Math 1330 Section 2.3 2.3: Rational Functions Definition: A rational function is a function that can be written in the form P(x) f(x) , where f and g are polynomials. Q(x) P(x) The domain of the rational function f(x) consists of all real numbers x Q(x) such that Q(x) 0. 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 f is all real numbers except those values for which Q(x) =0. x-intercept(s): The x intercept(s) of the function will be all values of x for which P(x) 0and Q(x) 0. y-intercept: The y intercept of the function is f(0) . Holes: The graph of the function will have a hole if thee is a like factor in the numerator and denominator. Vertical asymptotes: The graph of the function has a vertical asymptote at any value of x for which Q(x) 0but P(x) 0. 1 Math 1330 Section 2.3 x2 x 6 Example 1: Given f(x) , find the domain, x and y intercepts, x3 2x2 3x hole(s) if any and vertical asymptotes if any. 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. If the degree of the numerator is smaller than the degree of the denominator, then the graph of the function has a horizontal asymptote at y 0 . If the degree of the numerator is equal to the degree of the denominator, then the graph of the function has the horizontal asymptote at y = leading coefficient of the numerator / leading coefficient of the denominator If the degree of the numerator is greater than the degree of the denominator, then the graph of the function does not have a horizontal asymptote. 2 Math 1330 Section 2.3 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. x 4 a. f(x) x2 81 3 Math 1330 Section 2.3 x2 4x b. f(x) x2 16 x3 x2 2x Example 3: Given f(x) , find any hole(s), vertical and x2 2x 3 horizontal asymptotes. x3 2x2 x 2 Example 4: Given f(x) ,find the slant asymptote. x2 4 4 Math 1330 Section 2.3 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. 7. Determine the behavior of the function near the asymptotes: A. For a horizontal asymptote, which side of the line is the function on as x and as x ? B. For a vertical asymptote: for x values close to the asymptote, what is the sign of the function (is it positive or negative?). Check points on BOTH SIDES of the asymptote. C. For a slant asymptote, which side of the asymptote line is the function on? 5 Math 1330 Section 2.3 x 3 Example 5: Find all of the features of f(x) and graph the function. x 2 2x2 8 Example 6: Find all the features of f(x) and use them to x2 3x 2 graph the function. 6 Math 1330 Section 2.3 x2 16 Example 7: Find all the features of f(x) and use them to graph x 2 the function. 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 8: Does the function cross the horizontal asymptote? x2 2x 3 a. f(x) x2 x 6 1 2x b. f(x) x 1 7 Math 1330 Section 2.3 8 .