Presentation on theme: "3.4 Rational Functions I. A rational function is a function of the form Where p and q are polynomial functions and q is not the zero polynomial. The domain."— Presentation transcript:
3.4 Rational Functions I
A rational function is a function of the form Where p and q are polynomial functions and q is not the zero polynomial. The domain consists of all real numbers except those for which the denominator is 0.
Find the domain of the following rational functions: All real numbers except -6 and-2. All real numbers except -4 and 4. All real numbers.
y = L y = R(x) y x y = L y = R(x) y x Horizontal Asymptotes
x = c y x y x Vertical Asymptotes
If an asymptote is neither horizontal nor vertical it is called oblique. y x
Theorem Locating Vertical Asymptotes A rational function In lowest terms, will have a vertical asymptote x = r, if x - r is a factor of the denominator q.
Vertical asymptotes: x = -1 and x = 1 No vertical asymptotes Vertical asymptote: x = -4 Find the vertical asymptotes, if any, of the graph of each rational function.
1. If n < m, then y = 0 is a horizontal asymptote of the graph of R. 2. If n = m, then y = a n / b m is a horizontal asymptote of the graph of R. 3. If n = m + 1, then y = ax + b is an oblique asymptote of the graph of R. Found using long division. 4. If n > m + 1, the graph of R has neither a horizontal nor oblique asymptote. End behavior found using long division. Consider the rational function
Horizontal asymptote: y = 0 Horizontal asymptote: y = 2/3 Find the horizontal and oblique asymptotes if any, of the graph of