5.2 Rational Functions and Asymptotes MAT 204 SPRING 2009 5.2 Rational Functions and Asymptotes In this section, we will study the following topics: Graphs of rational functions Finding domains of rational functions Finding horizontal, vertical, and oblique asymptotes of rational functions

What is a Rational Function? MAT 204 SPRING 2009 What is a Rational Function? Like a rational number, which is a number that can be expressed as a ratio of two integers (a fraction), a RATIONAL FUNCTION IS A RATIO OF TWO POLYNOMIALS IN TERMS OF A SINGLE VARIABLE. E.g. Note that the value of the denominator cannot equal zero since division by zero is undefined.

The Domain of a Rational Function The domain of a rational function of x includes all real numbers EXCEPT those values of x that make the denominator zero. TO FIND THE DOMAIN OF A RATIONAL FUNCTION, set the denominator equal to zero and solve for x to determine which x-value(s) must be EXCLUDED from the domain. Example* Find the domain of each function. Solution: f(x) is undefined when x2 – 7x + 6 = ___ g(x) is undefined when x = ___

The Graph of a Rational Function: Vertical Asymptotes Example: The numerator must be placed inside parentheses since the entire expression is being divided by x. Notice that the graph has two branches. Also, notice how the graph seems to get closer and closer to the vertical line x = 0 but never touches it. Remember that f(x) is not defined at x = 0. The vertical line that the graph approaches is called a _____________________ ____________________.

The Graph of a Rational Function: Horizontal Asymptotes Example (continued): Also notice that the graph seems to approach an invisible horizontal line as x heads towards positive and negative infinity. This line is called the _____________________ ___________________. Horizontal asymptotes of rational functions can be found algebraically, but for now we will use our graphing calculators to find it numerically.

The Graph of a Rational Function: Horizontal Asymptotes Example (continued): One way to do this is to use your table. Look at the table on the left. I changed TABLE SETUP to Indpnt: Ask instead of Auto. Now I can enter whatever values of x I would like into the function. I want to see what happens as x gets very large and as x gets very small.

The Graph of a Rational Function: Vertical and Horizontal Asymptotes Example (continued): First I will look at increasingly larger values of x (heading towards ) to see what the functional value (the y-value) is getting closer to. As I enter larger and larger values of x, the y-value seems to get closer and closer to 2 (from above). Now I will look at what happens as x gets increasingly smaller (heading towards - ). As I entered smaller and smaller values of x, the y-value again seems to get closer and closer to 2 (from below). The horizontal asymptote in this example is the line _____________.

The Graph of a Rational Function: Range Example (continued): Look back at the graph of f(x). If you picture a dotted horizontal line passing through at y = 2, you can see how the branches of the graph get really, really close to this line as x gets very large (+) and as x gets very small (-), one from above and one from below, but they never reach the line. We can also see by this graph that as x gets closer to zero from the right, the y-values get larger and head towards . As x gets closer to zero from the left, the y-values get smaller and head towards -  . From this, we can determine the range for f(x). The RANGE would be the set of all real numbers except 2, which in interval notation would be ____________________________.

The Graph of a Rational Function: Vertical and Horizontal Asymptotes Example Use the graph of the functions to locate any vertical and horizontal asymptotes. a) b)

Finding Vertical Asymptotes Algebraically The vertical asymptotes of a rational function in simplest form exist at the x-values for which the function is undefined. (Remember, the function is undefined at the x-values for which the denominator is equal to zero.) So, to find the vertical asymptotes algebraically, set the denominator equal to zero and solve for x. (We used this technique when we found the domain of the function.)

Finding Vertical Asymptotes Algebraically Example* Solution: The function is undefined when x = _____ or x = _____ Therefore, the domain of f(x) is ________________________ The vertical asymptotes are the lines ________ and _________

Horizontal Asymptotes of a Rational Function MAT 204 SPRING 2009 Finding Horizontal Asymptotes Algebraically To determine the horizontal asymptote, we compare the degrees of the numerator and denominator of the rational function. Horizontal Asymptotes of a Rational Function Let f(x) be the rational function where n is the degree of N(x) and d is the degree of D(x) If n < d, the horizontal asymptote of the graph of f is the line y = 0. If n = d, the horizontal asymptote of the graph of f is the line is If n > d, the graph of f has NO HORIZONTAL ASYMPTOTE. (The graph may have an oblique asymptote…more on this soon.)

Finding Horizontal Asymptotes Algebraically MAT 204 SPRING 2009 Finding Horizontal Asymptotes Algebraically Example Identify any horizontal asymptotes of the graphs of the functions. a) b) c)

Choose the graph of the equation: B) C) D)

Oblique (Slant) Asymptotes We just learned that if the degree of the numerator is greater than the degree of the denominator, the graph has no horizontal asymptote. Taking this one step further, if the degree of the numerator is exactly ONE greater than the degree of the denominator, the graph has an OBLIQUE (SLANT) ASYMPTOTE. Example: oblique asymptote The graph of f has an oblique asymptote since the degree of the numerator is 2 and the degree of the denominator is 1. vertical asymptote

MAT 204 SPRING 2009 Oblique Asymptotes To find the equation of an oblique asymptote, you use LONG DIVISION to divide the numerator by the denominator. The quotient will give you the equation for the slant asymptote (Ignore the remainder.) The equation of the oblique asymptote will have the form y = mx + b. Note: A graph of a rational function that has an oblique asymptote will not have a horizontal asymptote but may have a vertical asymptote. You sketch the graph of a rational function with an oblique asymptote following the same guidelines I listed before, with the addition of using a dashed line to draw the oblique asymptote.

Oblique Asymptotes Example MAT 204 SPRING 2009 Oblique Asymptotes Example In the previous example, the oblique asymptote is found by dividing numerator by denominator: The quotient gives the equation of the oblique asymptote, namely, y = 2x – 11. This is how you write your answer.

Quote of the Day: “Rational functions are a pain in the asymptote!” MAT 204 SPRING 2009 End of Section 5.2 Quote of the Day: “Rational functions are a pain in the asymptote!”