Unit+4

1. Create a rational function whose vertical asymptotes add to zero and whose zeros add to zero. Describe the asymptote behavior and end behavior of the function you created using limit notation.  (x^2 – 4)/(x^2 – 9) As x approaches -3-, y approaches positive infinity. As x approaches -3+, y approaches negative infinity. As x approaches 3-, y approaches negative infinity. As x approaches 3+, y approaches positive infinity. As x approaches positive infinity, y approaches 1. As x approaches negative infinity, y approaches 1.
 * __Unit 4 Lesson 2 __**

2. True or false: A rational function has a vertical asymptote at x = c every time c is a zero of the denominator. If the statement is false justify your answer using mathematical terminology learned in class and examples of at least 2 functions that make this statement false.  False. For example, if the function is x – 2/(x^2 + x – 6) would make the statement false. (2, 0) is a 0 of this function, but there would be a hole at (2, 0) and x = 2 not a vertical asymptote, since y doesn’t go onto infinity in either direction. Another example would be x – 1/(x^2 – 1). There would be a hole at (1, 0) but x = 1 is not a vertical asymptote.

3. Describe how the graph of a nonzero rational function f(x) = (ax+b)/(cx+d) can be obtained from the graph y = 1/x. You can obtain f(x) from y = 1/x. In this case, you can divide ax + b by cx +d, and put your answer into fraction form of (remainder)/(x – h) + k (quotient). Depending on the remainder, you’ll have to stretch or shrink the hyperbolas. To find the vertical asymptote, it will be x = h, from the obtained fraction form, and the k will be the horizontal asymptote. Then you can draw in the hyperbolas in the same quadrants as 1/x, but they might be switched if there is a reflection. To get a more accurate sketch, you can find the y intercept by replacing x with 0 and finding the solution, and you can find the x intercept by setting that fraction form to 0 and solving for x.

4. Write a rational function with the following properties: (a) Vertical asymptotes: x = -5 and x = 2. (b) Horizontal asymptote: y = -3. (c) // y //-intercept 1. (-3x – 10)/(x^2 + 3x – 10)