Rational Functions 4.5
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1 Math 4 Pre-Calculus Name Date Rational Function Rational Functions 4.5 g ( ) A function is a rational function if f ( ), where g ( ) and ( ) h ( ) h are polynomials. Vertical asymptotes occur at -values for which the denominator h ( ) equals zero, but the numerator g ( ) zero for these same -values. Holes occur at the -values for which both the denominator h ( ) and the numerator ( ) the hole by evaluating the function at that -value after canceling out the common factors. Non-vertical asymptotes depend on the degrees of the denominator h ( ) and the numerator ( ) The range depends on the global behavior which can be determined by the asymptotes. 0 does not equal g equal zero. Find the y-value of g. Plug in 0 for to find y. In other words, find f ( ). The y-intercept is a point: ( 0,?). Find by setting y 0 and solving for. The zeros for a rational function are the zeros of the numerator (ecept those that are holes.) The -intercept(s) is(are) a point(s): (?, 0 ). Vertical Asymptotes The line a is a vertical asymptote for the graph of a function f if f ( ) or ( ) f as approaches a from either the left or the right. The graph will never cross a vertical asymptote. The multiplicity of the factor that causes a vertical asymptote determines the behavior on either side of that vertical asymptote. If the multiplicity is odd the graph behaves opposite on either side. If the multiplicity is even, the graph behaves the same on either side. Horizontal Asymptotes The line y c is a horizontal asymptote for the graph of a function f if ( ) may cross a horizontal asymptote, to find out if and where, set f ( ) c and solve for. f c, as or as. The graph Oblique Asymptotes The line y a + b is an oblique asymptote for the graph of a function f if f ( ) a + b, as or as. Rational Functions Bottom Degree Superior The degree of the denominator h ( ) is larger than the degree of the numerator g ( ) Non-vertical asymptotes (Horizontal): y 0 In mathematical terms, this means: as, f ( ) 0 and as f ( ), 0. Rational Functions Same Degree The degree of the denominator h ( ) is the same as the degree of the numerator g ( ) Non-vertical asymptotes (Horizontal): y r a t i o o f t h e l e a d i n g c o e f f i c i e n t s In mathematical terms, this means: as, f ( ) r a t i o and as f ( ), r a t i o. Rational Functions Top Degree Superior The degree of the denominator h ( ) is smaller than the degree of the numerator ( ) g. In this class we will only look at cases where the denominator s degree is smaller than the numerator s degree by one. Non-vertical asymptotes (Oblique): y a + b, where a + b is the quotient of the numerator divided by the denominator. In mathematical terms, this means: as, f ( ) a + b and as, ( ) f a + b.
2 1. Graph the equation. f ( ) 3. Graph the equation. f ( ) 3 3 9
3 3. Graph the equation. f ( ) Graph the equation. f ( ) ( 3)
4 5. Graph the equation. f ( ) 3 6. Graph the equation. f ( ) 3 1
5 Find an equation of a rational function f that satisfies the given conditions. 7. vertical asymptotes:, 4 horizontal asymptotes: y 0 -intercept: ( 1, 0 ) (in the book, this will be written as : -intercept: 1) f ( 0) 8. vertical asymptotes: 0, 1 horizontal asymptotes: 1 y -intercept: ( 1, 0 ) (in the book, this will be written as : -intercept: -1) hole at: 3
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