Solve and Graph Rational Functions and Expressions using the Personal Algebra Tutor ®

Main Screen:
Graph Rational Expressions

  1. In Task Box, select
    Rational Expressions - Graph f(x)
  2. In Solution Methods box, select Quadratic Formula
  3. Click Get Graph Data button
Takes you to next screen:
Input Editor screen
Rational Expressions Screen

Input Editor: Enter function.

  1. Click in Input
    text box.
    Enter function.
  2. Click OK button.
Takes you to the PAT Processor screen (not shown) where the function is factored. Then takes you to the:
Next screen
WinPAT Graph Paper

Rational Expressions Screen

Graph Paper screen

  1. Click Plot It button
    to draw graph
  2. Note Factored Expression
  3. Click on the
    Explanation
  4. Print Graph if desired
Takes you to the
Output screen
showing detailed solution
and explanation. Output screen.

Results

  • Function is graphed.
Solve Quadratic Equation Screen

Graphing Explanation screen.

Results

Shows the:
  1. Function is graphed.
  2. Vertical Asymptotes
  3. Domain
  4. Horizontal Asymptotes.
  5. Zeros
  6. Intercepts.
  7. Holes
Solve Quadratic Equation Screen

Graphing Rational Functions


A Rational Function is the ratio of two polynomial functions.

This lesson is to teach how to graph a rational function.

        P(x)
f(x) = -----
        Q(x)
where P(x) and Q(x) are Polynomial Functions. We will use the following to demonstrate the graphing procedure. P(x) = 2x2-10x+12 Q(x) = x2-5x+4 Therefore,
       2x2-10x+12
f(x) = -----------
	x2-5x+4
We could start off by creating a large table of x and f(x) values, but this would be a lot of work,
and some subtle features could be missed. Instead, we will analyze various characteristics of the function. Procedure
  • Factor the Numerator and Denominator.
    	
                      2x2-10x+12     2(x-2)(x-3)
     	f(x) = ------------- = -----------
    		   x2-5x+4        (x-1)(x-4)
    
  • Find the Zeros of the Numerator
    • If any factor of the numerator equals zero, then f(x) = 0.
    • Set each numerator factor to zero.
    • X-2 = 0 x-3 = 0
    • X = 2 x = 3
    • Therefore, at x = 2 and at x = 3, f(x) = 0.
    • Plot the points (2,0) and (3,0). < Action Step 1 >
  • Find the Zeros of the Denominator Function Q(x).
    • If any factor of the denominator of f(x) equals zero, then the function is undefined at that x value.
    • These points determine the Vertical Asymptotes, and are shown on the graph as dashed vertical lines.
    • Set each denominator factor to zero.
    • x-1 = 0 x-4 = 0
    • x = 1 x = 4
    • Therefore, at x = 1 and at x = 4, f(x) is undefined, and these points are not in the Domain.
    • Draw the two Vertical Asymptotes at x = 1 and x = 4. < Action Step 2 >
  • Determine the behavior of f(x) as x --> + infinity and infinity.
    • Here, only the highest order term of the numerator and denominator matter, so at very large values of x,
    • 		         2x2 = 2
      	   f(x) ==>  ______________
      				 x2
      
    • The equation of the horizontal line y = 2 is called the Horizontal Asymptote and is drawn as a dashed line.
    • At very large positive or negative values of x, the graph of this function approaches the Horizontal Asymptote, but never touches it.
    • Draw the two Horizontal Asymptote at y = 2. < Action Step 3 >
  • Significance of the Horizontal and Vertical Asymptotes
    • They define rectangular boundaries for each section of the graph.
    • The graph of a Rational Function cannot ever touch a Vertical Asymptote. In some Rational Functions, the graph may cross a horizontal asymptote.
    • Often, only one point in each region, can show the general shape of the function in that region. Additional points, can be added to more accurately produce the graph.
  • Supplementary Points to Plot.
    • Select values of x in each region defined by the Vertical Asymptotes. Select points between and beyond zeros and vertical asymptotes. Evaluate f(x) at these points, and plot them.
    • Selecting Points:
      • X = 0; f(0) = 12/4 = 3. Plot (0,3)
        • Plot the curve in upper left thru 0,3)
      • X = 5; f(5) = (2*3*2)/(4*1) = 12/4 = 3. Plot (5,3)
        • Plot the curve in upper right thru (5,3)
      • The Region between the two Vertical Asymptotes contains two points, the zeros, but it is not clear what the behavior is yet.
        • Select another point between them. Selecting x = 2.5.
        • Evaluate f(2.5) = (2(2.5-2)(2.5-3))/((2.5-1)(2.5-4))
        • F(2.5) =(2)(0.5)(-0.5)/((1.5)(-1.5) = -0.5/(-2.25) = 0.22
        • Plot (2.5,0.22)
        • Draw curve thru the three points, and using the Vertical Asymptotes as guides.
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