4.4 Rational Functions Objectives: Find the domain of a rational function. Find intercepts, vertical asymptotes, and horizontal asymptotes. Identify holes. Describe end behavior. Sketch complete graphs.

Rational Function A rational function is a function whose rule is the quotient of two polynomials.

Example #1 The Domain of Rational Functions Find the domain of the following rational functions: Set the denominator equal to 0, solve, and exclude those values. A) B)

Intercepts If a rational function has a y-intercept, it occurs at f (0). The x-intercepts occur where the numerator equals 0, not where the denominator equals 0.

Example #2 Intercepts of a Rational Function Find the intercepts of the following rational function: x-intercepts: y-intercept: This can be readily confirmed from a graph.

Holes & Vertical Asymptotes A greater or equal multiplicity in the numerator means a hole, a greater multiplicity in the denominator means an asymptote is located at that zero. *Note*: At least one of the same factor must occur in the numerator and denominator for a hole to exist.

Example #3 Holes & Vertical Asymptotes Use algebra to determine any holes or asymptotes in the graph of the following functions. The factor of (x + 2) has an equal multiplicity in the numerator and denominator so a hole is located where x = −2. The factor of (x – 3) has a greater multiplicity in the denominator, so a vertical asymptote is located where x = 3. The factor of x has an equal multiplicity in the numerator and denominator so a hole is located where x = 0. The factors of (x – 4) & (x + 1) have greater multiplicities in the denominator, so there are vertical asymptotes where x = 4 & x = −1.

Example #3 Holes & Vertical Asymptotes Use algebra to determine any holes or asymptotes in the graph of the following functions. The factor of (x + 6) only occurs in the numerator and not the denominator, so there is a zero where x = −6. The factors of (x + 2) & (x + 6) have greater multiplicities in the denominator, so a vertical asymptote is located where x = −2 & x = −6. The factor of (x + 7) occurs more in the numerator, so there is a hole where x = −7. The factor of (x2 + 4) does not have any real roots, so there are no vertical asymptotes.

Horizontal & Other Asymptotes Graphs of rational functions sometimes may cross horizontal asymptotes but never will cross vertical asymptotes.

Example #4 Graphing Rational Functions List any asymptotes and intercepts for the following rational functions, then sketch a graph. A.

Example #4 Graphing Rational Functions List any asymptotes and intercepts for the following rational functions, then sketch a graph. B.

Example #4 Graphing Rational Functions List any asymptotes and intercepts for the following rational functions, then sketch a graph. C.

Other Asymptotes When the degree of the numerator is greater than the degree of the denominator, then other special asymptotes will shape the graph. To find these special asymptotes, we must divide the numerator by the denominator. The equation of the quotient will approximate the equation of the special asymptotes.

Example #5 Slant Asymptotes Find any special asymptotes and sketch the graph. As x gets very large or very small the remainder approaches 0, so the equation of the slant asymptote is f(x) = x – 5. The original function also has a vertical asymptote at x = -4 and zeros at x = 3 and x = -2.

Example #6 Parabolic Asymptotes Find any special asymptotes and sketch the graph. As x gets very large or very small the remainder approaches 0, so the equation of the parabolic asymptote is f(x) = x2 + 3. The original function also has a vertical asymptote at x = -1 and a y-intercept at -1.

Example #5 A Complete Graph Find a complete graph for the following rational function. A complete graph of a rational function should show all intercepts, vertical asymptotes, and horizontal/special asymptotes. Any holes should also be correctly identified. For this graph, it has a zero at x = -2, vertical asymptotes at x = -4 & x = 3, and a horizontal asymptote at y = 0 since the degree of the denominator is greater than that of the numerator. To find the y-intercept, evaluate f(0).

Example #6 A Complete Graph Find a complete graph for the following rational function. For this function there is a hole at x = 4 since the multiplicities of that factor are equal. There are zeros at x = -2 and x = -1, and a vertical asymptote at x = 3. After the factor of (x – 4) cancels, it equals: This implies the graph has a special asymptote and will require division:

Example #6 A Complete Graph Find a complete graph for the following rational function. Now we can see that there is a slant asymptote at : Finally the y-intercept can be found by evaluating f(0):