rational_functions2 - Graphs of Rational Functions Prepared...

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Graphs of Rational Functions Prepared for Mth 163: Precalculus 1 Online By Richard Gill Through funding provided by a VCCS LearningWare Grant
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A rational function is a function that can be expressed in the form ) ( ) ( x g x f y = where both f(x) and g(x) are polynomial functions. Examples of rational functions would be: x x x x g x x x f x y 2 4 ) ( 3 2 ) ( 2 1 2 2 - - = - = - = Over the next few frames we will look at the graphs of each of the above functions.
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First we will look at This function has one value of x that is banned from the domain. What value of x do you think that would be? And why? If you guessed x = 2, congratulations. This is the value at which the function is undefined because x = 2 generates 0 in the denominator. Consider the graph of the function. What impact do you think this forbidden point will have on the graph? . 2 1 - = x y Think before you click.
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Now just because we cannot use x = 2 in our x-y table, it does not mean that we cannot use values of x that are close to 2. So before you click again, fill in the values in the table below. x 1.5 1.7 1.9 2.0 2 1 - = x y -2 As we pick values of x that are smaller than 2 but closer and closer to 2 what do you think is happening to y? If you said that y is getting closer and closer to negative infinity, nice job! -3.33 -10 undefined
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Now fill in the values in the rest of the table. x 1.5 1.7 1.9 2.0 2.1 2.3 2.5 2 1 - = x y -2 -3.33 -10 Und What about the behavior of the function on the other side of x = 2? As we pick values of x that are larger than 2 but closer and closer to 2 what do you think is happening to y? If you said that y is getting closer and closer to positive infinity, you are right on the money! 10 3.33 2
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Let’s see what the points that we have calculated so far would look like on graph. 2 1 - = x y The equation is x = 2 because every point on the line has an x coordinate of 2. This dotted vertical line is a crucial visual aid for the graph. Do you know what the equation of this dotted line is? 2.0 4.0 -10.0 -8.0 -6.0 -4.0 -2.0 2.0 4.0 6.0 8.0 10.0 x y (2.1, 10) (2.3, 3.33) (2.5, 2) (1.5, -2) (1.7, -3.33) (1.9, -10)
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2 1 - = x y 2.0 4.0 -10.0 -8.0 -6.0 -4.0 -2.0 2.0 4.0 6.0 8.0 10.0 x y (2.1, 10) (2.3, 3.33) (2.5, 2) (1.5, -2) (1.7, -3.33) (1.9, -10) Hint: it is one of the many great and imaginative words in mathematics.
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rational_functions2 - Graphs of Rational Functions Prepared...

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