Functions of Behavior

# Introduction to Behavior of Quadratic Functions on a Graph This class will introduce students to the topics of behavior of quadratic functions, or more specifically tangent functions. The term “quadratic function” refers to the set of positive/negative real-valued functions which can be graphed. A quadratic equation (also called a quadratic function) is a system of equations with one integral and one set of real and nominal values. Most of quadratic equation have singular solutions (i.e., solutions that are only the values of one of the terms in the equation). Therefore, it is usually much simpler and easier for a quadratic equation to be solved than an ordinary first-order polynomial equation.

In order to learn behavior of quadratic functions through mathematical modeling, students will be introduced to graphs. Graphs are typically used to represent data sets in mathematics, but they are also commonly used in many other fields. In the course of this class, the student will learn how to create and draw graphs in order to better understand and visualize the behavior of quadratic equations.

Specifically, we will focus on the topic of normal distributions, intercepts, slopes, kurtosis, momentum, density, momentum transfer, cumulative densities, as well as the law of conservation of energy and its derivatives. After learning how to create simple (and not too simple!) graphical representations of the data we studied in class, we will learn to create more complex (but still useful and interesting) graphs using the same graphical language. One common way to represent the behavior of quadratic functions on a graph is through the use of the symmetrical representation. Under the symmetrical representation, the x-axis of the graph is represented by a vertical line, while the y-axis is represented by a horizontal line. We can find many different shapes for the x and y axes, representing different values of the quadratic function.

Symmetrical graphs will help us learn the general shape of the function and how to identify vertical or horizontal outlots. The concept of symmetry is important in the study of the behavior of quadratic functions, since the overall shape of the curve or surface being plotted is essentially symmetrical.

The second common way to represent the behavior of quadratic functions on a graph is through the use of the graphical language called the graphical model of a system. The graphical model of a system is simply a way of modeling the real system using the axioms of algebra and calculus. The student should become familiar with different graphical models such as the parabolic, cylindrical, spherical, Hodge, log-normal, etc. This class should also teach students how to solve for a particular numerical value using a particular model.

The third way that the behavior of quadratic functions on a graphing chart can be understood is through the application of basic techniques taught in geometry. In this lesson, the student should learn how to measure angles and the spatial relationships between two points on the chart. Measuring angles using the horizontal and vertical lines on the chart and relating the data to the corresponding Cartesian formula will show the exact value of the equation at each point. Also, using the data to find the slope of a particular curve on the chart will demonstrate how the quadratic equation is changing as the value of the variable plotted increases.

The final way to understand the behavior of quadratic equations on a graph is through the implementation of some different analytic techniques. The student can implement the solutions of the quadratic equation using the right form (solve for x), a cubic spline, a parabola, or a cylindrical mesh. Graphical models can be used to solve the equations for x and y. Finally, integration of the data and finding the value of the function at each point can be illustrated by the use of the integral formula.

## Graphs and Behavior of Quadratic Functions

In this first lesson, you’ll explore the algebraic structure and behavior of quadratic functions which typically feature an arithmetic solution but without the constant terms. You will explore how to represent such a quadratic equation with graphing techniques and develop new methods to solve such equations by numerical computation. After this first lesson, you should be familiar with quadratic equations (either of the form ax*(x) for x = a where a is any real number preferably infinite, but not counting the zero), the definition of a quadratic equation, how to select solutions of these equations using standard methodologies, and concepts such as elliptical function, analytic continuation, and solutions of the form ax*(y) where y is any real number preferably infinite. The second lesson deals with the symmetries of a quadratic function. The symmetries of a function are its derivatives, which are functions of a real variable x such that t(x) is the Lagrange Equation of the real variable x. The symmetries of a function are where all the zero variables can be equal to one or all the derivatives of the function are zero.

In the definition of a quadratic equation, the sum of the values in the x-intercept is called the hyperbola and the sum of all the values in the x-term is called the hyperbola’s surface. It may help if you visualize the graph of the function as a piece of graph paper whose slope is the function of the hyperbola and whose horizontal axis represents the x-axis, and whose vertical axis represents the y-axis.

The third lesson focuses on creating high-quality graphs from mathematically-sound data. To make such graphs you must first create high-quality mathematical expressions for your graphs. Next, you must map them so that they fit naturally onto their corresponding plots. Finally, you must fit your data to the appropriate curve or surface. This gives rise to the lessons of finding the slope of the tangent lines and finding the intercept of the plotted functions.