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RATIONAL FUNCTIONS

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RATIONAL FUNCTIONS

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    1. RATIONAL FUNCTIONS GRAPHS & ANALYSIS

    2. Real Life A certain company burns coal to generate electricity. The cost (C) of removing p% of the smokestack pollutants is C = 80,000p 100 – p What is the domain?

    3. The Basic Rational Function A Rational function is formed by the ratio of two polynomials (fractional form): f(x) = N(x) D(x) Basic f(x) = 1 x Vertical asymptote x=0 where f(x) undefined so as x?0 f(x)??? Horizontal asymptote y = 0 since as x? ?? f(x)?0

    4. Asymptotes Divide the variable terms of highest exp (degree) The result shows horizontal or slant asymptotes Draw all asymptotes as dashed lines EX: F(x) = 3x3 – 2x +1 ? 3x3 ? 3/2 = y 2x3 – 4 2x3 EX: G(x) = 1 – x ? -x ? -1 = y x x EX: H(x) = 2x2 ? 0 = y EX: f(x)= 3x2 ? y=3x x3 x slant

    5. Asymptotes cont… Vertical Asymptotes occur at x value(s) that make f(x) undefined (Denominator = 0) Ex: F(x) = 4x+1 ? 4x+1 x2-x x(x-1) ? x = 0 & x = 1

    6. Graphing Guidelines Let x= 0 to find any y-intercepts unless x ? 0 Set N(x) = 0 and solve for any x-intercepts Set D(x) = 0 to find x value(s) for vertical asymptote(s) and use dashed line Divide leading terms to find remaining asym Find & plot at least one point between and beyond each x-int and vertical asymptote Use smooth curves to complete the graph

    7. Graphing Example Graph f(x) = 3x + 1 x X ? 0 so no y-int 3x+1 = 0 ? x = -1/3 is x-int X = 0 vertical asym 3x ? 3 = y horizontal asym x Plot pts between & beyond X-int and asymptote (-1,2) (-3,8/3)(1/3,6)(2,7/2)

    8. Graphing Example 2 Graph f(x) = 2(x2 – 9) x2 – 4 y-int: x-int: v-asym: Other asym: Additional points:

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