Math 1330 Section : Rational Functions Definition: A rational function is a function that can be written in the form f ( x ), where

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1 2.3: Rational Functions P( x ) Definition: A rational function is a function that can be written in the form f ( x ), where Q( x ) and Q are polynomials, consists of all real numbers x such that You will need to be able to find the following: Domain Intercepts Holes Vertical asymptotes Horizontal asymptotes Slant asymptotes Behavior near the vertical asymptotes Domain: The domain of is all real numbers except those values for which. x-intercept(s): All values of x for which. y-intercept: The y intercept of the function is. Holes: The graph of the function will have a hole if there is a common factor in the numerator and denominator. Vertical asymptotes: The graph of the function has a vertical asymptote at any value of x for which and. Example 1: Given the following function., Find the domain, x and y intercepts, hole(s) if any and vertical asymptotes if any. 1

2 Horizontal asymptotes: You can determine if a graph of a function has a horizontal asymptote by comparing the degree of the numerator with the degree of the denominator. Shorthand: degree of f = deg(f), numerator = N, denominator = D 1. If deg(n) > deg(d) then there is no horizontal asymptote. 2. If deg(n) < deg(d) then there is a horizontal asymptote and it is y = 0 (x-axis). 3. If deg(n) = deg(d) then there is a horizontal asymptote and it is, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator. Slant asymptote: The graph of the function may have a slant asymptote if the degree of the numerator is one more than the degree of the denominator. To find the equation of the slant asymptote, use long division to divide the denominator into the numerator. The quotient is the equation of the slant asymptote. Behavior near the vertical asymptotes: The graph of the function will approach either or on each side of the vertical asymptotes. To determine if the function values are positive or negative in each region, find the sign of a test value close to each side of the vertical asymptotes. On the graph this looks like: Example 2: Given the following functions find the vertical and horizontal asymptotes. 2

3 Example 3: Given the following functions find the vertical and horizontal asymptotes. Example 4: Given, find any hole(s), vertical and horizontal asymptotes. Example 5: Find the slant asymptote. 3

4 Graphing rational functions: 1. Factor numerator and denominator. 2. Find x-intercept(s) by setting numerator equal to zero. Note: if a factor cancels, it results in a hole instead of an x-intercept. 3. Find y-intercept (if any) by substituting x=0 into the original form of the function. Note: this is easier if you use the unfactored form. 4. Find horizontal asymptote (if any). There can be at most one horizontal asymptote. 5. Find vertical asymptotes (if any) by setting the denominator equal to zero. Remember: if a factor in the denominator cancels, it results in a hole instead of a vertical asymptote. 6. Use the x-intercepts and vertical asymptotes to divide the x-axis into intervals. Choose a test point in each interval to determine if the function is positive or negative there. This will tell you whether the graph approaches the vertical asymptote in an upward or downward direction. 7. Graph! Except for the breaks at the vertical asymptotes, the graph should be a nice smooth curve with no sharp corners Example 6: Find all of the features of and graph the function. 4

5 Example 7: Find all the features of and graph the function. Example 8: Find all the features of and graph the function. 5

6 Note: It is possible for the graph to cross the horizontal asymptote, maybe even more than once. To figure out whether it crosses (and where), set y equal to the y-value of the horizontal asymptote and then solve for x. Example 9: Does the function cross the horizontal asymptote? a. b. 6