2 RATIONAL FUNCTIONA rational function R is a function that can be written aswhere P and Q are polynomials.NOTE: Polynomials are rational functions.
3 ASYMPTOTESMany, though not all, rational functions have asymptotes. The two most common types of asymptotes are:vertical asymptoteshorizontal asymptotes
4 FINDING THE DOMAIN OF A RATIONAL FUNCTION The domain of a rational function is the set of all x-values that do not make the denominator zero.
5 CONTINUOUS FUNCTIONSA function is continuous if “it can be drawn without lifting your pencil from your paper.”A place where you must lift your pencil is called a discontinuity.For rational functions, discontinuities can be found by finding where the denominator is equal to zero.
6 VERTICAL ASYMPTOTES All vertical asymptotes occur at discontinuities. Theorem: A rational function R(x) = p(x)/q(x), in lowest terms, will have a vertical asymptote x = r if r is a real zeros of the denominator q. That is, if x − r is a factor of the denominator q of a rational function R(x) = p(x)/q(x), in lowest terms, the R will have a vertical asymptote x = r.NOTE: A graph can never cross a vertical asymptote.
7 MISSING POINTA missing point is another type of discontinuity. A missing point is also called “a hole-in-the-graph.”Theorem: A rational function R(x) = p(x)/q(x), that is not in lowest terms, will have a missing point at x = r if r is a real zero of the denominator q but is not a real zero of denominator after R has been put into lowest terms.
8 HORIZONTAL ASYMPTOTES A horizontal asymptote is a type of end behavior for rational functions.The line y = c is a horizontal asymptote of a rational function R if as the x-values get very small or very large, the y-values get close to c.NOTE: A graph may cross a horizontal asymptote.
9 FINDING HORIZONTAL ASYMPTOTES Letbe a rational function.y = 0 is the horizontal asymptote if n < m.y = an/bm is the horizontal asymptote if n = m.there is no horizontal asymptote if n > m.
10 OBLIQUE ASYMPTOTESAn oblique (or slant) asymptote is another type of end behavior for rational functions. Instead of the ends approaching a horizontal line, the ends approach a slanted line.Oblique asymptotes occur when the degree of the numerator is exactly one larger than the degree of the denominator.
11 FINDING THE OBLIQUE ASYMPTOTE To find the oblique asymptote, perform long division of polynomials. The oblique asymptote will bey = quotient.NOTE: The remainder is not used in finding the oblique asymptote.
12 ANALYZING THE GRAPH OF A RATIONAL FUNCTION Step 1: Find the domain of the rational function.Step 2: Locate the intercepts, if any, of the graph.Step 3: Test for symmetry.Step 4: Write R in lowest terms and find the real zeros of the denominator. This will determine the vertical asymptotes.Step 5: Locate the horizontal or oblique asymptotes, if any.
13 ANALYZING RATIONAL FUNCTIONS (CONCLUDED) Step 6: Determine where the graph is above the x-axis and where the graph is below the x-axis, using the zeros of the numerator and denominator to divided the x-axis into intervals.Step 7: Graph the asymptotes, if any, found in Steps 4 and 5. Plot the points found in Steps 2, 5, and 6. Use all the information to connect the points and graph R.