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.

• Solving Systems of Equations in Three Variables

  • Graphically, the solution is where the functions intersect.
  • The graphical method of solving a system of equations in three variables involves plotting the planes that are formed when graphing each equation in the system and then finding the intersection point of all three planes.
  • After that smaller system has been solved, whether by further application of the substitution method or by other methods, substitute the solutions found for the variables back into the first right-hand side expression.
  • Elimination by judicious multiplication is the other commonly-used method to solve simultaneous linear equations.
  • Using the elimination method, begin by subtracting the first equation from the second and simplifying:

• The Substitution Method

  • The substitution method is a way of solving a system of equations by expressing the equations in terms of only one variable.
  • The substitution method for solving systems of equations is a way to simplify the system of equations by expressing one variable in terms of another, thus removing one variable from an equation.
  • In the first equation, solve for one of the variables in terms of the others.
  • Thus, the solution to the system is: $(-2, -1)$, which is the point where the two functions graphically intersect.
  • Check the solution by substituting the values into one of the equations.

• 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.

• 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.

• 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 $x$:

• Inconsistent and Dependent Systems in Two Variables

  • We can apply the substitution or elimination methods for solving systems of equations to identify dependent systems.
  • We can apply the elimination method to evaluate these.
  • Recall that the graphical representation of an inconsistent system consists of parallel lines that have the same slope but different $y$-intercepts.
  • We can apply the elimination method to attempt to solve this system.
  • Explain when systems of equations in two variables are inconsistent or dependent both graphically and algebraically.

• Transformations of Functions

  • A transformation takes a basic function and changes it slightly with predetermined methods.  
  • This can be achieved by switching the sign of the input going into the function.  
  • Where $x_1$and $y_1$are the new expressions for the rotated function, $x_0$ and $y_0$ are the original expressions from the function being transformed, and $\theta$ is the angle at which the function is to be rotated.  
  • Scaling is a transformation that changes the size and/or the shape of the graph of the function.  
  • Note that until now, none of the transformations we discussed could change the size and shape of a function - they only moved the graphical output from one set of points to another set of points.  

• Nonlinear Systems of Equations and Problem-Solving

  • As with linear systems, a nonlinear system of equations (and conics) can be solved graphically and algebraically for all its variables.
  • The four types of conic section are the hyperbola, the parabola, the ellipse, and the circle; the circle is a special case of the ellipse.
  • If each equation is graphed, the solution for the system can be found at the point where all the functions meet.
  • The solutions for x can then be plugged into either of the original systems to find the value of y.
  • The parabola (blue) falls below the line (red) between x=-2 and x=3.