The graphical method

Examples of The graphical method in the following topics:

• Solving Systems Graphically

  • This is the graphical method.
  • Shown graphically, a set of equations solved with only one set of answers will have only have one point of intersection, as shown below.
  • Before successfully solving a system graphically, one must understand how to graph equations written in standard form, or $Ax+By=C$.
  • You can always use a graphing calculator to represent the equations graphically, but it is useful to know how to represent such equations formulaically on your own.
  • This is an example of a system of equations shown graphically that has two sets of answers that will satisfy both equations in the system.

• Solving Systems of Linear Inequalities

  • A system of inequalities can be solved graphically and non-graphically.
  • When using the graphical method for two variables, first plot all of the lines representing the inequalities, drawing a dotted line if it is either < or >, and a solid line if it is either $\leq$ or $\geq$.
  • This is referred to as the non-graphical method.
  • The non-graphical method is much more complicated, and is perhaps much harder to visualize all the possible solutions for a system of inequalities.
  • However, when you have several equations or several variables, graphing may be the only feasible method.

• Adding and Subtracting Vectors Graphically

  • The head-to-tail method of vector addition requires that you lay out the first vector along a set of coordinate axes.
  • Since vectors are graphical visualizations, addition and subtraction of vectors can be done graphically.
  • The graphical method of vector addition is also known as the head-to-tail method .
  • To subtract vectors the method is similar.
  • The head-to-tail method of vector addition requires that you lay out the first vector along a set of coordinate axes.

• Statistical Graphics

  • Statistical graphics are used to visualize quantitative data.
  • In addition, the choice of appropriate statistical graphics can provide a convincing means of communicating the underlying message that is present in the data to others.
  • If one is not using statistical graphics, then one is forfeiting insight into one or more aspects of the underlying structure of the data.
  • Statistical graphics have been central to the development of science and date to the earliest attempts to analyse data.
  • Since the 1970s statistical graphics have been re-emerging as an important analytic tool with the revitalisation of computer graphics and related technologies.

• A Graphical Interpretation of Quadratic Solutions

  • The roots of a quadratic function can be found algebraically or graphically.
  • These are two different methods that can be used to reach the same values, and we will now see how they are related.
  • Notice that these are the same values that when found when we solved for roots graphically.
  • Solve graphically and algebraically.
  • We have arrived at the same conclusion that we reached graphically.

• Appraisal methods

  • The following outlines some of the more commonly used methods, as well as some recently developed ones that can be useful for various feedback situations
  • Graphic rating scales: This method involves assigning some form of rating system to pertinent traits.
  • Behavioral methods: A broad category encompassing several methods with similar attributes.
  • These methods identify to what extent an employee displays certain behaviors, such as asking a customer to identify the usefulness of a sales representative's recommendation.
  • Significant planning will be required to develop appropriate methods for each business unit in an organization in order to obtain maximum performance towards the appraisal goals.

• Zeroes of Linear Functions

  • Graphically, where the line crosses the $x$-axis, is called a zero, or root.  
  • Zeros can be observed graphically.  
  • To find the zero of a linear function, simply find the point where the line crosses the $x$-axis.
  • The zero from solving the linear function above graphically must match solving the same function algebraically.
  • This is the same zero that was found using the graphing method.

• Guidelines for Plotting Frequency Distributions

  • These frequencies are often graphically represented in histograms.
  • A histogram is a graphical representation of tabulated frequencies , shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval.
  • The height of a rectangle is also equal to the frequency density of the interval, i.e., the frequency divided by the width of the interval.
  • Some theoreticians have attempted to determine an optimal number of bins, but these methods generally make strong assumptions about the shape of the distribution.
  • Define statistical frequency and illustrate how it can be depicted graphically.

• Inconsistent and Dependent Systems in Three Variables

  • Graphically, the ordered triple defines a point that is the intersection of three planes in space.
  • Graphically, the solutions fall on a line or plane that is the intersection of three planes in space.
  • Graphically, a system with no solution is represented by three planes with no point in common.
  • Or two of the equations could be the same and intersect the third on a line (see the example problem for a graphical representation).
  • Using the elimination method for solving a system of equation in three variables, notice that we can add the first and second equations to cancel $x$:

• Supply Function

  • A supply function is a model that represents the behavior of the producers and/or sellers in a market.
  • The relationship between the quantity produced and offered for sale and the price reflects opportunity cost.
  • Generally, it is assumed that there is a positive relationship between the price of the good and the quantity offered for sale.
  • Figure III.A.5 is a graphical representation of a supply function.
  • The equation for this supply function is Qsupplied= -10 + 2P.