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Published byArthur Nicholson Modified over 4 years ago

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**Rational Functions I; Rational Functions II – Analyzing Graphs**

Section 5.2 Section 5.3 Rational Functions I; Rational Functions II – Analyzing Graphs

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RATIONAL FUNCTION A rational function R is a function that can be written as where P and Q are polynomials. NOTE: Polynomials are rational functions.

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ASYMPTOTES Many, though not all, rational functions have asymptotes. The two most common types of asymptotes are: vertical asymptotes horizontal asymptotes

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**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.

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CONTINUOUS FUNCTIONS A 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.

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**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.

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MISSING POINT A 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.

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**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.

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**FINDING HORIZONTAL ASYMPTOTES**

Let be 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.

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OBLIQUE ASYMPTOTES An 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.

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**FINDING THE OBLIQUE ASYMPTOTE**

To find the oblique asymptote, perform long division of polynomials. The oblique asymptote will be y = quotient. NOTE: The remainder is not used in finding the oblique asymptote.

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**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.

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**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.

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