Lesson 78 Graphing rational functions Rational function A

Lesson 78 Graphing rational functions

Rational function A rational function is a function whose rule can be given as a rational expression. n This means that a rational expression has a variable in the denominator- they are undefined when the denominator is equal to zero. n A value of a variable for which an expression or a function is undefined is called an excluded value. n

Find the excluded values n n n n n y= m-4 3 m-12 Simplify the denominator by factoring out GCF 3(m-4) Set the denominator equal to 0, to find the excluded value 3(m-4)= 0 3 3 m-4 = 0 +4 +4 m= 4 is the excluded value

Find the excluded values n n n 1+5 a 30+5 a 2 x-8 4 x+10

Discontinuous functions Rational functions are discontinuous functions. n Discontinuous functions are functions that have a break or jump in the graph. n A break or jump in the graph can be due to an asymptote. n An asymptote is a boundary line that the graph of a function approaches but never touches or crosses n

Vertical asymptote n To determine vertical asymptotes, find the excluded value for the function

Horizontal asymptotes n Rational functions also have horizontal asymptotes. n When the function is in the form y= a c n x-b n the horizontal asymptote is at y = c n When + c = 0, the horizontal asymptote is the x-axis

Identify the asymptotes n n y= 2 x-8 to find vertical asymptote , find excluded value x-8 = 0, n so x = 8 is vertical asymptote n since c= 0 , n the horizontal asymptote is y = 0 n

Identify the asymptotes n n n y= 4 x+10 +3 y= 5 x-1 y= 6 x-3 -10

Graphing using asymptotes n n n The parent function for rational functions is y= 1 x By identifying the asymptotes and graphing them with dotted lines, and plotting a few points, the rational function can be graphed.

Identify the asymptote and graph n y= 1 x-4 -2 vertical asymptote is x = 4 n horizontal asymptote is y = -2 n Graph the asymptotes as dotted lines n Make a table of values (choose points on either side of the vertical asymptote) n x 0 2 3 5 6 n y -9/4 -5/2 -3 -1 -3/2 n Plot the points and connect them with smooth curves n

Identify the asymptotes and graph n y= 5 x-3 y= 3 x+2 n n n + 5