Section 5.2 (Sullivan, 8th ed.MAC1140/MAC11475.2: RATIONAL FUNCTIONSA rational function R(x) is a quotient of two polynomials p(x)/q(x). The domain consists of all real numbers except those for which the denominator q = 0. An asymptote is a line to which a certain part of the graph of a function gets closer and closer but never touches. A rational function in lowest terms will have a vertical asymptote at any value of x that would cause the denominator q(x) = 0. Proper rational functions, where the degree of the numerator is less than that of the denominator, will have a horizontal asymptote at y = 0. If the degree of the numerator is greater than or equal to that of the denominator, the horizontal or oblique asymptote can be found by long division; it will be the quotient without the remainder. The real zeros of a rational function are the values for which the numerator p(x) = 0. Graph using transformations; determine domain, range, asymptotes, and real zeros from the graph:1. R xx212.   R xx2123.   R xx 21224.  R xx 2122Domain: Domain: Domain: Domain: Range: Range: Range: Range: V.A.: V.A.: V.A.: V.A.:H. A.: H. A.: H. A.: H. A.:Zeros: Zeros: Zeros: Zeros: Find the domain, any vertical, horizontal, or oblique asymptotes, and the real zeros:Domain Asymptotes Zeros5.f xxx x( )( )( )  2 43 4 422 6.23 22 2( )2 6x xf xx x- +=+7.3 226 2 12( )3 5 2x x xf xx x+ +=- -Section 5.3 (Sullivan, 8th ed.MAC1140/MAC11478. Acceleration due to gravity, g, at a height h above sea level is given by: ( )( )14263.99 106.374 10g hh�=� +. Compare the acceleration due to gravity in Miami (sea level) to that in La Paz, Bolivia (3600 meters).Section 5.3 (Sullivan, 8th ed.MAC1140/MAC11475.3 Rational Functions II: Analyzing GraphsSteps for analyzing the graph of a rational function: [1] Find the domain before reducing it. [2]Reduce and locate any intercepts. X-intercepts are the real (not imaginary) zeros of the numerator: ifthe zero comes from a factor of odd multiplicity (odd exponent) the graph will cross the x-axis; if itcomes from a factor of even multiplicity it will touch and turn at that point. Y-intercepts, if any, areR(0). [3] Test for symmetry: If R(x) = R(-x) there is y-axis symmetry; if R(-x) = -R(x), it has originsymmetry. [4] Locate vertical asymptotes by finding the real zeros of the denominator: if the zerocomes from a factor of odd multiplicity, the graph on opposite sides of the asymptote will approachinfinity in opposite directions; if it comes from a factor of even multiplicity, it will approach infinity inthe same direction. [5] Find horizontal or oblique asymptotes (see Section 4.3); determine the points, ifany, at which the graph intersects these asymptotes by setting the function equal to the asymptote andsolving the equation. [6] Find other points, using the zeros of the numerator and denominator to dividethe x-axis into intervals. [7] Graph the asymptotes (steps 4 and 5) and plot the points (steps 2, 5, and 6)and connect the points using all the information.Analyze each of the following rational functions and graph:1. ( )229xR xx-=-2. ( )228 26 152 15x xR xx x+ +=- -DomainInterceptsSymmetryAsymptotes: Vertical Horizontal or ObliqueOther Points:Section 5.3 (Sullivan, 8th ed.MAC1140/MAC11473. 328( )2xf xx+=4. 428( )2xf xx+=DomainInterceptsSymmetryAsymptotes: Vertical Horizontal or ObliqueOther Points:Make up a rational function that might have the given graph:5. 6. 7.Vertical asymptotes Vertical asymptotes Vertical asymptotes Horizontal asymptote Oblique asymptote Horizontal asymptote X-intercepts X-intercepts X-intercepts.Section 5.3 (Sullivan, 8th ed.MAC1140/MAC1147( )R x =( )R x =( )R x =Section 5.3 (Sullivan, 8th ed.MAC1140/MAC1147TRY THESE (Sections 5.2, 5.3)A. B.1. For graph A above, find [a] the range and domain; [b] the intercepts, if any; [c] vertical asymptotes, if any; and [d] horizontal or oblique asymptotes.2. Use graph B above to graph the rational function using transformations: 35( )9xR xx x-=-C. D.Hole at x = 6Find the domain and any vertical, horizontal, or oblique asymptotes, and the real zeros & graph on graph C.8. 226( 4)( )3( 4 4)xf xx x- -=+ + DomainInterceptsSymmetryAsymptotes: Vertical Horizontal or ObliqueOther Points:9. Make up a rational function that might have the graph in graph D.Vertical asymptotes:Horizontal or Oblique asymptote X-interceptsSection 5.3 (Sullivan, 8th ed.MAC1140/MAC1147R(x)