Graphs of Rational Functions Prepared for Mth 163: Precalculus 1 Online By Richard Gill Through funding provided by a VCCS LearningWare Grant A rational function is a function that can be expressed in the form yпЂЅ f ( x) where both f(x) and g(x) are polynomial functions. g ( x) Examples of rational functions would be: yпЂЅ 1 xпЂ2 f ( x) пЂЅ 2x 3пЂ x x пЂ4 2 g ( x) пЂЅ x пЂ 2x 2 Over the next few frames we will look at the graphs of each of the above functions. First we will look at y пЂЅ 1 xпЂ2 . 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? Think before you click. 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 y пЂЅ 1.5 -2 1 xпЂ2 1.7 -3.33 1.9 -10 2.0 undefined 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! Now fill in the values in the rest of the table. 1 x y пЂЅ 1.5 -2 1.7 -3.33 1.9 -10 2.0 Und 2.1 10 2.3 3.33 2.5 2 xпЂ2 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! LetвЂ™s see what the points that we have calculated so far would look like on graph. y пЂЅ пЂ±пЂ°пЂ®пЂ° 1 This dotted vertical line is a crucial visual aid for the graph. Do you know what the equation of this dotted line is? xпЂ2 y (2.1, 10) пЂёпЂ®пЂ° пЂ¶пЂ®пЂ° (2.3, 3.33) пЂґпЂ®пЂ° пЂІпЂ®пЂ° (2.5, 2) пЂІпЂ®пЂ° пЂпЂІпЂ®пЂ° пЂпЂґпЂ®пЂ° (1.5, -2) (1.7, -3.33) пЂпЂ¶пЂ®пЂ° пЂпЂёпЂ®пЂ° пЂпЂ±пЂ°пЂ®пЂ° (1.9, -10) x пЂґпЂ®пЂ° The equation is x = 2 because every point on the line has an x coordinate of 2. y пЂЅ пЂ±пЂ°пЂ®пЂ° 1 Do you know what this dotted vertical line is called? xпЂ2 y (2.1, 10) пЂёпЂ®пЂ° пЂ¶пЂ®пЂ° (2.3, 3.33) пЂґпЂ®пЂ° пЂІпЂ®пЂ° (2.5, 2) пЂІпЂ®пЂ° пЂпЂІпЂ®пЂ° пЂпЂґпЂ®пЂ° (1.5, -2) (1.7, -3.33) пЂпЂ¶пЂ®пЂ° The line x = 2 is a vertical asymptote. пЂпЂёпЂ®пЂ° пЂпЂ±пЂ°пЂ®пЂ° x пЂґпЂ®пЂ° Hint: it is one of the many great and imaginative words in mathematics. (1.9, -10) yпЂЅ пЂ±пЂ°пЂ®пЂ° Our graph will get closer and closer to this vertical asymptote but never touch it. 1 xпЂ2 y (2.1, 10) пЂёпЂ®пЂ° пЂ¶пЂ®пЂ° (2.3, 3.33) пЂґпЂ®пЂ° пЂІпЂ®пЂ° (2.5, 2) пЂІпЂ®пЂ° пЂпЂІпЂ®пЂ° пЂпЂґпЂ®пЂ° (1.5, -2) (1.7, -3.33) пЂпЂ¶пЂ®пЂ° пЂпЂёпЂ®пЂ° пЂпЂ±пЂ°пЂ®пЂ° (1.9, -10) x пЂґпЂ®пЂ° If f(x) approaches positive or negative infinity as x approaches c from the right or the left, then the line x = c is a vertical asymptote of the graph of f. A horizontal asymptote is a horizontal line that the graph gets closer and closer to but never touches. The official definition of a horizontal asymptote: The line y = c is a horizontal asymptote for the graph of a function f if f(x) approaches c as x approaches positive or negative infinity. Huh?! DonвЂ™t you just love official definitions? At any rate, rational functions have a tendency to generate asymptotes, so lets go back to the graph and see if we can find a horizontal asymptote. y пЂЅ пЂ±пЂ°пЂ®пЂ° 1 xпЂ2 y (2.1,10) пЂёпЂ®пЂ° пЂ¶пЂ®пЂ° (2.3,3.33) пЂґпЂ®пЂ° пЂІпЂ®пЂ° (2.5,2) пЂІпЂ®пЂ° пЂпЂІпЂ®пЂ° пЂпЂґпЂ®пЂ° (1.5,-2) (1.7,-3.33) пЂпЂ¶пЂ®пЂ° пЂпЂёпЂ®пЂ° пЂпЂ±пЂ°пЂ®пЂ° Looking at the graph, as the x values get larger and larger in the negative direction, the y values of the graph appear to get closer and closer to what? (1.9,-10) x If you guessed that the y values appear to get closer and closer to 0, you may be onto something. LetвЂ™s look at a table of values for confirmation. Before you click again, take a minute to calculate the y values in the table below. What is your conclusion about the trend? x y пЂЅ 1 xпЂ2 0 -(1/2) -5 -(1/7) -20 -(1/22) -100 -(1/102) Conclusion: as the x values get closer and closer to negative infinity, the y values will get closer and closer to 0. Question: will the same thing happen as x values get closer to positive infinity? How about a guess? What do you think is going to happen to the y values of our function as the x values get closer to positive infinity? As x п‚® п‚Ґ, y пЂЅ As x п‚® п‚Ґ , yпЂЅ 1 xпЂ2 1 xпЂ2 п‚® ? п‚® 0 By looking at the fraction analytically, you can hopefully see that very large values of x will generate values of y very close to 0. If you are uneasy about this, expand the table in the previous slide to include values like x = 10, 100, or 1000. On the next frame then, is our final graph for this problem Note how the graph is very much dominated by its asymptotes. You can think of them as magnets for the graph. This problem was an exploration but in the future, it will be very important to know where your asymptotes are before you start plotting points. y y = 1/(x-2) Vertical Asymptote at x = 2 пЂґпЂ®пЂ° пЂІпЂ®пЂ° пЂпЂґпЂ®пЂ° пЂпЂІпЂ®пЂ° пЂІпЂ®пЂ° пЂґпЂ®пЂ° пЂ¶пЂ®пЂ° пЂёпЂ®пЂ° x пЂпЂІпЂ®пЂ° пЂпЂґпЂ®пЂ° пЂпЂ¶пЂ®пЂ° Horizontal Asymptote at y = 0. Next up is the graph of one of the functions that was mentioned back in frame #2. f ( x) пЂЅ 2x 3пЂ x LetвЂ™s see if we can pick out the asymptotes analytically before we start plotting points in an x-y table. Do we have a vertical asymptote? If so, at what value of x? We have a vertical asymptote at x = 3 because at that value of x, the denominator is 0 but the numerator is not. Congratulations if you picked this out on your own. The horizontal asymptote is a little more challenging, but go ahead and take a guess. Notice though that as values of x get larger and larger, the 3 in the denominator carries less and less weight in the calculation. f ( x) пЂЅ 2x 3пЂ x As the 3 вЂњdisappearsвЂќ, the function looks more and more likeвЂ¦ f ( x) пЂЅ 2x пЂx which reduces to y = -2. This means that we should have a horizontal asymptote at y = -2. We already have evidence of a vertical asymptote at x = 3. So we are going to set up the x-y table then with a few values to the left of x = 3 and a few values to the right of x = 3. To confirm the horizontal asymptote we will also use a few large values of x just to see if the corresponding values of y will be close to y = -2. 2x x y пЂЅ -5 -10/8 = -1.25 0 0 2.5 5/.5 = 10 3 Undefined 3пЂ x 3.5 7/-.5 = -14 5 10/-2 = -5 10 20/-7 = -2.86 50 100/-47= -2.13 Take a few minutes and work out the y values for this table. DonвЂ™t be lazy now, work them out yourself. As expected, y values tend to explode when they get close to the vertical asymptote at x = 3. Also, as x values get large, y values get closer and closer to the horizontal asymptote at y = -2. The graph is a click away. y y = 2x/(3-x) пЂёпЂ®пЂ° Here is the graph with most of the points in our table. пЂ¶пЂ®пЂ° пЂґпЂ®пЂ° пЂІпЂ®пЂ° пЂпЂ¶пЂ®пЂ° пЂпЂґпЂ®пЂ° пЂпЂІпЂ®пЂ° пЂІпЂ®пЂ° пЂпЂІпЂ®пЂ° пЂґпЂ®пЂ° пЂ¶пЂ®пЂ° пЂёпЂ®пЂ° пЂ±пЂ°пЂ®пЂ° x пЂ±пЂІпЂ®пЂ° Vertical asymptote at x = 3. пЂпЂґпЂ®пЂ° пЂпЂ¶пЂ®пЂ° пЂпЂёпЂ®пЂ° пЂпЂ±пЂ°пЂ®пЂ° пЂпЂ±пЂІпЂ®пЂ° пЂпЂ±пЂґпЂ®пЂ° Horizontal asymptote at y = -2. Believe it or not, you are now sophisticated enough mathematically to draw conclusions about the graph three ways: Analytically: finding asymptotes with algebra!! Numerically: supporting and generating conclusions with the x-y table!! Graphically: a visual look at the behavior of the function. If your conclusions from the above areas do not agree, investigate further to uncover the nature of the problem. We are going to finish this lesson with an analysis of the third function that was mentioned in the very beginning: x пЂ4 2 g ( x) пЂЅ x пЂ 2x 2 This is a rational function so we have potential for asymptotes and this is what we should investigate first. Take a minute to form your own opinion before you continue. Hopefully you began by setting the denominator equal to 0. x пЂ 2x пЂЅ 0 2 xпЂЁx пЂ 2 пЂ© пЂЅ 0 x пЂЅ 0, x пЂЅ 2 It appears that we may have vertical asymptotes at x = 0 and at x = 2. We will see if the table confirms this suspicion. x пЂ4 2 x yпЂЅ x пЂ 2x See anything peculiar? 2 -2 0 -1 -1 -.5 -3 -.1 -19 0 Und 1 3 1.5 2.33 1.9 2.05 2 und Notice that as x values get closer and closer to 0, the y values get larger and larger. This is appropriate behavior near an asymptote. But as x values get closer and closer to 2, the y values do not get large. In fact, the y values seem to get closer and closer to 2. Now, if x =2 creates 0 in the denominator why donвЂ™t we have an asymptote at x = 2? We donвЂ™t get a vertical asymptote at x = 2 because when x = 2 both the numerator and the denominator are equal to 0. In fact, if we had thought to reduce the function in the beginning, we could have saved ourselves a lot of trouble. Check this out: x пЂ4 2 y пЂЅ x пЂ 2x 2 пЂЅ пЂЁ x пЂ« 2 пЂ©пЂЁ x пЂ 2 пЂ© пЂЅ xпЂЁx пЂ 2 пЂ© xпЂ«2 x пЂ4 x 2 Does this mean that yпЂЅ x пЂ 2x 2 and yпЂЅ xпЂ«2 x are identical functions? Yes, at every value of x except x = 2 where the former is undefined. There will be a tiny hole in the graph where x = 2. x пЂ4 2 g ( x) пЂЅ x пЂ 2x 2 As was the case with the previous function, we concentrate on the ratio of the term with the largest power of x in the numerator to the term with the largest power of x in the denominator. As x gets largeвЂ¦ x пЂ4 2 x пЂ 2x 2 п‚» x 2 x 2 пЂЅ1 As we look for horizontal asymptotes, we look at y values as x approaches plus or minus infinity. The denominator will get very large but so will the numerator. You can verify this in the table. x пЂ4 2 x yпЂЅ 10 1.2 100 1.02 1000 1.002 x пЂ 2x 2 So, we have a horizontal asymptote at y = 1. To summarize then, we have a vertical asymptote at x = 0, a hole in the graph at x = 2 and a horizontal asymptote at y = 1. Here is the graph with a few of the points that we have in our tables. x пЂ4 2 g ( x) пЂЅ Hole in the graph. x пЂ 2 xy 2 пЂґпЂ®пЂ° Horizontal asymptote at y = 1. пЂІпЂ®пЂ° пЂпЂґпЂ®пЂ° пЂпЂІпЂ®пЂ° пЂІпЂ®пЂ° пЂпЂІпЂ®пЂ° пЂпЂґпЂ®пЂ° пЂґпЂ®пЂ° x пЂ¶пЂ®пЂ° Vertical asymptote at x = 0. Now you will get a chance to practice on exercises that use the topics that were covered in this lesson: Finding vertical and horizontal asymptotes in rational functions. Graphing rational functions with asymptotes. Good luck and watch out for those asymptotes!