2. HW: Handout due at the end of class Wednesday. Do Now: Take out your pencil, notebook, and calculator. 1)Sketch a graph of the following rational function. What are the asymptotes? What is the domain and range? Objectives: You will be able to find and determine the point of discontinuity algebraically and graphically. You will be able to find the vertical and horizontal asymptotes algebraically and graphically. Agenda: 1.Do Now 2.Hw Questions 3.Graphing rational functions day 3 What are the key features of a rational function? Tuesday, March 17, 2015

3. Properties of Rational Functions Definition: Rational Function The functions p and q are polynomials. The domain of a rational function is the set of all real numbers except those values that make the denominator, q(x), equal to zero.

4. Example 1: Domain of a Rational Function

5. Example 2: Domain of a Rational Function

6. Example 3: Domain of a Rational Function

7. Example 4: Domain of a Rational Function

8. Linear Asymptotes Lines in which a graph of a function will approach. Vertical Asymptote A vertical asymptote exists for any value of x that makes the denominator zero AND is not a value that makes the numerator zero. Example Asymptotes A vertical asymptotes exists at x = -5.

9. Vertical Asymptote Example Asymptotes A vertical asymptote does not exist at x = 3 as it is a value that also makes the numerator zero. A hole exists in the graph at x = 3.

10. Horizontal Asymptote A horizontal asymptote exists if the largest exponents in the numerator and the denominator are equal, Properties of Rational Functions If the largest exponent in the denominator is equal to the largest exponent in the numerator, then the horizontal asymptote is equal to the ratio of the coefficients. or if the largest exponent in the denominator is larger than the largest exponent in the numerator.

11. Asymptotes Horizontal Asymptote Example Properties of Rational Functions A horizontal asymptote exists at y = 0. A horizontal asymptote exists at y = 5/2.

12. Continuous vs. Discontinuous Rational Functions What value of x makes the denominator 0? Since, there is no value we call this a continuous graph. CONTINUOUS GRAPH-a graph with no jumps, breaks, or holes. (You can draw the graph and your pencil never leaves the paper). What value of x makes the denominator 0? Since, x cannot be -2, therefore the graph is discontinuous. DISCONTINUOUS GRAPH-a graph with a jump, break or hole or a combination of them. (You have to lift your pencil off your paper to draw the graph)

13. Points of Discontinuity A point on the graph for which the denominator of a rational function is zero. Removable Discontinuity- a hole in the graph. Example: Non Removable Discontinuity-an asymptote. Removable discontinuity at x=2 because you can remove the point of discontinuity by canceling out the (x-2)’s. If you can cancel it out then there is a hole at that point of discontinuity. Non-removable discontinuity at x=2 because you can’t manipulate the function to cancel out the point of discontinuity so therefore you have an vertical asymptote at x=2.

14. Examples Find the domain and points of discontinuity for each function. Determine what type of discontinuity, the x and y intercepts, and asymptotes.

15. Graphing a rational function