1. Rational Functions An introduction L. Waihman

2. A function is CONTINUOUS if you can draw the graph without lifting your pencil. A POINT OF DISCONTINUITY occurs when there is a break in the graph. Note the break in the graph when x=3. Why?

3. Look at the equation of the graph. Where is this equation undefined? We can factor the numerator and reduce the fraction to determine that the graph will be a line; however, the undefined point point of discontinuity remains, so there is a point of discontinuity here.

4. There are three basic kinds of discontinuity: point, jump, and infinite point, jump, and infinite. The greatest integer function is an example jump discontinuity of a jump discontinuity. Tangent, Cotangent, secant and cosecant infinite functions are all examples of infinite discontinuities discontinuities. The previous function was an example of a point of discontinuity.

5. rational function A rational function is the quotient of at least two polynomials. The graphs of rational functions frequently display infinite and point discontinuities. Rational functions have vertical asymptotes and may have horizontal asymptotes as well.

6. Let’s look at the parent function: If x = 0, then the entire function is undefined. Thus, there is a vertical asymptote at x=0. Looking at the graph, you can see that the value of the function , as the values of x 0 from the positive side; and the value of the function - , as the values of x 0 from the negative side. These are the limits of the function and are written as:

7. Domain The domain is then limited to: The domain is then limited to: To find the domain of a rational function, To find the domain of a rational function, set the denominator equal to zero. set the denominator equal to zero. The domain will always be all real The domain will always be all real numbers except those values found by numbers except those values found by solving this equation. solving this equation.

8. Determine the domain of these rational functions:

9. Recall that a vertical asymptote occurs when there is a value for which the function is undefined. This means, if there are no common factors, anywhere the denominator equals zero.

10. Remember that asymptotes are lines. When you label a vertical asymptote, you must write the equation of the vertical line. Just make x equal everything it couldn’t in the domain. State the vertical asymptotes:

11. Let’s say x is any positive number. As that value increases, the value of the entire function decreases; but, it will never become zero or negative. So this part of the graph will never cross the x-axis. We express this using limit notation as: What if x is a negative number? As that value decreases, the value of the entire function increases; but, it will never become zero or positive. So this part of the graph will never cross the x-axis, either and:

12. Thus, the line is a horizontal asymptote. As x ∞, f(x) 0, and as x  ∞, f(x) 0. Given: is a polynomial of degree n, is a polynomial of degree m, and, 3 possible conditions determine a horizontal asymptote: If n<m, then is a horizontal asymptote. If n>m, then there is NO horizontal asymptote. If n=m, then is a horizontal asymptote, where c is the quotient of the leading coefficients.

13. Horizontal Asymptotes BOBO BOTN EATS D/C Bigger On Bottom y=0 Bigger On Top – None Exponents Are The Same; Divide the Coefficients

14. Find the horizontal asymptote: Exponents are the same; divide the coefficients Bigger on Top; None Bigger on Bottom; y=0

15. Suppose that you were asked to graph: 1 st, determine where the graph is undefined. (Set the denominator to zero and solve for the variable.) 2 nd, find the x-intercept by setting the numerator = to 0 and solving for the variable. So, the graph crosses the x-axis at vertical asymptote There is a vertical asymptote here. Draw a dotted line at:

16. 3 rd, find the y-intercept by letting x=0 and solving for y. 4 th, find the horizontal asymptote. (Note: the exponents are the same so divide the coefficients - EATS D/C) 3/1 = 3 So, the graph crosses the y-axis at The horizontal asymptote is:

17. Now, put all the information together and sketch the graph:

18. Graph: 1 st, find the vertical asymptote. 2 nd, find the x-intercept. 3 rd, find the y-intercept. 4 th, find the horizontal asymptote. 5 th, sketch the graph.

19. Graph: 1 st, factor the entire equation: 2 nd, find the x-intercepts: 3 rd, find the y-intercept: 4 th, find the horizontal asymptote: 5 th, sketch the graph. Then find the vertical asymptotes:

20. Graph: Notice that in this function, the degree of the numerator is larger than the denominator. Thus n>m and there is no horizontal asymptote. However, if n is one more than m, the rational function will have a slant asymptote. To find the slant asymptote, divide the numerator by the denominator: The result is. Notice that as the values of x increase, the fractional part decreases (goes to 0), so the function approaches the line. Thus the line is a slant asymptote.

21. Graph: 1 st, find the vertical asymptote. 2 nd, find the x-intercepts: and 3 rd, find the y-intercept: 4 th, find the horizontal asymptote. none 5 th, find the slant asymptote: 6 th, sketch the graph.