Section 8.4 – Graphing Rational Functions EQ: How do I graph a rational function using the vertical and horizontal asymptotes?

A Rational Function An equation of the form𝑓 𝑥 = 𝑎(𝑥) 𝑏(𝑥) , where a(x) and b(x) are polynomial functions and b(x) ≠ 0

Vertical and Horizontal Asymptotes Vertical Asymptotes Whenever b(x) = 0 Horizontal Asymptotes (at most one) If the degree of 𝑎 𝑥 is greater than the degree of 𝑏 𝑥 , there is no horizontal asymptote If the degree of 𝑎 𝑥 is less than the degree of 𝑏 𝑥 , the horizontal asymptote is the line y = 0 If the degree of 𝑎 𝑥 equals the degree of 𝑏 𝑥 , the horizontal asymptote is the line 𝑦= 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑎(𝑥) 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑏(𝑥)

Graph by finding the following Vertical Asymptote(s), if any: Set the denominator = 0 and solve Horizontal Asymptote(s), if any. Compare the degree of the numerator (m) to the degree of the denominator (n) m < n: HA is y = 0 m = n: HA is (LC of numerator/LC of denominator) m > n: HA does not exist y – intercept: Let x = 0 and solve x – intercept(s): Set the numerator = 0 and solve for x Choose other x – values, 2 on each side of an asymptote

Example 1 What are the asymptotes of the function 𝑓 𝑥 = 𝑥−3 𝑥−2 ? Vertical Asymptote: Set the denominator = 0 and solve. VA: x = 2 Horizontal Asymptote: compare the degree of the numberator (m) to the degree of the denominator (n). HA: 1𝑥 1 1𝑥 1 = 1 1 →𝑦=1

Example 2 What are the asymptotes of the function 𝑓 𝑥 = 2𝑥−7 𝑥+4 ? Vertical Asymptote: Set the denominator = 0 and solve. VA: x = -4 Horizontal Asymptote: compare the degree of the numberator (m) to the degree of the denominator (n). HA: 2𝑥 1 1𝑥 1 = 2 1 →𝑦=2

Example 3 Graph the function, then state the domain and range and any asymptote equations. a) 𝑓 𝑥 = 𝑥 𝑥 2 −9 VA: Set the denominator = 0 and solve. 𝑥 2 −9=(𝑥−3)(𝑥+3) 𝑥=−3, 3 HA: 𝑥 𝑥 2 →𝑚<𝑛→𝑦=0 *Use calculator to find other points, then connect the points Domain: all real, 𝑥≠−3, 3 Range: all real, 𝑦≠0

Example 3 Graph the function, then state the domain and range and any asymptote equations. b) 𝑓 𝑥 = 𝑥 2 𝑥 2 −1 VA: Set the denominator = 0 and solve. 𝑥 2 −1=(𝑥−1)(𝑥+1) 𝑥=−1, 1 HA: 𝑥 2 𝑥 2 →𝑚=𝑛→𝑦=1 *Use calculator to find other points, then connect the points Domain: all real, 𝑥≠−1, 1 Range: all real, 𝑦≠1

Point Discontinuity A point where a graph is undefined, looks like a hole in the graph

Point of discontinuity is at x = 3 Example 4 What is the point of discontinuity of the following functions? a) 𝑥 2 −9 𝑥−3 (𝑥+3)(𝑥−3) 𝑥−3 Point of discontinuity is at x = 3

Point of discontinuity is at x = -1 Example 4 What is the point of discontinuity of the following functions? b) 2 𝑥 2 +2 𝑥+1 2(𝑥+1)(𝑥−1) 𝑥+1 Point of discontinuity is at x = -1

Point of discontinuity is at x = -5 Example 4 What is the point of discontinuity of the following functions? c) 𝑥 2 +4𝑥−5 𝑥+5 (𝑥+5)(𝑥−1) 𝑥+5 Point of discontinuity is at x = -5

Point of discontinuity is at x = 2 Example 5 Graph 𝑓 𝑥 = 𝑥 2 −4 𝑥−2 Hole at x = 2 (𝑥+2)(𝑥−2) 𝑥−2 Point of discontinuity is at x = 2 𝑓 𝑥 =𝑥+2

Point of discontinuity is at x = -4 Example 6 Graph 𝑓 𝑥 = 𝑥 2 −16 𝑥+4 (𝑥+4)(𝑥−4) 𝑥+4 Point of discontinuity is at x = -4 𝑓 𝑥 =𝑥−4 Hole at x = -4