the development of a goiter, an enlargement of the thyroid gland ( Figure 2 ).
The addition of small amounts of iodine to table salt (iodized salt) has essentially eliminated this health concern in the United States, but as much as 40% of the world’s population is still at risk of iodine deficiency. The iodine atoms are added as anions and each has a 1− charge and a mass number of 127.
Check Your Learning
Determine the numbers of protons, neutrons, and electrons in one of these iodine anions.
Answer
The atomic number of iodine (53) tells us that a neutral iodine atom contains 53 protons in its nucleus and 53 electrons outside its nucleus. Because the sum of the number of protons and neutrons equals the mass number (127), the number of neutrons is 74 (127 − 53 = 74). Since the iodine is added as a 1− anion, the number of electrons is 54 [53 – (1–) = 54].
Check Your Learning
An ion of platinum has a mass number of 195 and contains 74 electrons. How many protons and neutrons does it contain, and what is its charge?
78 protons; 117 neutrons; charge is 4+
Chemical Symbols
A chemical symbol is an abbreviation that we use to indicate an element or an atom of an element. For example, the symbol for mercury is Hg ( Figure 3 ). We use the same symbol to indicate one atom of mercury (microscopic domain) or to label a container of many atoms of the element mercury (macroscopic domain).
The symbols for several common elements and their atoms are listed in Table 2 . Some symbols are derived from the common name of the element; others come from the name in another language (usually Latin). Symbols have one or two letters. To avoid confusion with other notations, only the first letter of a symbol is capitalized. For example, Co is the symbol for the element cobalt, but CO is the formula for the compound carbon monoxide, which contains atoms of the elements carbon (C) and oxygen (O). All known elements and their symbols are in the periodic table.
| Some Common Elements and Their Symbols | |||
|---|---|---|---|
| Element | Symbol | Element | Symbol |
| aluminum | Al | iron | Fe (from ferrum ) |
| bromine | Br | lead | Pb (from plumbum ) |
| calcium | Ca | magnesium | Mg |
| carbon | C | mercury | Hg (from hydrargyrum ) |
| chlorine | Cl | nitrogen | N |
| chromium | Cr | oxygen | O |
| cobalt | Co | potassium | K (from kalium ) |
| copper | Cu (from cuprum ) | silicon | Si |
| fluorine | F | silver | Ag (from argentum ) |
| gold | Au (from aurum ) | sodium | Na (from natrium ) |
| helium | He | sulfur | S |
| hydrogen | H | tin | Sn (from stannum ) |
| iodine | I | zinc | Zn |
Traditionally, the discoverer(s) of a new element names the element. However, until the name is recognized by the International Union of Pure and Applied Chemistry (IUPAC), the recommended name of the new element is based on the Latin word(s) for its atomic number. For example, element 106 was called unnilhexium (Unh), element 107 was called unnilseptium (Uns), and element 108 was called unniloctium (Uno) for several years. These elements are now named after scientists (or occasionally locations); for example, element 106 is now known as seaborgium (Sg) in honor of Glenn Seaborg, a Nobel Prize winner who was active in the discovery of several heavy elements.
Visit the IUPAC site to learn more about the International Union of Pure and Applied Chemistry and explore its periodic table at the bottom of the web page.
Isotopes
The symbol for a specific isotope of any element is written by placing the mass number as a superscript to the left of the element symbol ( Figure 4 ). The atomic number is sometimes written as a subscript also on the left of the symbol. However, since this number defines the element’s identity, as does its symbol, it is often omitted. For example, magnesium exists as a mixture of three isotopes, each with an atomic number of 12 and with mass numbers of 24, 25 and 26, respectively. These isotopes can be identified as 24 Mg, 25 Mg and 26 Mg. These isotope symbols are read as “element, mass number” and can be symbolized consistent with this reading. For instance, 24 Mg is read as “magnesium 24,” and can be written as “magnesium-24” or “Mg-24.” 25 Mg is read as “magnesium 25,” and can be written as “magnesium-25” or “Mg-25.” All magnesium atoms have 12 protons in their nucleus. They differ only because a 24 Mg atom has 12 neutrons in its nucleus, a 25 Mg atom has 13 neutrons and a 26 Mg atom has 14 neutrons.
Information about the naturally occurring isotopes of elements with atomic numbers 1 through 10 is given in Table 3 . Note that in addition to standard names and symbols, the isotopes of hydrogen are often referred to using common names and accompanying symbols. Hydrogen-2, symbolized 2 H, is also called deuterium and sometimes symbolized D. Hydrogen-3, symbolized 3 H, is also called tritium and sometimes symbolized T.
| Nuclear Compositions of Atoms of the Very Light Elements | ||||||
|---|---|---|---|---|---|---|
| Element | Symbol | Atomic Number | Number of Protons | Number of Neutrons | Mass (amu) | % Natural Abundance |
| hydrogen | $_{1}^{1}\text{H}$ (protium) | 1 | 1 | 0 | 1.0078 | 99.989 |
| $_{1}^{2}\text{H}$ (deuterium) | 1 | 1 | 1 | 2.0141 | 0.0115 | |
| $_{1}^{3}\text{H}$ (tritium) | 1 | 1 | 2 | 3.01605 | (trace amounts) | |
| helium | $_{2}^{3}\text{He}$ | 2 | 2 | 1 | 3.01603 | 0.00013 |
| $_{2}^{4}\text{He}$ | 2 | 2 | 2 | 4.0026 | 100 | |
| lithium | $_{3}^{6}\text{Li}$ | 3 | 3 | 3 | 6.0151 | 7.59 |
| $_{3}^{7}\text{Li}$ | 3 | 3 | 4 | 7.0160 | 92.41 | |
| beryllium | $_{4}^{9}\text{Be}$ | 4 | 4 | 5 | 9.0122 | 100 |
| boron | $^{10}_{ 5}\text{B}$ | 5 | 5 | 5 | 10.0129 | 19.9 |
| $_{ 5}^{11}\text{B}$ | 5 | 5 | 6 | 11.0093 | 80.1 | |
| carbon | $_{6}^{12}\text{C}$ | 6 | 6 | 6 | 12.0000 | 98.89 |
| $_{6}^{13}\text{C}$ | 6 | 6 | 7 | 13.0034 | 1.11 | |
| $_{6}^{14}\text{C}$ | 6 | 6 | 8 | 14.0032 | (trace amounts) | |
| nitrogen | $_{7}^{14}\text{N}$ | 7 | 7 | 7 | 14.0031 | 99.63 |
| $_{7}^{15}\text{N}$ | 7 | 7 | 8 | 15.0001 | 0.37 | |
| oxygen | $_{8}^{16}\text{O}$ | 8 | 8 | 8 | 15.9949 | 99.757 |
| $_{8}^{17}\text{O}$ | 8 | 8 | 9 | 16.9991 | 0.038 | |
| $_{8}^{18}\text{O}$ | 8 | 8 | 10 | 17.9992 | 0.205 | |
| fluorine | $_{9}^{19}\text{F}$ | 9 | 9 | 10 | 18.9984 | 100 |
| neon | $_{10}^{20}\text{Ne}$ | 10 | 10 | 10 | 19.9924 | 90.48 |
| $_{10}^{21}\text{Ne}$ | 10 | 10 | 11 | 20.9938 | 0.27 | |
| $_{10}^{22}\text{Ne}$ | 10 | 10 | 12 | 21.9914 | 9.25 |
Use this Build an Atom simulator to build atoms of the first 10 elements, see which isotopes exist, check nuclear stability and gain experience with isotope symbols.
Atomic Mass
The atomic mass of a single atom is approximately equal to its mass number (a whole number) due to the mass of each proton and each neutron contributing approximately one amu each to an atom, while each electron contributes far less. However, the average masses of atoms of most elements are not whole numbers because most elements exist naturally as mixtures of two or more isotopes.
The mass of an element shown in a periodic table or listed in a table of atomic masses is a weighted-average mass of all the isotopes present in a naturally occurring sample of that element. This is equal to the sum of each individual isotope’s mass multiplied by its fractional abundance.
In case you are wondering
MASS SPECTROMETRYChemistry End of Chapter Exercises
1. In what way are isotopes of a given element always different? In what way(s) are they always the same?
2. Write the symbol for each of the following ions:
(a) the ion with a 1+ charge, atomic number 55, and mass number 133
(b) the ion with 54 electrons, 53 protons, and 74 neutrons
(c) the ion with atomic number 15, mass number 31, and a 3− charge
(d) the ion with 24 electrons, 30 neutrons, and a 3+ charge
3. Write the symbol for each of the following ions:
(a) the ion with a 3+ charge, 28 electrons, and a mass number of 71
(b) the ion with 36 electrons, 35 protons, and 45 neutrons
(c) the ion with 86 electrons, 142 neutrons, and a 4+ charge
(d) the ion with a 2+ charge, atomic number 38, and mass number 87
4. Open the Build an Atom simulation and click on the Atom icon.
(a) Pick any one of the first 10 elements that you would like to build and state its symbol.
(b) Drag protons, neutrons, and electrons onto the atom template to make an atom of your element.
(c) State the numbers of protons, neutrons, and electrons in your atom, as well as the net charge and mass number.
(d) Click on “Net Charge” and “Mass Number,” check your answers to (c), and correct, if needed.
(e) Predict whether your atom will be stable or unstable. State your reasoning.
(f) Check the “Stable/Unstable” box. Was your answer to (e) correct? If not, first predict what you can do to make a stable atom of your element, and then do it and see if it works. Explain your reasoning.
5. Open the Build an Atom simulation
(a) Drag protons, neutrons, and electrons onto the atom template to make a neutral atom of Oxygen-16 and give the isotope symbol for this atom.
(b) Now add two more electrons to make an ion and give the symbol for the ion you have created.
6. Open the Build an Atom simulation
(a) Drag protons, neutrons, and electrons onto the atom template to make a neutral atom of Lithium-6 and give the isotope symbol for this atom.
(b) Now remove one electron to make an ion and give the symbol for the ion you have created.
7. Determine the number of protons, neutrons, and electrons in the following isotopes that are used in medical diagnoses:
(a) atomic number 9, mass number 18, charge of 1−
(b) atomic number 43, mass number 99, charge of 7+
(c) atomic number 53, atomic mass number 131, charge of 1−
(d) atomic number 81, atomic mass number 201, charge of 1+
(e) Name the elements in parts (a), (b), (c), and (d).
8. The following are properties of isotopes of two elements that are essential in our diet. Determine the number of protons, neutrons and electrons in each and name them.
(a) atomic number 26, mass number 58, charge of 2+
(b) atomic number 53, mass number 127, charge of 1−
9. Give the number of protons, electrons, and neutrons in neutral atoms of each of the following isotopes:
(a) $^{10}_{ 5}\text{B}$
(b) $^{199}_{80}\text{Hg}$
(c) $^{63}_{29}\text{Cu}$
(d) $^{13}_{6}\text{C}$
(e) $^{77}_{34}\text{Se}$
10. Give the number of protons, electrons, and neutrons in neutral atoms of each of the following isotopes:
(a) $^{7}_{3}\text{Li}$
(b) $^{125}_{52}\text{Te}$
(c) $^{109}_{74}\text{Ag}$
(d) $^{15}_{7}\text{N}$
(e) $^{31}_{15}\text{P}$
Glossary
- anion
- negatively charged atom or molecule (contains more electrons than protons)
- atomic mass
- average mass of atoms of an element, expressed in amu
- atomic mass unit (amu)
- (also, unified atomic mass unit, u, or Dalton, Da) unit of mass equal to 1 12 112 of the mass of a 12 C atom
- atomic number (Z)
- number of protons in the nucleus of an atom
- cation
- positively charged atom or molecule (contains fewer electrons than protons)
- chemical symbol
- one-, two-, or three-letter abbreviation used to represent an element or its atoms
- Dalton (Da)
- alternative unit equivalent to the atomic mass unit
- fundamental unit of charge
- (also called the elementary charge) equals the magnitude of the charge of an electron (e) with e = 1.602 × 10 −19 C
- ion
- electrically charged atom or molecule (contains unequal numbers of protons and electrons)
- isotope
- same number of protons and different number of neutrons
mass number (A)
sum of the numbers of neutrons and protons in the nucleus of an atomunified atomic mass unit (u)
alternative unit equivalent to the atomic mass unit


14.12: The Four Basic Forces
Learning Objectives
- State the four basic forces.
- Explain the Feynman diagram for the exchange of a virtual photon between two positive charges.
- Define QED.
- Describe the Feynman diagram for the exchange of a between a proton and a neutron.
As first discussed in Problem-Solving Strategies and mentioned at various points in the text since then, there are only four distinct basic forces in all of nature. This is a remarkably small number considering the myriad phenomena they explain. Particle physics is intimately tied to these four forces. Certain fundamental particles, called carrier particles, carry these forces, and all particles can be classified according to which of the four forces they feel. The table given below summarizes important characteristics of the four basic forces.
| Force | Approximate relative strength | Range | +/− 1 | Carrier particle |
|---|---|---|---|---|
| Gravity |
\(10^{-38}\) |
\(\infty\) |
+ only | Graviton (conjectured) |
| Electromagnetic |
\(10^{-2}\) |
\(\infty\) |
\(+/-\) |
Photon (observed) |
| Weak force |
\(10^{-13}\) |
\(<10^{-18} \mathrm{~m}\) |
\(+/-\) |
\(W^{+}, W^{-}, Z^{0}\) (observed 2 ) |
| Strong force |
\(1\) |
\(<10^{-15} \mathrm{~m}\) |
\(+/-\) |
Gluons (conjectured 3 ) |
Although these four forces are distinct and differ greatly from one another under all but the most extreme circumstances, we can see similarities among them. (In GUTs: the Unification of Forces , we will discuss how the four forces may be different manifestations of a single unified force.) Perhaps the most important characteristic among the forces is that they are all transmitted by the exchange of a carrier particle, exactly like what Yukawa had in mind for the strong nuclear force. Each carrier particle is a virtual particle—it cannot be directly observed while transmitting the force. Figure \(\PageIndex{1}\) shows the exchange of a virtual photon between two positive charges. The photon cannot be directly observed in its passage, because this would disrupt it and alter the force.
Figure \(\PageIndex{1}\) shows a way of graphing the exchange of a virtual photon between two positive charges. This graph of time versus position is called a Feynman diagram , after the brilliant American physicist Richard Feynman (1918–1988) who developed it.
Figure \(\PageIndex{3}\) is a Feynman diagram for the exchange of a virtual pion between a proton and a neutron representing the same interaction as in The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited . Feynman diagrams are not only a useful tool for visualizing interactions at the quantum mechanical level, they are also used to calculate details of interactions, such as their strengths and probability of occurring. Feynman was one of the theorists who developed the field of quantum electrodynamics (QED), which is the quantum mechanics of electromagnetism. QED has been spectacularly successful in describing electromagnetic interactions on the submicroscopic scale. Feynman was an inspiring teacher, had a colorful personality, and made a profound impact on generations of physicists. He shared the 1965 Nobel Prize with Julian Schwinger and S. I. Tomonaga for work in QED with its deep implications for particle physics.
Why is it that particles called gluons are listed as the carrier particles for the strong nuclear force when, in The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited , we saw that pions apparently carry that force? The answer is that pions are exchanged but they have a substructure and, as we explore it, we find that the strong force is actually related to the indirectly observed but more fundamental gluons . In fact, all the carrier particles are thought to be fundamental in the sense that they have no substructure. Another similarity among carrier particles is that they are all bosons (first mentioned in Patterns in Spectra Reveal More Quantization ), having integral intrinsic spins.
There is a relationship between the mass of the carrier particle and the range of the force. The photon is massless and has energy. So, the existence of (virtual) photons is possible only by virtue of the Heisenberg uncertainty principle and can travel an unlimited distance. Thus, the range of the electromagnetic force is infinite. This is also true for gravity. It is infinite in range because its carrier particle, the graviton, has zero rest mass. (Gravity is the most difficult of the four forces to understand on a quantum scale because it affects the space and time in which the others act. But gravity is so weak that its effects are extremely difficult to observe quantum mechanically. We shall explore it further in General Relativity and Quantum Gravity ). The \(W^{+}, W^{-}\), and \(Z^{0}\) particles that carry the weak nuclear force have mass, accounting for the very short range of this force. In fact, the \(W^{+}, W^{-}\), and \(Z^{0}\) are about 1000 times more massive than pions, consistent with the fact that the range of the weak nuclear force is about 1/1000 that of the strong nuclear force. Gluons are actually massless, but since they act inside massive carrier particles like pions, the strong nuclear force is also short ranged.
The relative strengths of the forces given in the Table \(\PageIndex{1}\) are those for the most common situations. When particles are brought very close together, the relative strengths change, and they may become identical at extremely close range. As we shall see in GUTs: the Unification of Forces , carrier particles may be altered by the energy required to bring particles very close together—in such a manner that they become identical.
Summary
- The four basic forces and their carrier particles are summarized in the Table \(\PageIndex{1}\) .
- Feynman diagrams are graphs of time versus position and are highly useful pictorial representations of particle processes.
- The theory of electromagnetism on the particle scale is called quantum electrodynamics (QED).
Footnotes
Glossary
- Feynman diagram
- a graph of time versus position that describes the exchange of virtual particles between subatomic particles
- gluons
- exchange particles, analogous to the exchange of photons that gives rise to the electromagnetic force between two charged particles
- quantum electrodynamics
- the theory of electromagnetism on the particle scale


1.2: Art of the Past and the Origins of Creativity
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A common saying is, 'history always repeats itself'. Art history and art movements also repeat; artists are influenced by the past and present; the Romans copied the Greeks, and earlier styles inspired art movements. The early Cycladic (1.2) carved simple forms with static arms and legs, David (1.3) by Michelangelo uses contrapposto positioning and realistic details, and the yellow modern art sculpture (1.4) is free form yet based upon a human figure and the interpretation in different millennia, referred to as 'artistic' recurrence.
Artists get their ideas from many places, have you ever wondered where? Why are they creating art, and what is the driving force behind their creation? Is it political, sacred, dreamscape, ceremonial, cultural, expression, therapy, illustrative, historical, literature, poetry, musical, theatrical, nature, narrative, exposing, thought-provoking, or experimental, whatever it is, it must come from within the artist. The process is part of the journey, and the journey is the process. If a mistake happens, paint over it. There is no such thing as a mistake in art; it is just not finished. Art is only complete when the artist believes it is finished!
Over 4 million years, humans evolved into the unique Homo Sapiens—modern human brains have increased in size leading to creative and organized groups of prehistoric people. The discovery of fire was perhaps one of the greatest achievements of the Stone Age, and scholarly evidence supports the practice of fire around 400,000 years ago, although some scholars believe it might have been much earlier. The human brain divided between the humanities and science began to evolve as complementary divisions eons ago. Fire is at the heart of our creativity.
Everyone loves a campfire; it brings people together when the skies darken, and long nights ensue. Research indicates that adding protein to a diet allowed the brain to increase in size. Protein, an essential element for the evolution of the human brain, fire likewise allowed the cooking of protein, eliminating the restriction of raw foods. This increase of protein in the prehistoric diet lead to the development of a larger front lobe brain creating more empathy, problem-solving, and social intelligence. Did the stories told around the chiaroscuro firelight, with singing, dancing, and chanting, lead to drawing on the walls of caves or rock outcrops, recording their epic hunts or daily lives? The unification of small nomadic groups of people required rules and regulations which may have begun around the campfire.
According to carbon dating methods, the earliest known artwork is 30,000 years old (Venus of Willendorf 1.7) and associated with those who created cave art. Previously, anthropologists considered Mesopotamia as the "Cradle of Civilization" home to the oldest culture in the world. Cultural anthropologists have defined the Cradle of Civilization to signify the period of written language, agriculture, raising animals, and public buildings. This list was used by most cultures and now defines Ancient Egypt along the Nile, Mesopotamia, the Indus Valley, Ancient China in the Yellow River region, the Central Andes and Mesoamerica as centers where different civilizations started independently.
Creativity continued to grow among cultures, and depending on the natural resources, grand palaces, tombs, and structures rose out of the earth. Art appreciation usually focuses on Egypt, Greek, Roman, Renaissance, and modern art. This textbook covers cultures well beyond classical art and delves into the numerous societies of people and their culture from the six continents. Exploring art outside the norm, understand the origins of creativity, and how it connects the art of the past in all cultures across the world is critical in art appreciation.


3.29: Linear Regression (2 of 4)
Learning Objectives
- For a linear relationship, use the least squares regression line to model the pattern in the data and to make predictions.
We continue our discussion of linear relationships with a focus on how to find the best line to summarize a linear pattern in data. Specifically, we do the following:
- Develop a measurement for identifying the best line to summarize the data.
- Use technology to find the best line.
Let’s begin with a simple data set with only five data points.
Which line appears to be a better summary of the linear pattern in the data?
Let’s make some observations about how these lines relate to the data points.
The line on the left passes through two of the five points. The point (12, 7) is very close to the line. The points (8, 13) and (17, 14) are relatively far from the line.
The line on the right does not pass through any of the points. It appears to pass through the middle of the distribution of the data. The points (8, 13) and (17, 14) are closer to this line than to the line on the left. But the other data points are farther from this line.
Which line is the best summary of the positive linear association we see in the data? Well, we may not agree on this, so we need a measurement of “best fit.”
Here’s the basic idea: The closer the line is to all of the data points, the better the line summarizes the pattern in the data. Notice when the line is close to the data points, it gives better predictions. A good prediction means the predicted y -value from the line is close to the actual y -value for the data point.
Here are the scatterplots again. For each data point, we drew a vertical line segment from the point to the summary line. The length of each vertical line segment is the amount that the predicted y -value deviates from the actual y -value for that data point. We think of this as the error in the prediction . We want to adjust the line until the overall error for all points together is as small as possible.
The most common measurement of overall error is the sum of the squares of the errors, or SSE (sum of squared errors) . The line with the smallest SSE is called the least-squares regression line . We call this line the “line of best fit.”
Here are the scatterplots again. As before, each vertical line represents the error in a prediction. For each data point, the squared error is equal to the area of a yellow square. The least-squares regression line is the line with the smallest SSE, which means it has the smallest total yellow area.
Using the least-squares measurement, the line on the right is the better fit. It has a smaller sum of squared errors. When we compare the sum of the areas of the yellow squares, the line on the left has an SSE of 57.8. The line on the right has a smaller SSE of 43.9.
But is the line on the right the best fit? The answer is no. The line of best fit is the line that has the smallest sum of squared errors (SSE). For this data set, the line with the smallest SSE is y = 6.72 +0.26x. The SSE is 41.79.
Now you try it with a new data set. Use the following simulation to adjust the line. See if you can find the least-squares regression line. (Try to find the line that makes the SSE as small as possible.)
Click here to open this simulation in its own window.
Contributors and Attributions
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution


9.9: Summary
Summary: Utilizing Library Resources for Academic Success
Chapter Takeaways
Information Literacy Understanding how to locate, evaluate, and effectively use information is crucial for academic success. Information literacy involves recognizing when information is needed and having the ability to find, evaluate, and use that information effectively. This skill set is essential for conducting research and completing assignments accurately.
Library Resources College libraries offer a wealth of resources beyond just books. These include journals, electronic books, multimedia resources, and access to various databases. Librarians are valuable allies in navigating these resources and can provide guidance on finding the materials you need for your coursework and research.
Databases Databases are searchable collections of information, often containing peer-reviewed journal articles, which are essential for academic research. Learning how to use databases effectively will save you time and ensure you are accessing high-quality, reliable sources. Familiarize yourself with databases relevant to your field of study and utilize the library's online guides and tutorials.
Academic Integrity Maintaining academic integrity is fundamental in your academic journey. This means avoiding plagiarism, understanding the importance of original work, and giving proper credit to the sources you use. It’s essential to understand your institution’s policies on academic honesty and the consequences of violating them.
Citing Sources Proper citation is a cornerstone of academic integrity. Knowing how to cite sources correctly in various formats (APA, MLA, Chicago, etc.) helps you avoid plagiarism and lends credibility to your work. Most libraries offer guides and tools to help with citing sources accurately.
Artificial Intelligence The role of artificial intelligence (AI) in research is growing. AI can help in data analysis, finding patterns in research data, and even in generating new hypotheses. However, it's important to understand the ethical implications of using AI in research, ensuring transparency and fairness in its application.
Research Studies Conducting and understanding research studies is a critical skill. Whether you are reading studies or conducting your own, it is important to understand research methodologies, data analysis, and how to critically evaluate the findings. Libraries often provide resources and workshops on how to conduct effective research.
In summary, mastering the use of library resources and understanding the principles of information literacy, academic integrity, and the proper use of artificial intelligence are essential for college success. Utilizing these tools effectively will enhance your academic performance and prepare you for future scholarly endeavors.
Career Connection
Using the library as a nursing student is an essential part of developing strong information literacy skills, which are crucial for academic and professional success. Libraries offer access to a wide range of resources, including textbooks, journals, and databases that provide reliable and up-to-date information. Understanding how to navigate these resources, especially databases like CINAHL (Cumulative Index to Nursing and Allied Health Literature), Gale Health and Wellness, and Medline. is fundamental. These databases are specifically tailored for nursing and healthcare disciplines, offering peer-reviewed articles, clinical studies, and evidence-based practice guidelines. By mastering these tools, nursing students can efficiently find relevant research, supporting their coursework and clinical practice with the best available evidence.
Academic integrity is another vital aspect of using the library effectively. Nursing students must learn to properly cite sources and avoid plagiarism, ensuring their work maintains ethical standards. The library provides guidance on these practices, often through workshops or tutorials. Additionally, engaging with research studies in medicine helps students stay current with advancements in the field, fostering a deeper understanding of complex medical issues and improving patient care. By critically appraising and synthesizing research findings, nursing students can contribute to evidence-based practice and continuous improvement in healthcare outcomes. Utilizing the library not only enhances academic performance but also prepares students for the rigorous demands of the nursing profession.
Rethinking
After this chapter, how comfortable are you with:
- Understanding Information Literacy and why it matters?
- Determining how and when to use library resources?
- Learning to craft a research question using keywords and search strategies?
- Conduct appropriate research?
- Building citations and understand how not to plagiarize your sources?
- Using Artifical Intelligence ethically and appropriately?


11.3: Parabolas
By the end of this section, you will be able to:
- Graph vertical parabolas
- Graph horizontal parabolas
- Solve applications with parabolas
Before you get started, take this readiness quiz.
-
Graph: \(y=-3 x^{2}+12 x-12\).
If you missed this problem, review Example 9.47. -
Solve by completing the square: \(x^{2}-6 x+6=0\).
If you missed this problem, review Example 9.12. -
Write in standard form: \(y=3 x^{2}-6 x+5\).
If you missed this problem, review Example 9.59.
Graph Vertical Parabolas
The next conic section we will look at is a parabola . We define a parabola as all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.
A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.
Previously, we learned to graph vertical parabolas from the general form or the standard form using properties. Those methods will also work here. We will summarize the properties here.
Vertical Parabolas
|
General form \(y=a x^{2}+b x+c\) |
Standard Form \(y=a(x-h)^{2}+k\) |
|
|---|---|---|
| Orientation | \(a>0\) up; \(a<0\) down | \(a>0\) up; \(a<0\) down |
| Axis of Symmetry | \(x=-\dfrac{b}{2 a}\) | \(x=h\) |
| Vertex |
Substitute \(x=-\dfrac{b}{2 a}\) and
solve for \(y .\) |
\((h, k)\) |
| \(y\)-intercept | Let \(x=0\) | Let \(x=0\) |
| \(x\)-intercepts | Let \(y=0\) | Let \(y=0\) |
The graphs show what the parabolas look like when they open up or down. Their position in relation to the \(x\)- or \(y\)-axis is merely an example.
To graph a parabola from these forms, we used the following steps.
How to Graph Vertical Parabolas \(y=a x^{2}+b x+c\) or \(f(x)=a(x-h)^{2}+k\) using Properties.
- Step 1: Determine whether the parabola opens upward or downward.
- Step 2. Find the axis of symmetry.
- Step 3. Find the vertex.
- Step 4. Find the \(y\)-intercept. Find the point symmetric to the \(y\)-intercept across the axis of symmetry.
- Step 5. Find the \(x\)-intercepts.
- Step 6. Graph the parabola.
The next example reviews the method of graphing a parabola from the general form of its equation.
Graph \(y=-x^{2}+6 x-8\) by using properties.
Solution :
| \( \begin{align*} \color{red}{y} &\color{red}{=} a x^{2}+b x+c \\[4pt] \color{black}{y} &=-x^{2}+6 x-8 \end{align*}\) | |
| Since \(a\) is \(-1\), the parabola opens downward. | |
| To find the axis of symmetry, find \(x=-\dfrac{b}{2 a}\). | \( \begin{align*} x &=-\dfrac{b}{2 a}\\[4pt] x &=-\dfrac{6}{2(-1)} \\[4pt] x &= 3 \end{align*}\) |
| The axis of symmetry is \(x=3\). | |
| The vertex is on the line \(x=3\). | \(y=-x^{2}+6 x-8\) |
| Let \(x=3\). | |
| \(\begin{align*} y &=-9+18-8 \\[4pt] y &=1 \end{align*}\) | |
| The vertex is \((3,1)\). | |
| The \(y\)-intercept occurs when \(x=0\). | \(y=-x^{2}+6 x-8\) |
| Substitute \(x=0\). | \(y=-\color{red}{0}^{\color{black}{2}}+6 \cdot \color{red}{0} \color{black}{-} 8\) |
| Simplify. | \(y=-8\) |
| The \(y\)-intercept is \((0,-8)\). | |
| The point \((0,−8)\) is three units to the left of the line of symmetry. The point three units to the right of the line of symmetry is \((6,−8)\). | Point symmetric to the \(y\)-intercept is \((6,−8)\). |
| The \(x\)-intercept occurs when \(y=0\). | \(y=-x^{2}+6 x-8\) |
| Let \(y=0\). | \(\color{red}{0} \color{black}{=}-x^{2}+6 x-8\) |
| Factor the GCF. | \(0=-\left(x^{2}-6 x+8\right)\) |
| Factor the trinomial. | \(0=-(x-4)(x-2)\) |
| Solve for \(x\). | \(x=4, \quad x=2\) |
| The \(x\)-intercepts are \((4,0),(2,0)\). | |
| Graph the parabola. |
Graph \(y=-x^{2}+5 x-6\) by using properties.
- Answer
Graph \(y=-x^{2}+8 x-12\) by using properties.
- Answer
The next example reviews the method of graphing a parabola from the standard form of its equation, \(y=a(x-h)^{2}+k\).
Write \(y=3 x^{2}-6 x+5\) in standard form and then use properties of standard form to graph the equation.
Solution :
| Rewrite the function in \(y=a(x-h)^{2}+k\) form by completing the square. | \(\begin{align*} y &=3 x^{2}-6 x+5 \\[4pt] y &=3\left(x^{2}-2 x\right)+5 \\[4pt] y &=3\left(x^{2}-2 x+1\right) + 5-3 \\[4pt] y &=3(x-1)^{2}+2 \end{align*}\) |
| Identify the constants \(a, h, k\). | \(a=3, h=1, k=2\) |
| Since \(a=2\), the parabola opens upward. | |
| The axis of symmetry is \(x=h\). | The axis of symmetry is \(x=1\). |
| The vertex is \((h,k)\). | The vertex is \((1,2)\). |
| Find the \(y\)-intercept by substituting \(x=0\), | \( \begin{align*} y &=3(x-1)^{2}+2 \\[4pt] y &=3 \cdot 0^{2}-6 \cdot 0+5 \\[4pt] y &=0 \end{align*} \) |
| \(y\)-intercept \((0,5)\) | |
| Find the point symmetric to \((0,5)\) across the axis of symmetry. | \((2,5)\) |
| Find the \(x\)-intercepts. | \(\begin{aligned} y &=3(x-1)^{2}+2 \\[4pt] 0 &=3(x-1)^{2}+2 \\[4pt] -2 &=3(x-1)^{2} \\[4pt] -\dfrac{2}{3} &=(x-1)^{2} \\[4pt] \pm \sqrt{-\dfrac{2}{3}} &=x-1 \end{aligned}\) |
| The square root of a negative number tells us the solutions are complex numbers. So there are no \(x\)-intercepts. | |
| Graph the parabola. |
- Write \(y=2 x^{2}+4 x+5\) in standard form and
- use properties of standard form to graph the equation.
- Answer
-
- \(y=2(x+1)^{2}+3\)
- Write \(y=-2 x^{2}+8 x-7\) in standard form and
- use properties of standard form to graph the equation.
- Answer
-
- \(y=-2(x-2)^{2}+1\)
Graph Horizontal Parabolas
Our work so far has only dealt with parabolas that open up or down. We are now going to look at horizontal parabolas. These parabolas open either to the left or to the right. If we interchange the \(x\) and \(y\) in our previous equations for parabolas, we get the equations for the parabolas that open to the left or to the right.
Horizontal Parabolas
|
General form \(x=a y^{2}+b y+c\) |
Standard form \(x=a(y-k)^{2}+h\) |
|
|---|---|---|
| Orientation | \(a>0\) right; \(a<0\) left | \(a>0\) right; \(a<0\) left |
| Axis of Symmetry | \(y=-\dfrac{b}{2 a}\) | \(y=k\) |
| Vertex |
Substitute \(y=-\dfrac{b}{2 a}\) and
solve for \(x .\) |
\((h, k)\) |
| \(x\)-intercepts | Let \(x=0\) | Let \(x=0\) |
| \(y\)-intercept | Let \(y=0\) | Let \(y=0\) |
The graphs show what the parabolas look like when they to the left or to the right. Their position in relation to the \(x\)- or \(y\)-axis is merely an example.
Looking at these parabolas, do their graphs represent a function? Since both graphs would fail the vertical line test, they do not represent a function.
To graph a parabola that opens to the left or to the right is basically the same as what we did for parabolas that open up or down, with the reversal of the \(x\) and \(y\) variables.
- Step 1: Determine whether the parabola opens to the left or to the right.
- Step 2: Find the axis of symmetry.
- Step 3: Find the vertex.
- Step 4: Find the \(x\)-intercept. Find the point symmetric to the \(x\)-intercept across the axis of symmetry.
- Step 5: Find the \(y\)-intercepts.
- Step 6: Graph the parabola.
Graph \(x=2 y^{2}\) by using properties.
Solution :
| Since \(a=2\), the parabola opens to the right. | |
| To find the axis of symmetry, find \(y=-\dfrac{b}{2 a}\) | \(y=-\dfrac{b}{2 a}\) |
| \(y=-\dfrac{0}{2(2)}\) | |
| \(y=0\) | |
| The axis of symmetry is \(y=0\). | |
| The vertex is on the line \(y=0\). | \(x=2 y^{2}\) |
| Let \(y=0\). | |
| \(x=0\) | |
| The vertex is \((0,0)\). |
Since the vertex is \((0,0)\), both the \(x\)- and \(y\)-intercepts are the point \((0,0)\). To graph the parabola we need more points. In this case it is easiest to choose values of \(y\).
We also plot the points symmetric to \((2,1)\) and \((8,2)\) across the \(y\)-axis, the points \((2,−1),(8,−2)\).
Graph the parabola.
Graph \(x=y^{2}\) by using properties.
- Answer
Graph \(x=-y^{2}\) by using properties.
- Answer
In the next example, the vertex is not the origin.
Graph \(x=-y^{2}+2 y+8\) by using properties.
Solution :
| Since \(a=-1\), the parabola opens to the left. | |
|
To find the axis of symmetry,
find \(y=-\dfrac{b}{2 a}\) |
\(y=-\dfrac{b}{2 a}\) |
| \(y=-\dfrac{2}{2(-1)}\) | |
| \(y=1\) | |
| The axis of symmetry is \(y=1\). | |
| The vertex is on the line \(y=1\). | \(x=-y^{2}+2 y+8\) |
| Let \(y=1\). | |
| \(x=9\) | |
| The vertex is \((9,1)\). | |
| The \(x\)-intercept occurs when \(y=0\). | \(x=-y^{2}+2 y+8\) |
| \(x=8\) | |
| The \(x\)-intercept is \((8,0)\). | |
|
The point \((8,0)\) is one unit below the line of
symmetry. The symmetric point one unit above the line of symmetry is \((8,2)\) |
Symmetric point is \((8,2)\). |
| The \(y\)-intercept occurs when \(x=0\). | \(x=-y^{2}+2 y+8\) |
| Substitute \(x=0\). | \(0=-y^{2}+2 y+8\) |
| Solve. | \(y^{2}-2 y-8=0\) |
| \((y-4)(y+2)=0\) | |
| \(y=4, \quad y=-2\) | |
| The \(y\) -intercepts are \((0,4)\) and \((0,-2)\). | |
| Connect the points to graph the parabola. |
Graph \(x=-y^{2}-4 y+12\) by using properties.
- Answer
Graph \(x=-y^{2}+2 y-3\) by using properties.
- Answer
In Table 11.2.4, we see the relationship between the equation in standard form and the properties of the parabola. The How To box lists the steps for graphing a parabola in the standard form \(x=a(y-k)^{2}+h\). We will use this procedure in the next example.
Graph \(x=2(y-2)^{2}+1\) using properties.
Solution :
| Identify the constants \(a, h, k\). | \(a=2, h=1, k=2\) |
| Since \(a=2\), the parabola opens to the right. | |
| The axis of symmetry is \(y=k\). | The axis of symmetry is \(y=2\). |
| The vertex is \((h,k)\). | The vertex is \((1,2)\). |
| Find the \(x\)-intercept by substituting \(y=0\). |
\(x=2(y-2)^{2}+1\)
\(x=2(0-2)^{2}+1\) \(x=9\) |
| The \(x\)-intercept is \((9,0)\). | |
| Find the point symmetric to \((9,0)\) across the axis of symmetry. | \((9,4)\) |
| Find the \(y\)-intercepts. Let \(x=0\). | \(\begin{aligned} x &=2(y-2)^{2}+1 \\ 0 &=2(y-2)^{2}+1 \\-1 &=2(y-2)^{2} \end{aligned}\) |
| A square cannot be negative, so there is no real solution. So there are no \(y\)-intercepts. | |
| Graph the parabola. |
Graph \(x=3(y-1)^{2}+2\) using properties.
- Answer
Graph \(x=2(y-3)^{2}+2\) using properties.
- Answer
In the next example, we notice the a is negative and so the parabola opens to the left.
Graph \(x=-4(y+1)^{2}+4\) using properties.
Solution :
| Identify the constants \(a, h, k\). | \(a=-4, h=4, k=-1\) |
| Since \(a=-4\), the parabola opens to the left. | |
| The axis of symmetry is \(y=k\). | The axis of symmetry is \(y=-1\). |
| The vertex is \((h,k)\). | The vertex is \((4,-1)\). |
| Find the \(x\)-intercept by substituting \(y=0\). |
\(x=-4(y+1)^{2}+4\)
\(x=-4(0+1)^{2}+4\) \(x=0\) |
| The \(x\)-intercept is \((0,0)\). | |
| Find the point symmetric to \((0,0)\) across the axis of symmetry. | \((0,-2)\) |
| Find the \(y\)-intercepts. | \(x=-4(y+1)^{2}+4\) |
| Let \(x=0\). | \(\begin{aligned} 0 &=-4(y+1)^{2}+4 \\-4 &=-4(y+1)^{2} \\ 1 &=(y+1)^{2} \\ y+1 &=\pm 1 \end{aligned}\) |
| \(y=-1+1 \quad y=-1-1\) | |
| \(y=0 \quad\quad y=-2\) | |
| The \(y\)-intercepts are \((0,0)\) and \((0,-2)\). | |
| Graph the parabola. |
Graph \(x=-4(y+2)^{2}+4\) using properties.
- Answer
Graph \(x=-2(y+3)^{2}+2\) using properties.
- Answer
The next example requires that we first put the equation in standard form and then use the properties.
Write \(x=2 y^{2}+12 y+17\) in standard form and then use the properties of the standard form to graph the equation.
Solution :
| \(x=2 y^{2}+12 y+17\) | |
| Rewrite the function in \(x=a(y-k)^{2}+h\) form by completing the square. | \(x=2\left(y^{2}+6 y\right)+17\) |
| \(x=2(y+3)^{2}-1\) | |
| Identify the constants \(a, h, k\). | \(a=2, h=-1, k=-3\) |
| Since \(a=2\), the parabola opens to the right. | |
| The axis of symmetry is \(y=k\). | The axis of symmetry is \(y=-3\). |
| The vertex is \((h,k)\). | The vertex is \((-1,-3)\). |
| Find the \(x\)-intercept by substituting \(y=0\). |
\(x=2(y+3)^{2}-1\)
\(x=2(0+3)^{2}-1\) \(x=17\) |
| The \(x\)-intercept is \((17,0)\). | |
| Find the point symmetric to \((17,0)\) across the axis of symmetry. | \((17,-6)\) |
|
Find the \(y\)-intercepts. Let \(x=0\). |
\(\begin{aligned} x &=2(y+3)^{2}-1 \\ 0 &=2(y+3)^{2}-1 \\ 1 &=2(y+3)^{2} \\ \dfrac{1}{2} &=(y+3)^{2} \\ y+3 &=\pm \sqrt{\dfrac{1}{2}} \\ y &=-3 \pm \dfrac{\sqrt{2}}{2} \end{aligned}\) |
| \(y=-3+\dfrac{\sqrt{2}}{2} \quad y=-3-\dfrac{\sqrt{2}}{2}\) | |
| \(y \approx-2.3 \quad y \approx-3.7\) | |
| The \(y\)-intercepts are \(\left(0,-3+\dfrac{\sqrt{2}}{2}\right),\left(0,-3-\dfrac{\sqrt{2}}{2}\right)\). | |
| Graph the parabola. |
- Write \(x=3 y^{2}+6 y+7\) in standard form and
- Use properties of the standard form to graph the equation.
- Answer
-
- \(x=3(y+1)^{2}+4\)
- Write \(x=-4 y^{2}-16 y-12\) in standard form and
- Use properties of the standard form to graph the equation.
- Answer
-
- \(x=-4(y+2)^{2}+4\)
Solve Applications with Parabolas
Many architectural designs incorporate parabolas. It is not uncommon for bridges to be constructed using parabolas as we will see in the next example.
Find the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.
Solution :
We will first set up a coordinate system and draw the parabola. The graph will give us the information we need to write the equation of the graph in the standard form \(y=a(x-h)^{2}+k\).
|
Let the lower left side of the bridge be the origin of the coordinate grid at the point \((0,0)\). Since the base is \(20\) feet wide the point \((20,0)\) represents the lower right side.
The bridge is 10 feet high at the highest point. The highest point is the vertex of the parabola so the \(y\)-coordinate of the
|
|
| Identify the vertex, \((h,k)\). | \((h, k)=(10,10)\) |
| \(h=10, \quad k=10\) | |
|
Substitute the values into the standard form. The value of \(a\) is still unknown. To find the value of \(a\) use one of the other points on the parabola. |
\(\begin{aligned} y &=a(x-h)^{2}+k \\ y &=a(x-10)^{2}+10 \\(x, y) &=(0,0) \end{aligned}\) |
| Substitute the values of the other point into the equation. |
\(y=a(x-10)^{2}+10\)
\(0=a(0-10)^{2}+10\) |
| Solve for \(a\). | \(\begin{aligned} 0 &=a(0-10)^{2}+10 \\-10 &=a(-10)^{2} \\-10 &=100 a \\ \dfrac{-10}{100} &=a \\ a &=-\dfrac{1}{10} \end{aligned}\) |
| \(y=a(x-10)^{2}+10\) | |
| Substitute the value for \(a\) into the equation. | \(y=-\dfrac{1}{10}(x-10)^{2}+10\) |
Find the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.
- Answer
-
\(y=-\dfrac{1}{20}(x-20)^{2}+20\)
Find the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.
- Answer
-
\(y=-\dfrac{1}{5} x^{2}+2 x y=-\dfrac{1}{5}(x-5)^{2}+5\)
Access these online resources for additional instructions and practice with quadratic functions and parabolas.
- Quadratic Functions
- Introduction to Conics and Graphing Horizontal Parabolas
Key Concepts
- Parabola: A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.
Vertical Parabolas
|
General form \(y=a x^{2}+b x+c\) |
Standard Form \(y=a(x-h)^{2}+k\) |
|
|---|---|---|
| Orientation | \(a>0\) up; \(a<0\) down | \(a>0\) up; \(a<0\) down |
| Axis of Symmetry | \(x=-\dfrac{b}{2 a}\) | \(x=h\) |
| Vertex |
Substitute \(x=-\dfrac{b}{2 a}\) and
solve for \(y .\) |
\((h, k)\) |
| \(y\)-intercept | Let \(x=0\) | Let \(x=0\) |
| \(x\)-intercepts | Let \(y=0\) | Let \(y=0\) |
- How to graph vertical parabolas \(y=a x^{2}+b x+c\) or \(f(x)=a(x-h)^{2}+k)\) using properties.
- Determine whether the parabola opens upward or downward.
- Find the axis of symmetry.
- Find the vertex.
- Find the \(y\)-intercept. Find the point symmetric to the \(y\)-intercept across the axis of symmetry.
- Find the \(x\)-intercepts.
- Graph the parabola.
Horizontal Parabolas
|
General form \(x=a y^{2}+b y+c\) |
Standard form \(x=a(y-k)^{2}+h\) |
|
|---|---|---|
| Orientation | \(a>0\) right; \(a<0\) left | \(a>0\) right; \(a<0\) left |
| Axis of Symmetry | \(y=-\dfrac{b}{2 a}\) | \(y=k\) |
| Vertex |
Substitute \(y=-\dfrac{b}{2 a}\) and
solve for \(x .\) |
\((h, k)\) |
| \(x\)-intercepts | Let \(x=0\) | Let \(x=0\) |
| \(y\)-intercept | Let \(y=0\) | Let \(y=0\) |
How to graph horizontal parabolas \(x=a y^{2}+b y+c\) or \(x=a(y-k)^{2}+h\) using properties.
- Determine whether the parabola opens to the left or to the right.
- Find the axis of symmetry.
- Find the vertex.
- Find the \(x\)-intercept. Find the point symmetric to the \(x\)-intercept across the axis of symmetry.
- Find the \(y\)-intercepts.
- Graph the parabola.
Glossary
- parabola
- A parabola is all points in a plane that are the same distance from a fixed point and a fixed line.


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Self-Test 1
-
Which term best describes a material that allows electrical energy to pass through it?
-
Resister
-
Insulator
-
Conductor
-
Connector
-
Which of the following best describes the term
electrical cable
?
-
Any wire
-
Any insulated wire
-
Multiple wires grouped together in a common insulation
-
Multiple insulated wires grouped together in a common sheathing
-
What is the primary purpose of insulation on wires?
-
To protect the wire
-
To make installation easier
-
To prevent unwanted current flow
-
So the wire can be colour coded
-
When is stranded wire used?
-
On short wires
-
When cost is a factor
-
On straight runs of wire
-
When flexibility is needed
-
Wire is sized by gauge: the higher the number, the larger the wire.
-
True
-
False
-
What should be done to stranded wire prior to bending?
-
It should be curled.
-
It should be tinned.
-
It should be wound.
-
It should be twisted.
-
Solderless connections are more costly than soldering.
-
True
-
False
-
What is one advantage to a twist-on connector?
-
It is permanent.
-
One size fits all.
-
It will not come apart.
-
It can be removed easily.
-
It doesn’t matter what material the conductor is made from when making a connection.
-
True
-
False
-
What type of connector should be used to create a permanent splice?
-
Twist
-
Winged
-
Crimp-on
-
Set screw


6.2: Add and Subtract Radical Expressions
Rules for radical multiplication and division have a simplicity and ease which lulls students into thinking addition and subtraction will follow suit. However, rules for addition and subtraction have more complication and less flexibility. Let’s start with the most common pitfall:
\(\sqrt[n]{a} + \sqrt[n]{b} \neq \sqrt[n]{a+b}\)
Note: The symbol \(\neq\) says “does not equal.”
These two examples have even more temptation to go wrong due to the squaring and the variables:
\(\sqrt{x^2 + y^2} \neq x+y\)
Also,
\(\sqrt{x^2 - y^2} \neq x-y\)
This doesn’t change any property that was previously introduced! The following still holds true! (\(x ≥ 0\) and \(y ≥ 0\) if \(n\) is even)
\(\sqrt{x^2} + \sqrt{y^2} = x + y\)
And
\(\sqrt{x^2} - \sqrt{y^2} = x - y\)
And in general,
\(\sqrt[n]{x^n} + \sqrt[n]{y^n} = x ± y\)
Use a calculator to convince yourself:
\(\sqrt{6} + \sqrt{2} \neq \sqrt{8}\)
On the left side, find the value \(\sqrt{6} + \sqrt{2}\) to the nearest hundredth:
\(\sqrt{6} + \sqrt{2} ≈ 3.86\)
On the right side,
\(\sqrt{8} ≈ 2.83\)
Now we see radicands cannot be added:
\(3.86 \neq 2.83\)
Use the order of operations as the problem is presented. Any other order of computation could be a pitfall.
\(\textcolor{green}{\checkmark} \sqrt[3]{27} + \sqrt[3]{8} = 3+ 2 =5 \)
The order of operations tells us to take the cube root first, then add. Use the order shown and it will be correct!
Using “Like Terms” to Add or Subtract “Like Radicands”
Adding and subtracting radicals starts with simplifying radicals. Use as often as possible the property \(\sqrt[n]{a^n} = a\) to simplify radicals. We will continue to assume the variables take on nonnegative values only. Let’s explore how gathering like terms can be used with radicals.
\(2x + x = 3x\) \(x =\) any real number
Gathering like terms is a technique you’ve used countless times before. Why not use with radicals? Radicals are real numbers, too! This approach opens our toolbox for adding or subtracting radical expressions. We can add radicals if the radicand is identical and index \(n\) also matches.
Suppose \(x = \sqrt{5}\). Gather like radicals:
\(2\sqrt{5} + \sqrt{5} = 3\sqrt{5}\)
Two radical terms are said to be like radicals if they have the same index and the same radicand.
Which of the three expressions can be simplified? Explain.
- \(4 \sqrt[3]{10} + 7 \sqrt[3]{10}\)
- \(3 \sqrt{2} + 3 \sqrt{11}\)
- \(\sqrt[3]{6} + \sqrt{6}\)
Solution
- This is the only given expression of the three that can be simplified.
\(4 \sqrt[3]{10} + 7 \sqrt[3]{10} = 11\sqrt[3]{10}\)
\(4 \sqrt[3]{10} \text{ and } 7 \sqrt[3]{10}\) both contain \(\sqrt[3]{10}\) as the like radicals.
- Cannot be simplified.
\(3 \sqrt{2} \neq 3 \sqrt{11}\)
The radicands are not equal.
- Cannot be simplified.
\(\sqrt[3]{6} \neq \sqrt{6}\)
The indices are not equal.
Simplify First
Don’t see a match? Make sure your radicals are completely simplified before you attempt to add or subtract radicals. The following examples demonstrate how simplification can help you find like radicals.
Simplify \(\sqrt{75} + 2 \sqrt{12} − \sqrt{3}\)
Solution
\(\begin{array} &\sqrt{75} + 2 \sqrt{12} − 5 \sqrt{3}& &\text{At first glance, none of the radicals are like radicals. However, both \(\sqrt{75}\) and \(\sqrt{12}\) can be simplified.} \\ & = \sqrt{25 \cdot 3} + 2 \sqrt{4 \cdot 3} − \sqrt{3} &\text{Factor perfect squares within each radicand.} \\ &= 5\sqrt{3} + 2 \cdot 2 \sqrt{3} − \sqrt{3} &\text{Simplify.} \\ &= 5\sqrt{3} + 4\sqrt{3} − \sqrt{3} &\text{Combine like radicals.} \\ (5 + 4 − 1)\sqrt{3} &= 8\sqrt{3} & \end{array}\)
Add \(\sqrt[3]{54x^2} + \sqrt[3]{3x^2} + \sqrt[3]{16x^2}\)
Solution
\(\begin{array} &&\sqrt[3]{54x^2} + \sqrt[3]{3x^2} + \sqrt[3]{16x^2} &\text{Simplify each radical.} \\ &= \sqrt[3]{27 \cdot 2x^2} + \sqrt[3]{3x^2} + \sqrt[3]{8 \cdot 2x^2} &\text{Factor the radicands.} \\ &= 3 \textcolor{red}{\sqrt[3]{2x^2}} + \sqrt[3]{3x^2} + 2 \textcolor{red}{\sqrt[3]{3x^2}} &\text{Two terms contain like radicals. Simplify. } \\ &= 5 \sqrt[3]{3x^2} + \sqrt[3]{3x^2} & \end{array}\)
Find the area and the perimeter of the rectangle shown.
Length \(= (6 \sqrt{3} + \sqrt{5})\) feet
Width \(= (\sqrt{3} + 5\sqrt{5})\) feet
Solution
Use FOIL:
\(\begin{array} &&\text{Area } &= (6 \sqrt{3} + \sqrt{5})(\sqrt{3} + 5\sqrt{5}) &= \underbrace{(6 \sqrt{3})(\sqrt{3})}_{\text{F}} + \underbrace{(6 \sqrt{3})(5\sqrt{5})}_{\text{O}} + \underbrace{(\sqrt{5})(\sqrt{3})}_{\text{I}} + \underbrace{(\sqrt{5})(5\sqrt{5})}_{\text{L}} \\ &&&= 6 \cdot 3 + 6 \cdot 5 \cdot \sqrt{3 \cdot 5} + \sqrt{5 \cdot 3} + 5 \cdot 5 \\&&&= 18 + 30\sqrt{15} + \sqrt{15} +25 \\&&&= 43 +31\sqrt{15} \text{ ft}^2 \end{array}\)
\(\begin{array} &&\text{Perimeter } &= \overbrace{2(6 \sqrt{3} + \sqrt{5}) + 2(\sqrt{3} + 5\sqrt{5})}^{P = 2L + 2W} &= \underbrace{(2 \cdot 6 \sqrt{3} + 2 \cdot \sqrt{5})}_{\text{Distributive Property}} + \underbrace{(2 \cdot \sqrt{3} + 2 \cdot 5\sqrt{5})}_{\text{Distributive Property}} \\ &&&= (12 \sqrt{3} + 2 \sqrt{5}) + (2 \sqrt{3} + 2 \cdot 5\sqrt{5}) \\&&&= \underbrace{12 \sqrt{3} + 2\sqrt{3}}_{\text{Like Radicals}} + \underbrace{2\sqrt{5}+10\sqrt{5}}_{\text{Like Radicals}} \\&&&= 14\sqrt{3} + 12\sqrt{5} \text{ ft}^2 \end{array}\)
Try It! (Exercises)
For #1-4, add or subtract the expressions, if possible. Assume the variables represent nonnegative numbers.
- \(7\sqrt{5} − 2\sqrt{5} + \sqrt{5}\)
- \(3\sqrt{3x} − 4\sqrt{3x} + 6\sqrt{6x} − 8\sqrt{6x}\)
- \(\sqrt[3]{9y} + \sqrt[4]{9y} − \sqrt[3]{2y} + 2\sqrt[4]{9y}\)
- \(7b \sqrt[5]{16b^2} − 4b\sqrt[3]{16b^2} − 3b\sqrt[5]{3b^2}\)
For #5-8, simplify each radical expression, then add or subtract the expressions, if possible. Assume the variables represent nonnegative numbers.
- \(2\sqrt{3a^3} + 5a\sqrt{3a} − \sqrt{27a^3}\)
- \(\sqrt[3]{56c^6} − 2c^2 \cdot \sqrt[3]{189} + \sqrt[3]{7c^2} \)
- \(3\sqrt[4]{4p^7} − 3p\sqrt[4]{324p^3} + 2\sqrt[4]{64p^7}\)
- \(2b\sqrt[5]{192b^3} − \sqrt[5]{64b} + 3\sqrt[5]{6b^8} + \sqrt[5]{486b}\)
For #9-14, evaluate each of the following. Simplify wherever possible. Assume the variables represent nonnegative numbers.
- \(\sqrt{12}(\sqrt{2} − \sqrt{18})\)
- \((7\sqrt[3]{10} + 3\sqrt[3]{4})(2\sqrt[3]{20} − \sqrt[3]{50})\)
- \((9\sqrt{2x} + 6\sqrt{3x})^2\)
- \((\sqrt{26} − 4\sqrt{10})(\sqrt{26} + 4\sqrt{10})\)
- \((\sqrt[3]{24} − \sqrt[3]{54})^2\)
- \(5(3\sqrt{7} − 1)(4\sqrt{14} + 1)\)
For #15-16, find the area and the perimeter of each rectangle. The length and width are each marked in the figures and the units are feet.
- A square has side lengths = \((\sqrt[4]{4} + 2)\) feet. Find each of the area and the perimeter of the square.
- If a square has area \(= 540\) \(\text{ft}^2\). What is the perimeter of the square?


Chapter 5: Formal semantics
In this chapter, we describe a formal semantics for SL and for QL. The word ‘semantics’ comes from the greek word for ‘mark’ and means ‘related to meaning.’ So a formal semantics will be a mathematical account of meaning in the formal language.
A formal, logical language is built from two kinds of elements: logical symbols and non-logical symbols. Connectives (like ‘&’) and quantifiers (like ‘∀’) are logical symbols, because their meaning is specified within the formal language. When writing a symbolization key, you are not allowed to change the meaning of the logical symbols. You cannot say, for instance, that the ‘¬’ symbol will mean ‘not’ in one argument and ‘perhaps’ in another. The ‘¬’ symbol always means logical negation. It is used to translate the English language word ‘not’, but it is a symbol of a formal language and is defined by its truth conditions.
The sentence letters in SL are non-logical symbols, because their meaning is not defined by the logical structure of SL. When we translate an argument from English to SL, for example, the sentence letter \(M\) does not have its meaning fixed in advance; instead, we provide a symbolization key that says how \(M\) should be interpreted in that argument. In QL, the predicates and constants are non-logical symbols.
In translating from English to a formal language, we provided symbolization keys which were interpretations of all the non-logical symbols we used in the translation. An interpretation gives a meaning to all the non-logical elements of the language.
It is possible to provide different interpretations that make no formal difference. In SL, for example, we might say that \(D\) means ‘Today is Tuesday’; we might say instead that \(D\) means ‘Today is the day after Monday.’ These are two different interpretations, because they use different English sentences for the meaning of \(D\). Yet, formally, there is no difference between them. All that matters once we have symbolized these sentences is whether they are true or false. In order to characterize what makes a difference in the formal language, we need to know what makes sentences true or false. For this, we need a formal characterization of truth .
When we gave definitions for a sentence of SL and for a sentence of QL, we distinguished between the
object language
and the
metalanguage
. The object language is the language that we are
talking about
: either SL or QL. The metalanguage is the language that we use to talk about the object language: English, supplemented with some mathematical jargon. It will be important to keep this distinction in mind.


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acupressure
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practice of stimulating or putting pressure on specific areas of the body
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aromatherapy
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use of essential oils to aid in relaxation
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biofeedback
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mind-body practice where a sensor monitors different pain reactions in the body while the person uses techniques to control those reactions.
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counter pressure
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act of giving sustained pressure to the back, hips, sacrum, or other joints
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doulas
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trained support laypersons who provide emotional, physical, and social support during pregnancy, labor, and postpartum
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effleurage
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light stroking massage using the tips of the fingers in slow, long strokes
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epidural anesthesia
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regional anesthesia produced by injection and infusion of a local anesthetic (bupivacaine [Marcaine]) and a narcotic (fentanyl) around the spinal nerves to block pain from T10 to S5 during labor
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failed intubation
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difficult airway with lack of success on more than two attempts
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focal points
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used when the laboring person focuses on a picture or an object during the contraction to direct attention away from the contraction
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hydrotherapy
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process of the laboring person submerging in water for pain relief
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imagery
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uses the mind-body connection to focus the laboring person’s awareness on a positive image
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pelvic tilts
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rocking of the pelvis by moving the back up, then sinking the back down
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pudendal block
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local anesthetizing of the pudendal nerve that provides sensation to the perineum, anus, vulva, and clitoris


8.9.6: Chapter Test
Decide whether randomization is being used in the selection of these samples. If it is, identify the type of random sample (simple, systematic, cluster, or stratified).
A vinyl record dealer is trying to price a large collection she’s thinking of buying. She looks at every tenth record on the shelf and notes the value.
A court clerk is charged with identifying one hundred people for a jury pool for upcoming legal hearings. He has an alphabetized list of registered voters in his jurisdiction, so he uses a random number generator to pick one hundred names from the list.
A state agricultural officer is worried about the spread of a parasitic disease among cattle. He chooses 30 cattle farms at random and tests each cow on those 30 farms for the disease.
A sample of donations to a blood bank contained these blood types:
| A | O | A | O | A | A | AB | O |
| O | A | AB | A | O | O | A | A |
| O | O | B | O | B | O | O | O |
| B | A | O | A | O | O | A | A |
Create a frequency distribution for these data.
Create a bar chart without technology to visualize the data in question 4.
The table below shows the number of players on the rosters of each English Premier League team. Find the mode, median, and mean without technology.
| Team | Active Players | Team | Active Players |
|---|---|---|---|
| Manchester United F.C. | 27 | Newcastle United F.C. | 28 |
| Manchester City F.C. | 24 | Aston Villa F.C. | 24 |
| Chelsea F.C. | 25 | Fulham F.C. | 27 |
| Arsenal F.C. | 28 | Southampton F.C. | 24 |
| Tottenham Hotspur F.C. | 27 | Wolverhampton Wanderers F.C. | 22 |
| Liverpool F.C. | 30 | Brighton & Hove Albion | 31 |
| Crystal Palace | 29 | Burnley F.C. | 22 |
| Everton F.C. | 23 | West Bromwich Albion F.C. | 26 |
| Leicester City | 25 | Leeds United F.C. | 25 |
| West Ham United F.C. | 24 | Sheffield United F.C. | 27 |
Using the data in “MBB2019” , find the mean, median, and mode of the total number of rebounds made by each team (TRB).
Using the data in “MBB2019” , find the range and standard deviation of the total number of rebounds made by each team (TRB).
Use the data in “MBB2019” to answer the following:
What number of total rebounds (TRB) is at the 3rd quartile?
At what percentile is the University of Evansville, which had 1095 total rebounds?
Answer the following about data that are distributed normally with mean 50 and standard deviation 5:
What proportion of the data are between 35 and 65?
What’s the standardized score of the data value 37?
At what percentile would the data value 58 fall?
What data value would be at the 40th percentile?
Use the data in “MBB2019” to answer the following:
What is the standardized score for the number of total rebounds (TRB) recorded by Mississippi State University (TRB = 1215)?
At what percentile would we estimate Mississippi State’s TRB value fall, using the normal distribution?
At what percentile does Mississippi State’s TRB value actually fall?
Use the data in “NBA2019” to answer the following:
What is the correlation coefficient between the number of two-point field goals made per game (2P) and number of points scored per game (PTS)?
What is the equation of the regression line we would use to predict PTS from 2P?
Write a sentence to interpret the slope of the regression line.
Predict the number of points per game for a player who makes 6 two-point field goals per game.
J.J. Redick made 2.8 field goals per game, and averaged 18.1 points per game. Did he score more or fewer points per game than expected? By how much?


6.3: Electrospray Ionization (ESI) Mass Spectrometry
Electrospray Ionization Mass Spectrometry is a technique used to determine molecular weights for proteins, peptides, and other biological macromolecules such as oligosaccharides 1 . Originally described by Canadian-American Physicist Sir Arthur J. Dempster in an article titled "A new method of positive ray analysis." 7 His work was the first modern mass spectrometer, which used positive rays to determine the mass-to-charge ratio of various isotopes of lithium and magnesium. Dempster showed that it was possible to determine the isotopes' relative proportions and atomic weights using this method 7 . Mass spectrometry in general is useful for structure elucidation (when combined with chromatography separation techniques), peptide sequencing (when combined with ion traps), and quantification (when combined with a triple-quadrupole mass analyzer) for example, and is mainly limited by the ability of the ions generated to remain stable until their arrival at the detector. Electrospray Ionization describes the method by which the macromolecules are ionized via a “soft” ionization, which does not fragment nor harshly degrade the macromolecules and ionizes by multiple charging 1 . Rather, the macromolecules are ionized into small droplets 2 . The droplets are then desolvated further, effectively decreasing the droplet size into molecules with protons 1 . The protonated/desolvated molecules enter the mass analyzer and subsequently the detector to determine the mass/charge ratio 1 . The main advantage is that the samples can easily be introduced to the instrument in solution with the ability to detect very small masses (atomic masses) to very large molecules (MDaltons) with detection limits down to the pico-, femto-, and attomole levels 5 . Furthermore, this analytical technique has become of great importance within the last decade or so due to advancements enabling even higher sensitivity and its ability to be combined with high-throughput automation for -"omics" studies with high applicability in drug discovery.
Sample Preparation
Samples for ESI-MS are typically purified, which is ideal for the instrument because mixtures of components with different physicochemical properties are not good analytes for the technique (e.g. glycans vs. peptides) 1 . Methods such as high-performance liquid chromatography, capillary electrophoresis, and liquid-solid column chromatography are commonly employed for the purification step, and subsequently injected into the ESI-MS 1 . Usually, the purification method is attached directly to the capillary needle 1 .
Electrospray Ionization (ESI)
A mechanical syringe pump injects the liquid sample (usually less than one mM in a polar volatile solvent) into the capillary needle, and thereby nebulizing it into a fine mist 1 . Typically an analyte will undergo three major processes after injection and transfers into the gas phase: it will produce charged droplets from the high-voltage cappillary tip, repeatedly evaporate solvent from the charged droplet followed by droplet disintegration into much smaller droplets, and lastly transfer of the ion into the gas-phase 5 . The capillary needle typically has an inner diameter of ~0.1mm and outer diameter of ~0.2mm, and a low flow rate around 1 to 20 μL/min 1 . A voltage is applied to the tip of the capillary of around 2-6 kV, where the surrounding source-sampling cone is located around 1-3 cm from the spray needle tip (see Figure \(\PageIndex{1}\)) 2 .
The strong electric field causes the nebulized particles to carry a charge, thus, becoming electrospray droplets where a drying or sheath gas flows around the capillary improving nebulization 1 .
The electric field also directs the spray toward the mass spectrometer while droplets diminish in size (solvent evaporation) 1 . This process is described pictorally in Figure \(\PageIndex{3}\) below. Once the electrospray droplets pass through the heating capillary (Figure \(\PageIndex{2}\), labeled as desolvating capillary in Figure \(\PageIndex{1}\)), the ions completely desolvate 1 . The heating capillary is typically around 0.2mm inner diameter, 60mm in length, at a temperature controlled in a range from 100-300 o C for desolvation and continual droplet shrinkage 1 . At this point two forces become dominant: surface tension in the droplets acting to retain the shape of the droplet, and Coulomb force of repulsion between like charges on the surface which act to break down the shape of the droplet 2 .
As the droplets travel through the heating capillary, they have a high enough electric field density that causes like charges to repel one another, increasing surface tension (first to second step in Figure \(\PageIndex{3}\)) 2 . The droplets then reach the Rayleigh limit, which describes the limit of the number of charges that can be present on a charged droplet before fission occurs and is broken down (third step in Figure \(\PageIndex{3}\)) 1 . It is at this point the surface tension can no longer sustain the Coulomb force of repulsion, and a “Coulomb explosion” or “Coulomb fission” occurs (fourth step in Figure \(\PageIndex{3}\)) 1 . The parent droplets disintegrate into much smaller droplets of positive or negative charge, with a much higher mass-to-charge ratio 1 . These small droplets have high mass-to-charge ratios because smaller portions of parent charges are carried off and distributed in many offspring droplets compared to the mass carried off from parent droplets. An example time history is shown below of methanol droplets produced by electrospray in Figure \(\PageIndex{4}\).
The decomposition of a droplet from ESI occurs in manner similar to the inset of Figure \(\PageIndex{4}\), and is dictated by the Rayleigh equation:
\[q^2 = 8 \pi^2 \epsilon_o \gamma D^3 \label{1}\]
Where q is charge, epsilon 'naught' is the permissivity of the medium, gamma is the surface tension of the droplet, and D is the droplet diameter.
Mass Analyzers
From the ion beam, the mass analyzer takes different types of ions and separates them based on their mass-to-charge ratio. Afterwards, the ions are passed onto the detector. There are many types such as magnetic (B)/electric (E) sector mass analyzers, linear quadrupole ion trap (LIT), three-dimensional qudrupole ion trap (QIT), orbitrap ( Mass Analyzer Orbitrap ), time-of-flight mass analyzer (TOF, Mass Analyzer Time of Flight ), and ion cyclotron resonance mass analyzer (ICR) 1 . They all take advantage of dynamic of static magnetic/electric fields based on Lorentz force law (Equation (1), a charge experiences electric and magnetic forces when traveling through magnetic/electric fields) 3 and Newton's second law of motion (Equation (2), objects accelerate based on their mass and net forces acting on the object) 4 .
\[F = qE + qv x B \label{2}\]
\[F = m x a \label{3}\]
Where in Equation (2), F is the Lorentz force, q is the charge of the ion, E is the electric field, v is the velocity of the ion, and B is the magnetic field. In Equation (3), F is Force, m is mass, and a is acceleration.
Table 1 below briefly summarizes comparisons between different mass analyzers. The mass analyzer uses electrostatic lenses (see Figure \(\PageIndex{1}\)) to direct the beam into the analyzer, and is kept at a high vacuum (around 10 -3 torr to 10 -6 torr pressure) to prevent any undesired molecular interactions between the ions and the atmosphere 1 .
Table 1: Comparison of mass analyzers (de Hoffmann & Stroobant, 2007)
Five main characteristics for the measuring performance of a mass analyzer are the mass range limit, analysis speed, transmission, mass accuracy, and resolution
5
. According to de Hoffmann and Stroobant, mass range is the mass-to-charge (m/z) limit the mass analyzer can measure ions, expressed in units of Th (Thomson), or u (unified atomic mass unit) for ions with an elementary charge (z=1), where
Where m is the mass of the first peak, divided by the difference, ∆m, between the neighboring peaks 2 . The data is improved with more resolution (higher R) 5 .
Detectors
A common type of detector used in conjunction with a quadrupole mass analyzer for example, is an electron multiplier (EM) detector 5 . In this type of detector electrons are accelerated to a high velocity, enhancing detection efficiency 5 . To achieve this, an electrode called a conversion dynode is held at a high potential from + 3 kV to + 30 kV, opposite to the ions’ charges 5 . Once ions strike the conversion dynode, several secondary particles are emitted 5 . Secondary particles are typically negative and positive ions, electrons, and neutral particles 5 . For example, if positive ions strike the conversion dynode, negative ions and electrons are the secondary particles of interest, likewise, if negative ions strike the conversion dynode the secondary particles are positive ions 5 . The secondary particles are then amplified in a cascade effect to produce a current 5 .
Figure \(\PageIndex{8}\) shows a schematic for a discrete dynode electron multiplier detector 5 . The first dynode starts at a higher magnitude potential (but negative), causing the secondary particles to emit electrons 5 . The secondary particles and electrons accelerate to each subsequent dynode (due to the lower potential) until the electrons reach the output of the electron multiplier, which is held at ground potential 5 . A cascade of electrons are created at the end, and the current produced and amplified by conventional electronic amplification 5 .
Electrospray Ionization on Membrane Proteins
Membrane proteins are of particular interest because of their importance in cell signaling, transport, adhesion 6 , and intercellular interactions. Some advantages of mass spectrometry methods in studying membrane proteins are that freezing or crystallization (as in X-ray crystallography) are not required, and that it is possible to study these proteins in their native (unfolded) state 6 . Utilization of electrospray ionization (ESI) mass spectrometry (MS) on membrane proteins typically falls under two categories: native MS and labeling MS 6 . Native MS involves maintaining non-covalent interactions, which preserves tertiary and quaternary structures, and is carried out in vacuo 6 . Labeling MS methods involve chemical crosslinking, hydrogen-deuterium exchange, and hydroxyl radical footprinting (HRFP) to name a few 6 . For each method, proteins are labeled in solution (peptides reacted with chemical labels), undergo proteolysis, and are subsequently measured/quantified via MS method 6 (see Figure \(\PageIndex{9}\) below).
Considerations for ESI on Membrane Proteins
Careful consideration in optimizing the parameters are absolutely necessary when analyzing biological materials in mass spectrometric methods. Two examples of these parameters are the collision voltage and selection of detergent 6 . The collision voltage is that which is applied to molecular ions, accelerating them into the collision cell with an inert gas 5 . Optimization of collision voltage involves selecting a voltage that enables fragment ions to be observed, but also well resolved. This voltage goes hand-in-hand with careful consideration of the buffer/detergent. Ideally, the buffer/detergent needs to be able to efficiently solubilize the protein, and also be easily removed to allow the protein to be properly desolvated (see Figures 3 and 4) 6 . Detergents are used for membrane proteins because of their amphiphilic nature, similar to the membrane proteins themselves 6 . Where these two parameters come together are when the membrane protein-detergent complex transfers into the gas phase: the collision voltage must be high enough to desolvate the membrane protein from the detergent, and the detergent must not strongly solvate the membrane protein 6 . Strong solvating detergents require higher energies which risk destabilizing the protein prior to detection in the mass analyzer.
If the membrane protein is not liberated from the detergent, perhaps due to low collision voltage, the membrane proteins' signals maybe be suppressed by noise from the remaining detergent (see Figure \(\PageIndex{1}\)0b and 10f). Figure \(\PageIndex{1}\)0 below also demonstrates the concepts of selecting proper collision voltage combined with an appropriate detergent.
Summary
-
Electrospray (ESI) is a soft-ionization technique that has gained popularity in biological applications as it is capable of maintaining non-covalent interactions of proteins.
- Use of appropriate buffers/detergents/optimized instrument conditions renders ESI-mass spectrometry (MS) to be a powerful and sensitive tool to elucidate membrane protein structures.
- ESI can be coupled with a variety of mass analyzers (see Table 1), providing users of mass spectrometry flexibility in analysis.
- ESI can also be coupled downstream from a variety of chromatographic techniques such as HPLC to enhance mass spectrometry-based studies on complex molecular structures.
References
- Banerjee, S., & Mazumdar, S. (2012). Electrospray Ionization Mass Spectrometry: A Technique to Access the Information beyond the Molecular Weight of the Analyte. International Journal of Analytical Chemistry.
- Murphy, J. (2016, October 8). Electrospray Ionization Mass Spectrometery. Retrieved from Chemistry LibreTexts(TM): https://chem.libretexts.org/Core/Ana...s_Spectrometry
- Bird, R. B., et al. (2001). Transport Phenomena, Wiley.
- Young, H. D., Freedman, R. A., & Sears, F. W. (2004). Sears and Zemansky's university physics (11th ed.). San Francisco, Calif.: Pearson/Addison Wesley.
- de Hoffmann, E., & Stroobant, V. (2007). Mass Spectrometry Principles and Applications. West Sussex, England: John Wiley & Sons, Ltd.
- A.N. Calabrese, S.E. Radford, Methods (2018). https://doi.org/10.1016/j.ymeth.2018.02.020
- Dempster, A. J. (1921). "Positive Ray Analysis of Lithium and Magnesium." Physical Review 18(6): 415-422.
- McFarland, S. (Ed.). (2008, April 19). File:Fission.jpg. Retrieved June 13, 2018, from http://www.appropedia.org/File:Fission.jpg
- Bird, R. B., et al. (2001). Transport Phenomena, Wiley.
- (2003). University Physics With Modern Physics,11/e, Pearson Education.
Contributors and Attributions
- Christopher Ranque


Educational games are usually decision-making activities that include rules, a goal, conditions or constraints, competition, challenge, strategies, and feedback. Games can be as simple as answering questions to win Tic-Tac-Toe or filling in crossword answers to more complex games that require interactions with other learners.
Educational games:
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should encourage the development of specific skills
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The skills can be in specific subject areas such as science and math or general skills like literacy, problem solving, critical thinking, and decision-making.
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Success should be based on whether the specific learning outcomes have been met, rather than on good hand-eye coordination.
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can be used to teach many different skills
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The example shown in figure 22.4 illustrates how a game can be used to teach keyboarding skills.
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One difficulty is that games tend to require more explanations of the goals, learning outcomes, and directions than tutorials or drill and practice methods. Without guidance, learning is less effective.
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can be an effective, motivational, and fun way to learn
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To be effective, the game must be challenging, students must be actively involved, and students must be given feedback and guidance with respect to the learning outcomes.
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Research has shown that many learners like to learn through educational games.
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Some educational games are a part of simulations that involve competition and/or cooperation.
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Both males and females can enjoy and learn from games suited to their interests.
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are sometimes a waste of time
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Some products are fancy but do not teach well.
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Evaluate a game before purchasing it to ensure that the game teaches an important skill effectively. Some games may lead to violent and aggressive behaviours.
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Some people erroneously believe that games cannot be effective teaching tools.


10.5: Putting It Together- Retail Operations and Managing the Store
So human resources is a huge part of the retail business landscape. From screening potential employees, interviewing, hiring and training, each step of the process requires skill and planning.
New employees must be onboarded effectively to feel comfortable in their new role. Then continued training to maintain and improve skills must be organized, tracked and implemented. Employees make a retail business what it is! Service is the key, and excellent staff give great service.
Contributors and Attributions
CC licensed content, Original
- Putting It Together: Retail Operations and Managing the Store. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution