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 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) 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

2.7 Graphs of Rational Functions 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