2.7 Graphs of Rational Functions

2.7 Graphs of Rational Functions After completing this section, you should be able to: Describe the graphs of rational functions Identify horizontal and vertical asymptotes Predict the end behavior of rational functions

Consider this … • Predict some of the characteristics of the graph of y = (x+2)2(x-2).

Rational Functions Let f and g by polynomial functions with g(x)  0. Then the function given by y = f(x) g(x) is a rational function.

Let’s get real about rational functions • The domain of a rational function is the set of all real numbers except the zeros of its denominator. • Every rational function is continuous on its domain.

Ex 1 Find the domain of the function f. Use limits to describe the behavior of f at value(s) of x not in its domain. a. f(x) = 1 b. f(x) = -3 . x + 3 x – 1

Exploring Graphs of Rational Functions • Complete the exploration on p. 238 • You have 5 min. • Discuss your results with a neighbor • Plan to share your answers in 2 min.

Vertical Asymptotes Finding a vertical asymptote • Find the zero of the denominators. • These values are POTENTIAL vertical asymptotes. • Factor the numerator and denominator, if necessary. • Cancel any common factors. (These will be holes in the graph NOT asymptotes). • The remain zeros in the denominator will be the values of x which are vertical asymptotes expressed as lim f(x) = ±∞ x  a.

Horizontal Asymptotes describe End Behavior of a Function The graphs of y = f(x)/g(x) = (anxn + ..)/(bmxm+…) have the following characteristics: • If n < m, the end behavior asymptote is the horizontal asymptote y = 0. • If n = m, the end behavior asymptote is the horizontal asymptote y = an/bm(ratio of the leading coefficients) • If n > m, the end behavior asymptote is the quotient polynomial function y = q(x), where f(x) = g(x)q(x) + r(x). (There is NO horizontal asymptote - it’s a diagonal line, called a slant asymptote) Note: Determine the limits as x approaches infinity of f(x) to find horizontal asymptotes.

Ex 2 Describe how the graph of the given function can be obtained by transforming the reciprocal function g(x) = 1/x. Identify the horizontal and vertical asymptotes and use limits to describe the corresponding behavior. Sketch the graph of the function. a) f(x) = 2x – 1 b) f(x) = 3x – 2 x + 3 x – 1

Ex 3 Find the horizontal and vertical asymptotes of f(x). Use limits to describe the corresponding behavior. a) f(x) = 2x2 – 1 b) f(x) = x – 3 x2 + 2 x2 + 3x

Intercepts • Recall finding x-intercepts occur when f(x) (the y-value) is equal to 0 whereas • The y-intercepts occur when the domain (the x-value) is equal to 0.

Ex 4 Find the asymptotes and intercepts of the function, and graph the function. a) f(x) = x2 – 2x - 3 b) f(x) = 3 x + 2 x3 – 4x

Ex 5 Find the intercepts, asymptotes, use limits to describe the behavior at the vertical asymptotes, and analyze and draw the graph of the given rational function. a) f(x) = x + 1 b) f (x) = 2 x2 – 3x – 10 x2 + 4x + 3

Tonight’s Assignment • p. 246-247 4 – 60 m. of 4

2.7 Graphs of Rational Functions