Section 7.2

 A rational function, f is a quotient of polynomials. That is, where P(x) and Q(x) are polynomials and Q(x) ≠ 0.

 Determine the domain of the following a) b) c)

The line x = k is a vertical asymptote of the graph of f if f(x)  ∞ or f(x)  –∞ as x approaches k from either the left or the right. x = 2

Let f be a rational function given by written in lowest terms. To find a vertical asymptote, set the denominator, q(x), equal to 0 and solve. If k is a zero of q(x), then x = k is a vertical asymptote.

 Find the vertical asymptotes of the function and then graph the function on your graphing calculator. a) b) c)

The line y = k is a horizontal asymptote of the graph of f if f(x)  k as x approaches either ∞ or –∞.

Horizontal Asymptote (a)If the degree of the numerator is less than the degree of the denominator, then y = 0 (the x-axis) is a horizontal asymptote. (b) If the degree of the numerator equals the degree of the denominator, then y = a/b is a horizontal asymptote, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator. (c)If the degree of the numerator is greater than the degree of the denominator, then there are no horizontal asymptotes.

 Find the horizontal asymptote of the functions below. a) b) c)

To determine whether the graph will intersect its horizontal asymptote at y = k, set the f(x) = k and solve. If there is no solution the graph will not cross the asymptote.

Determine algebraically if the graph of the function will cross its horizontal asymptote.

 To graph a rational function, f (x)=P(x)/Q(x) 1. Determine the domain of the function and restrict any x-values as needed. 2. Find and plot the y-intercept (evaluate f (0)). 3. Find and plot any x-intercepts (solve P(x)=0). 4. Find any vertical asymptotes (solve Q(x)=0), if there is any. 5. Find the horizontal asymptote, if there is one. Determine whether the graph will intersect its Horizontal asymptote y = b by solving f(x) = b, where b is the y-value of the horizontal asymptote. 6. Plot at least one point between x-intercepts and vertical asymptotes to determine the behavior of the graph. 7. Complete the sketch.

 Sketch the graph of

If f(x) = p(x)/q(x), then it is possible that, for some number k, both p(k) = 0 and q(k) = 0. In this case, the graph of f may not have a vertical asymptote at x=k; rather it may have a “hole” at x=k. Slide

Find all asymptotes and/or “holes” for the function Sketch the graph of