Chapter 2 – Polynomial and Rational Functions 2.6/7 – Graphs of Rational Functions and Asymptotes.

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Presentation transcript:

Chapter 2 – Polynomial and Rational Functions 2.6/7 – Graphs of Rational Functions and Asymptotes

Rational Functions A rational function is a division of two polynomials: The Reciprocal Function: Asymptote = a straight line approached by a curve as a variable in the equation for the curve approaches infinity. Remember: The domain is all real numbers EXCEPT whatever makes the denominator equal to zero!

Graphing Rational Functions Rules for graphing rational functions: 1. Simplify f(x) by factoring. See if anything cancels out. 2. Find and plot the Y-INTERCEPTS (if any) by evaluating f(0). 3. Set the NUMERATOR equal to zero to find any X-INTERCEPTS and plot them. Factor the numerator if possible! 4. Find any VERTICAL ASYMPTOTES or HOLES by setting the DENOMINATOR equal to zero. Plot the vertical asymptotes using dashed lines. Find the zeros of the denominator! Holes occur when a factor cancels out in the numerator/denominator

Graphing Rational Functions Rules for graphing rational functions: 5. Find and sketch the HORIZONTAL ASYMPTOTE of the graph using dashed lines. Let n=(degree of numerator) and m=(degree of denominator). ◦If n<m, the graph has a horizontal asymptote at y=0. ◦If n=m, divide the leading coefficients of the numerator and denominator. That value is the horizontal asymptote. ◦If n>m, there is no horizontal asymptote, but a slant asymptote instead.

Graphing Rational Functions Rules for graphing rational functions: 6. Plot at least one point between and beyond each x-intercept and vertical asymptote. 7. Draw the graph using smooth curves. NOTE: The curve MAY cross a horizontal asymptote, but it will NEVER cross a vertical asymptote!

Find the vertical asymptote(s) of: 1. x = 0 2. x = 3/2 3. x = 5/2 4. x = -5

Find the horizontal asymptote(s) of: 1. y=0 2. y=2 3. y=3 4. y=3 and y=10 5. y=15 and y=-2

Find the domain of: 1. (-∞, 3), (3, ∞) 2. (-∞, -1), (-1, ∞) 3. (-∞, -1), (-1, 3), (3, ∞) 4. (-1, 3) 5. (-∞, 0), (0, ∞) 6. (-∞, 1), (1, ∞)

Find the x-intercept(s) of: 1. (3, 0) 2. (3, 0) and (1, 0) 3. (1, 0) 4. (-1, 0) 5. (0, 0) 6. None