2. A rational function is the quotient of two polynomials Rational Functions: A rational function has the form where P(x) and Q(x) are polynomials. The domain of f is the set of all real numbers x with Q(x) ≠ 0.

3. If there are common factors between your two polynomials you can factor and cancel these factors and write the fraction in simplest form. However, we cannot forget that x ≠ 3. This will introduce a “hole.”

4. Find the domain and axis intercepts of the rational function f given by By inspection of the denominator we can see that x ≠ 2. Factoring the rational function gives us Substitution of 0 for x gives us the y intercept which is: (0, f(0)) = (0, 1) The denominator of a rational function (after simplified) will tell us where the vertical asymptotes will appear. X ≠ 2 therefore x = 2 is a vertical asymptote

5. The x-intercepts are determined from your numerator of your rational function (x+2)(x-1) set each factor equal to zero and solve x = -2, 1 A t-table, multiplicity, sign graph or substitution of a couple of values from the domain into the function will help determine where the graph begins with respect to the x axis.

6. Vertical Asymptotes The vertical line x = a is a vertical asymptote for the graph of f if Examples: page 152 figure 2.30.

7. Sketch the graph of Factor: Since (x-1) cancelled out it will be represented by a “hole” in your graph. The denominator determines the vertical asymptotes: x = 0 and x = -2 The numerator of the simplified expression determines the x-intercept: x = -1 The y intercept is: none – since the y-axis is a vertical asymptote if you tried to substitute a zero in for x it would be undefined.

8. To determine where a graph begins, ends etc you can use a sign graph using your simplified expression x + 1 - - - - - - - - - - - - - - -0 + + + + + + + + + + + + x - - - - - - - - - - - - - - - - - - - - -0 + + + + + + + + x + 2 - - - - - - - - - 0 + + + + + + + + + + + + + + + + fcn - - - - - - - - - 0 + + 0 - - - - - 0 + + + + + + + + ______________________________________ -2 -1 0 In the function row the “-” signs indicate that the graph will lie below the x-axis on that interval. The “+” signs indicate that the graph will lie above the x-axis on that interval. Bringing all of your data together you are now ready to graph your function. Graph together on white board.

9. The horizontal line y = a is a horizontal asymptote to the graph of f if A rational function can have at most one horizontal asymptote Determining if/where you have a horizontal asymptote: If the degree of the polynomial in the numerator is equal to the degree of the polynomial in the denominator then you have a horizontal asymptote at the ratio of the leading coefficients If the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator then you have a horizontal asymptote at y = 0. If the degree of the polynomial in the numerator is more than the degree of the polynomial in the denominator then you do not have a horizontal asymptote.

10. Determine if and where a horizontal asymptote exists in the following examples: Degrees are the same: y = -3/7 Degree in the denominator is greater than that of the numerator: y = 0 Degree in the numerator is greater than that of the denominator: no horizontal asymptote. Degree in the numerator is greater than that of the denominator: no horizontal asymptote.

11. Sketch the graph of Since the degrees of both polynomials are the same there is a horizontal asymptote at y = 1 vertical asymptotes: x = -2, 2 x – intercepts: -1, 1 x+1 - - - - - - - 0 + + + + + + + + + + + + x-1 - - - - - - - - - - - - - - - 0 + + + + + + x-2 - - - - - - - - - - - - - - - - - - - - 0 + + + x+2 - - 0 + + + + + + + + + + + + + + + fcn + + 0 - - - 0 + + + + + 0 - - - 0 + + + ______________________________ -2 -1 0 1 2