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 * Lesson 2**

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.

f(x) = x / ( x^2- 4 ); zeros = 0, vertical asymptotes = +2, -2 lim x --> -2-; f(x) --> -inf lim x --> -2+;f(x) --> +inf lim x --> 2-; f(x) --> -inf lim x --> 2+; f(x) --> +inf

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; f(x) = (x-2)/(x-4) has zeros 2 and 4, but it does not have a vertical asymptote at 2. Similarly, f(x) = (x+1)/(x-3) doesn't have a vertical asymptote at its zero -1.

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.

Start by applying polynomial division to f(x) to receive a fraction in the form of 1/(x-a) + b. The a value indicates how many units to the right you shift the graph; the b value indicates how many units up or down the graph is shifted. For example, 1/(x-2) + 5 would indicate a graph of 1/x shifted 2 units to the right and 5 units up.

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.

f(x) = -10/(x^2+3x-10) - 3