Lesson 3 4 Rational Functions Form of a

Lesson 3. 4 Rational Functions • Form of a rational function • Vertical Asymptotes • Graphing Rational Functions • Horizontal Asymptotes

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. Examples:

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

Example: Find the domain and axis intercepts of the rational function f given by Solution: 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

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.

Vertical Asymptotes The vertical line x = a is a vertical asymptote for the graph of f if Examples: page 152 figure 2. 30. NOTE: If a factor cancels out it becomes a “hole” in the graph. Factors in the denominator that do not cancel out become vertical asymptotes

Example: Sketch the graph of Solution: 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.

To determine where a graph begins, ends etc you can use a sign graph using your simplified expression x+1 x x+2 fcn - - - -0 + + + - - - - - -0 + + + + -----0++++++++ ----- 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.

Horizontal Asymptotes 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.

Example: 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.

Example: Sketch the graph of Solution: 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 x-2 x+2 fcn -------0++++++ ----------0+++ -- 0++++++++ ++0 ---0+++++0 --- 0+++ _______________ -2 -1 0 1 2