Biography:

One of their most recent publications is 3 - Graphing and Linear Systems. Which was published in journal .

More information about Charles P. McKeague research including statistics on their citations can be found on their Copernicus Academic profile page.

Charles P. McKeague's Articles: (29)

3 - Graphing and Linear Systems

Publisher SummaryThis chapter discusses graphing and linear systems. It focuses on the development of the method of solving linear equations in two variables. The solution set for a system of equations is the set of all ordered pairs that satisfy both equations. The solution set to a system of linear equations comprises of the following elements: (1) one ordered pair when the graphs of the two equations intersect at only one point, (2) no ordered pairs when the graphs of the two equations are parallel lines, and (3) an infinite number of ordered pairs when the graphs of the two equations coincide.

7 - Roots and Radicals

Publisher SummaryThis chapter discusses roots and radicals. It presents operations on and simplification of radicals. Finding the square root of a number is the reverse of raising a number to the second power. The more familiarity with the properties of exponents, the better equipped to deal with roots and radicals. The chapter highlights the properties of exponents and the operations on polynomials. Every positive real number x has two square roots, one positive and one negative. The positive square root is written as x. The negative square root of x is written as x. In both the cases, the square root of x is a number squared to get x. The cube root of x is written 3x and is the number cubed to get x.

8 - More Quadratic Equations

Publisher SummaryThis chapter focuses on the solutions of quadratic equations that cannot be factored. It presents a method of solving quadratic equations that can be used on all quadratic equations irrespective of whether they are factorable. It presents the application of this method to the general quadratic equation ax2 + bx + c = 0. The chapter also presents graphs of second-degree equations. It explains the addition, subtraction, multiplication, and division of complex numbers. It also discusses the graphing of parabolas. The graph of an equation of the form y = ax2 + bx + c, a ≠ 0, is a parabola.

7 - Linear Equations and Inequalities

Publisher SummaryThis chapter presents the concept of graphing some straight lines. It presents the definition of the slope of a line and the use of this definition to find two special forms of the equation of a straight line. The chapter discusses how to find the graph of a straight line given the equation of that line and how to find the equation of a line given the coordinates of two points on the line or given one point and the slope of the line. A rectangular coordinate system is made by drawing two real number lines at right angles to each other. The two number lines, called axes, cross each other at 0. This point is called the origin. Positive directions are to the right and up. Negative directions are down and to the left. The horizontal number line is called the x-axis and the vertical number line is called the y-axis. A linear inequality in two variables is any expression that can be put in the form ax + by < c, where a, b, and c are real numbers (a and b are not both 0). The inequality symbol can be any one of the following four: <, ≤, >, and ≥. The solution set for a linear inequality is a section of the coordinate plane. The boundary for the section is found by replacing the inequality symbol with an equal sign and graphing the resulting equation.

10 - Relations and Functions

Publisher SummaryThis chapter discusses two main concepts, that is, relations and functions. Relations and functions have many applications in the real world. When one says that the price of gasoline is increasing because there is more demand for it this year, one is expressing a relationship between the price of gasoline and the demand for it. It is implied that the price of gasoline is a function of the demand for it. Mathematics becomes a part of this problem when one expresses, with an equation, the exact relationship between the two quantities. A relation is any set of ordered pairs. The set of all first coordinates is called the domain of the relation, and the set of all second coordinates is said to be the range of the relation. A function is a relation in which no two different ordered pairs have the same first coordinates. The domain and range of a function are the sets of first and second coordinates, respectively. The chapter discusses some basic definitions associated with functions and relations. It presents a new notation associated with functions called function notation and it considers combinations of functions and exponential functions. The chapter discusses the inverse of a function.

4 - Rational Expressions

Publisher SummaryThis chapter is focuses on simplifying a certain kind of algebraic expression, called “rational expressions” because they are to algebra what rational numbers are to arithmetic. For rational expressions, multiplying the numerator and denominator by the same non-zero expression might change the form of the rational expression, but it would produce an expression equivalent to the original one. The same is true when dividing the numerator and denominator by the same non-zero quantity. The division of a polynomial by a monomial is the simplest kind of polynomial division. This kind of division is similar to long division with whole numbers. Division of a polynomial by a monomial is accomplished by dividing each term of the polynomial by the monomial. The product of two rational expressions is the product of their numerators over the product of their denominators. The quotient of two rational expressions is the product of the first and the reciprocal of the second. The least common denominator (LCD), for a set of denominators is the smallest expression that is divisible by each of the denominators.

6 - Quadratic Equations

Publisher SummaryThis chapter discusses quadratic equations. A new method of solving quadratic equations has been developed; this new method is called “completing the square.” Completing the square on a quadratic equation allows obtaining the solutions, regardless of whether the equation can be factored. The key to understanding the method of completing the square lies in recognizing the relationship between the last two terms of any perfect square trinomial whose leading coefficient is 1. The quadratic formula is a useful tool in mathematics. It allows solving all types of quadratic equations. The chapter describes the discriminant, which is used to find the kind of solutions a quadratic equation has without solving the equation. The zero-factor property is used to build equations from their solutions. The zero-factor property and long division with polynomials are used to solve some third-degree equations. The quadratic formula gives the solutions to any quadratic equation in standard form.

8 - Systems of Linear Equations

Publisher SummarySystems of linear equations are used in many different disciplines. Systems of linear equations can be used to solve multiple-loop circuit problems in electronics, kinship patterns in anthropology, genetics problems in biology, and profit-and-cost problems in economics. There are many other applications as well. Solving a system of linear equations by graphing is the least accurate method. If the coordinates of the point of intersection are not integers, it can be difficult to read the solution set from the graph. There is another method of solving a linear system that does not depend on the graph. It is called the addition method. A system of linear equations consists of two or more linear equations considered simultaneously. The solution set to a linear system in two variables is the set of ordered pairs that satisfy both equations. The solution set to a linear system in three variables consists of all the ordered triples that satisfy each equation in the system.

9 - The Conic Sections

Publisher SummaryThis chapter focuses on four special types of graphs and their associated equations. The four types of graphs are called “conic sections” because each can be found by slicing a cone with a plane. There are many applications associated with conic sections. The planets orbit the sun in elliptical orbits. Many of the comets that come in contact with the gravitational field surrounding the earth travel in parabolic or hyperbolic paths. Flashlight and searchlight mirrors have elliptical or parabolic shapes because of the way surfaces with those shapes reflect light. The chapter presents graphs of ellipses and hyperbolas. The graphs that are centered about the origin have been considered. The most convenient method of solving a system that contains one or two second-degree equations is by substitution, although the addition method can be used at times.

12 - Sequences and Series

Publisher SummaryA sequence is a function whose domain is the set of positive integers. The general term is used to define the other terms of the sequence. That is, if the formula for the general term, “an” is given, one can find any other term in the sequence. The sum of a number of terms in a sequence is called a series. A sequence can be finite or infinite depending on whether the sequence ends at the nth term. An arithmetic progression is a sequence of numbers in which each term is obtained from the preceding term by adding the same amount each time. A sequence of numbers in which each term is obtained from the previous term by multiplying by the same amount each time is called a geometric progression. This chapter focuses on the classification of sequences, called geometric progressions.

3 - Exponents and Polynomials

Publisher SummaryThis chapter discusses exponents and polynomials. The chapter presents the simplification of those products that involve more than one base and more than one exponent. Actually, almost all of the expressions can be simplified by applying the definition of positive integer exponents; that is, by writing all expressions in expanded form and then simplifying by applying ordinary arithmetic. This process is usually very time-consuming. The chapter further presents two properties of exponents that show how exponents affect division. A term or monomial is a constant or the product of a constant and one or more variables raised to whole-number exponents. The numerical part of each monomial is called the numerical coefficient. The degree of a polynomial with one variable is the highest power to which the variable is raised in any one term.

7 - Linear Equations and Inequalities

Publisher SummaryThis chapter discusses linear equations and inequalities. Mathematics is a language that can describe certain aspects of the world better than English. One important aspect of the world is the idea of a path, track, orbit, or course. Mathematics can be used to describe these paths very accurately. The simplest of paths, which is a straight line, can be described by an linear equation in two variables. A pair of numbers written in a specific order is called an ordered pair. The first number in the ordered pair is always associated with the variable x, the second number with the variable y. The first number is called the x-coordinate of the ordered pair, and the second number is called the y-coordinate of the ordered pair.

8 - Systems of Linear Equations

Publisher SummaryThis chapter discusses systems of linear equations. Systems of linear equations are used extensively in many different disciplines. They can be used to solve multiple-loop circuit problems in electronics, kinship patterns in anthropology, genetics problems in biology, and profit-and-cost problems in economics. The chapter presents three different methods of solving linear systems in two variables. Two of these methods are extended to include solutions to systems in three variables. A fourth method of solving linear systems involves what are known as determinants. Two linear equations that have no solutions in common are said to be inconsistent, while two linear equations that have all their solutions in common are said to be dependent.

9 - The Conic Sections

Publisher SummaryThis chapter discusses four special types of graphs and their associated equations: parabolas, circles, ellipses, and hyperbolas. They are called conic sections because each can be found by slicing a cone with a plane. There are many applications associated with conic sections. The planets orbit the sun in elliptical orbits. Many of the comets that come in contact with the gravitational field surrounding the earth travel in parabolic or hyperbolic paths. Flashlight and searchlight mirrors have elliptical or parabolic shapes because of the way surfaces with those shapes reflect light. The arches of many bridges are in the shape of parabolas. Objects fired into the air travel in parabolic paths. The equations associated with these graphs are all second-degree equations in two variables.

11 - Logarithms

Publisher SummaryThis chapter presents the applications of a new notation for exponents. The properties of logarithms are actually the properties of exponents. Logarithms were earlier used extensively to simplify tedious calculations. As handheld calculators are so common now, logarithms are seldom used in connection with computations. Nevertheless, there are many other applications of logarithms to both science and higher mathematics. A common logarithm is a logarithm with a base of 10. As common logarithms are used so frequently, it is customary, in order to save time, to omit notating the base. The chapter presents two special identities that arise from the definition of logarithms. Each is useful in proving the properties for evaluating some simple logarithmic expressions.

12 - Sequences and Series

Publisher SummaryThis chapter discusses sequences and series. The chapter focuses on two main types of sequences—arithmetic and geometric sequences—and describes the nth partial sum of a sequence. Sequences are used frequently in many different branches of mathematics and science. Many real-life situations can be described in terms of arithmetic and geometric sequences. The binomial theorem has some useful applications that are not limited to mathematics. It is used in statistics, biology, physics, and other disciplines. A sequence of numbers in which each term is obtained from the previous term by multiplying by the same amount each time is called a geometric sequence.

6 - Equations

Publisher SummaryThis chapter discusses how to solve simple equations that have the form of the linear and quadratic equations. It also describes the equations that contain multiples of angles. These equations are solved with the help of the basic trigonometric identities and the double- and half-angle formulas. The solution set for an equation is the set of all numbers which, when used in place of the variable, make the equation a true statement. Solving linear or first-degree equations is accomplished by applying two important properties: (1) the addition property of equality and (2) the multiplication property of equality. According to the addition property of equality, adding the same quantity to both sides of an equation will not change the solution set. According to the multiplication property of equality, multiplying both sides of an equation by the same nonzero quantity will not change the solution set. The chapter presents examples that demonstrate how to use these two properties to solve a linear equation in one variable.

7 - Triangles

Publisher SummaryThis chapter describes the law of sines that presents the relationship between the sides and angles in any triangle. The chapter presents the derivation of a formula that relates the sides and angles of any triangle. This formula is called the law of cosines and is used to solve for the missing parts of triangles. The law of sines states that the ratio of the sine of an angle to the length of the side opposite that angle is constant in any triangle. It is noteworthy that the law of sines, along with some fancy electronic equipment, was used to obtain the results of some of the field events in one of the Olympic Games.

8 - Complex Numbers and Polar Coordinates

Publisher SummaryPolar coordinates are used to name points in the plane and are an alternative to rectangular coordinates. This chapter presents a relationship between polar and rectangular coordinates. It presents a graph of equations in which the variables are given in polar coordinates. A complex number is any number that can be written in the form a + bi, where a and b are real numbers and i2 = −1, The form a + bi is called standard form for complex numbers. The number a is called the real part of the complex number. The number b is called the imaginary part of the complex number. If b ≠ 0, then a + bi is also an imaginary number. The absolute value or modulus of the complex number z = a + bi is the distance from the origin to the point (a, b). If this distance is denoted by r, then .

5 - Rational Exponents and Roots

Publisher SummaryThis chapter discusses the fractional exponents, roots, and complex numbers. The exponents and polynomials are very useful in understanding the concepts developed here. Radical expressions and complex numbers behave like polynomials. Many of the formulas that describe the characteristics of objects in the universe involve roots. The formula for the length of the diagonal of a square involves a square root. The length of time it takes a pendulum to swing through one complete cycle depends on the square root of the length of the pendulum. The formulas that describe the changes in length, mass, and time for objects traveling at velocities close to the speed of light also contain roots.

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